Speeding Up Known Integrations

• To: mathgroup at smc.vnet.net
• Subject: [mg120967] Speeding Up Known Integrations
• From: Sean Violante <sean.violante07 at imperial.ac.uk>
• Date: Fri, 19 Aug 2011 06:34:46 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

```I have a large collection of iterated integrals to evaluate.
These are basically sums of terms of the form.

t_1^n1 E^{ k1 t_1}]    t_2^n2 E^{ k2 t_2}  t_3^n3 E^{ k3 t_3}

Is there any way of speeding up this evaluation?  I was wondering
whether gathering up terms in the above form, and then applying the
integral to each would work out faster, or is this effectively what
Integrate is doing anyway?

If faster, how would you suggest gathering up terms?
I have tried a few simple things eg  multiplying by exp[n (k+c)W0t2]
( where n is the largest negative power of Exp[ k t] and Exp [c t] ie so
expression  becomes polynomial)
and then using CoefficientList [ %,{W0t2, e^k W0t2, e^c W0t2} ]
However, I never quite manage to pull out all the exponential terms

Sean

below is an example ( where Integrate1 will be replaced by integrate).
It takes about 100 seconds on my machine.
Integrate1[
1/StdF1[t] \[ExponentialE]^(-t \[Kappa] +
2 W0t2 \[Kappa]) (VC0 + \[ExponentialE]^(-c W0t2)
VCC + \[ExponentialE]^(-W0t2 \[Kappa])
VCK)^2 Integrate1[\[ExponentialE]^(
W0t1 \[Kappa]) (VC0 + \[ExponentialE]^(-c W0t1)
VCC + \[ExponentialE]^(-W0t1 \[Kappa])
VCK) \[Rho] ((\[ExponentialE]^(-t \[Kappa] +
W0t1 \[Kappa]) (VpC0 + \[ExponentialE]^(-c W0t1)
VpCC) \[Kappa] Subscript[a, 1] Subscript[\[Xi], 2])/(
c StdF1[t] - \[Kappa] StdF1[t]) + (\[ExponentialE]^(-c t +
c W0t1) (VpC0 + \[ExponentialE]^(-c W0t1)
VpCC) \[Kappa] Subscript[a, 1] Subscript[\[Xi],
2])/(-c StdF1[t] + \[Kappa] StdF1[
t]) + (\[ExponentialE]^(-c t +
c W0t1) (VpC0 + \[ExponentialE]^(-c W0t1) VpCC) Subscript[
a, 2] Subscript[\[Xi], 2])/StdF1[t]), {W0t1, 0,
W0t2}] Subscript[a, 1] Subscript[\[Xi], 1], {W0t2, 0, t}] \!
\*SubsuperscriptBox[\(\[Xi]\), \(1\), \(2\)] /.

```

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