Optimal. Leaf size=125 \[ \frac{3 \text{Chi}(2 b x)}{b^4}-\frac{3 x^2 \text{Shi}(b x) \sinh (b x)}{b^2}-\frac{6 \text{Shi}(b x) \sinh (b x)}{b^4}+\frac{6 x \text{Shi}(b x) \cosh (b x)}{b^3}-\frac{x^2}{b^2}-\frac{x^2 \sinh ^2(b x)}{2 b^2}-\frac{3 \log (x)}{b^4}-\frac{4 \sinh ^2(b x)}{b^4}+\frac{2 x \sinh (b x) \cosh (b x)}{b^3}+\frac{x^3 \text{Shi}(b x) \cosh (b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.204213, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {6542, 12, 5372, 3310, 30, 6548, 2564, 6546, 3312, 3301} \[ \frac{3 \text{Chi}(2 b x)}{b^4}-\frac{3 x^2 \text{Shi}(b x) \sinh (b x)}{b^2}-\frac{6 \text{Shi}(b x) \sinh (b x)}{b^4}+\frac{6 x \text{Shi}(b x) \cosh (b x)}{b^3}-\frac{x^2}{b^2}-\frac{x^2 \sinh ^2(b x)}{2 b^2}-\frac{3 \log (x)}{b^4}-\frac{4 \sinh ^2(b x)}{b^4}+\frac{2 x \sinh (b x) \cosh (b x)}{b^3}+\frac{x^3 \text{Shi}(b x) \cosh (b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6542
Rule 12
Rule 5372
Rule 3310
Rule 30
Rule 6548
Rule 2564
Rule 6546
Rule 3312
Rule 3301
Rubi steps
\begin{align*} \int x^3 \sinh (b x) \text{Shi}(b x) \, dx &=\frac{x^3 \cosh (b x) \text{Shi}(b x)}{b}-\frac{3 \int x^2 \cosh (b x) \text{Shi}(b x) \, dx}{b}-\int \frac{x^2 \cosh (b x) \sinh (b x)}{b} \, dx\\ &=\frac{x^3 \cosh (b x) \text{Shi}(b x)}{b}-\frac{3 x^2 \sinh (b x) \text{Shi}(b x)}{b^2}+\frac{6 \int x \sinh (b x) \text{Shi}(b x) \, dx}{b^2}-\frac{\int x^2 \cosh (b x) \sinh (b x) \, dx}{b}+\frac{3 \int \frac{x \sinh ^2(b x)}{b} \, dx}{b}\\ &=-\frac{x^2 \sinh ^2(b x)}{2 b^2}+\frac{6 x \cosh (b x) \text{Shi}(b x)}{b^3}+\frac{x^3 \cosh (b x) \text{Shi}(b x)}{b}-\frac{3 x^2 \sinh (b x) \text{Shi}(b x)}{b^2}-\frac{6 \int \cosh (b x) \text{Shi}(b x) \, dx}{b^3}+\frac{\int x \sinh ^2(b x) \, dx}{b^2}+\frac{3 \int x \sinh ^2(b x) \, dx}{b^2}-\frac{6 \int \frac{\cosh (b x) \sinh (b x)}{b} \, dx}{b^2}\\ &=\frac{2 x \cosh (b x) \sinh (b x)}{b^3}-\frac{\sinh ^2(b x)}{b^4}-\frac{x^2 \sinh ^2(b x)}{2 b^2}+\frac{6 x \cosh (b x) \text{Shi}(b x)}{b^3}+\frac{x^3 \cosh (b x) \text{Shi}(b x)}{b}-\frac{6 \sinh (b x) \text{Shi}(b x)}{b^4}-\frac{3 x^2 \sinh (b x) \text{Shi}(b x)}{b^2}-\frac{6 \int \cosh (b x) \sinh (b x) \, dx}{b^3}+\frac{6 \int \frac{\sinh ^2(b x)}{b x} \, dx}{b^3}-\frac{\int x \, dx}{2 b^2}-\frac{3 \int x \, dx}{2 b^2}\\ &=-\frac{x^2}{b^2}+\frac{2 x \cosh (b x) \sinh (b x)}{b^3}-\frac{\sinh ^2(b x)}{b^4}-\frac{x^2 \sinh ^2(b x)}{2 b^2}+\frac{6 x \cosh (b x) \text{Shi}(b x)}{b^3}+\frac{x^3 \cosh (b x) \text{Shi}(b x)}{b}-\frac{6 \sinh (b x) \text{Shi}(b x)}{b^4}-\frac{3 x^2 \sinh (b x) \text{Shi}(b x)}{b^2}+\frac{6 \int \frac{\sinh ^2(b x)}{x} \, dx}{b^4}+\frac{6 \operatorname{Subst}(\int x \, dx,x,i \sinh (b x))}{b^4}\\ &=-\frac{x^2}{b^2}+\frac{2 x \cosh (b x) \sinh (b x)}{b^3}-\frac{4 \sinh ^2(b x)}{b^4}-\frac{x^2 \sinh ^2(b x)}{2 b^2}+\frac{6 x \cosh (b x) \text{Shi}(b x)}{b^3}+\frac{x^3 \cosh (b x) \text{Shi}(b x)}{b}-\frac{6 \sinh (b x) \text{Shi}(b x)}{b^4}-\frac{3 x^2 \sinh (b x) \text{Shi}(b x)}{b^2}-\frac{6 \int \left (\frac{1}{2 x}-\frac{\cosh (2 b x)}{2 x}\right ) \, dx}{b^4}\\ &=-\frac{x^2}{b^2}-\frac{3 \log (x)}{b^4}+\frac{2 x \cosh (b x) \sinh (b x)}{b^3}-\frac{4 \sinh ^2(b x)}{b^4}-\frac{x^2 \sinh ^2(b x)}{2 b^2}+\frac{6 x \cosh (b x) \text{Shi}(b x)}{b^3}+\frac{x^3 \cosh (b x) \text{Shi}(b x)}{b}-\frac{6 \sinh (b x) \text{Shi}(b x)}{b^4}-\frac{3 x^2 \sinh (b x) \text{Shi}(b x)}{b^2}+\frac{3 \int \frac{\cosh (2 b x)}{x} \, dx}{b^4}\\ &=-\frac{x^2}{b^2}+\frac{3 \text{Chi}(2 b x)}{b^4}-\frac{3 \log (x)}{b^4}+\frac{2 x \cosh (b x) \sinh (b x)}{b^3}-\frac{4 \sinh ^2(b x)}{b^4}-\frac{x^2 \sinh ^2(b x)}{2 b^2}+\frac{6 x \cosh (b x) \text{Shi}(b x)}{b^3}+\frac{x^3 \cosh (b x) \text{Shi}(b x)}{b}-\frac{6 \sinh (b x) \text{Shi}(b x)}{b^4}-\frac{3 x^2 \sinh (b x) \text{Shi}(b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.162853, size = 93, normalized size = 0.74 \[ -\frac{-4 \text{Shi}(b x) \left (b x \left (b^2 x^2+6\right ) \cosh (b x)-3 \left (b^2 x^2+2\right ) \sinh (b x)\right )+3 b^2 x^2+b^2 x^2 \cosh (2 b x)-12 \text{Chi}(2 b x)-4 b x \sinh (2 b x)+8 \cosh (2 b x)+12 \log (x)}{4 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.056, size = 126, normalized size = 1. \begin{align*}{\frac{{x}^{3}\cosh \left ( bx \right ){\it Shi} \left ( bx \right ) }{b}}-3\,{\frac{{x}^{2}{\it Shi} \left ( bx \right ) \sinh \left ( bx \right ) }{{b}^{2}}}+6\,{\frac{x\cosh \left ( bx \right ){\it Shi} \left ( bx \right ) }{{b}^{3}}}-6\,{\frac{{\it Shi} \left ( bx \right ) \sinh \left ( bx \right ) }{{b}^{4}}}-{\frac{{x}^{2} \left ( \cosh \left ( bx \right ) \right ) ^{2}}{2\,{b}^{2}}}+2\,{\frac{x\cosh \left ( bx \right ) \sinh \left ( bx \right ) }{{b}^{3}}}-{\frac{{x}^{2}}{2\,{b}^{2}}}-4\,{\frac{ \left ( \cosh \left ( bx \right ) \right ) ^{2}}{{b}^{4}}}-3\,{\frac{\ln \left ( bx \right ) }{{b}^{4}}}+3\,{\frac{{\it Chi} \left ( 2\,bx \right ) }{{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3}{\rm Shi}\left (b x\right ) \sinh \left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{3} \sinh \left (b x\right ) \operatorname{Shi}\left (b x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sinh{\left (b x \right )} \operatorname{Shi}{\left (b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3}{\rm Shi}\left (b x\right ) \sinh \left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]