Optimal. Leaf size=154 \[ -\frac{\text{Chi}(2 a+2 b x)}{2 b^2}-\frac{a (a+b x) \text{Shi}(a+b x)^2}{2 b^2}-\frac{a \text{Shi}(2 a+2 b x)}{b^2}+\frac{\text{Shi}(a+b x) \sinh (a+b x)}{b^2}+\frac{a \text{Shi}(a+b x) \cosh (a+b x)}{b^2}+\frac{\log (a+b x)}{2 b^2}+\frac{\cosh (2 a+2 b x)}{4 b^2}+\frac{x (a+b x) \text{Shi}(a+b x)^2}{2 b}-\frac{x \text{Shi}(a+b x) \cosh (a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.327381, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 14, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.4, Rules used = {6538, 6542, 5617, 6741, 6742, 2638, 3298, 6546, 3312, 3301, 6534, 6540, 5448, 12} \[ -\frac{\text{Chi}(2 a+2 b x)}{2 b^2}-\frac{a (a+b x) \text{Shi}(a+b x)^2}{2 b^2}-\frac{a \text{Shi}(2 a+2 b x)}{b^2}+\frac{\text{Shi}(a+b x) \sinh (a+b x)}{b^2}+\frac{a \text{Shi}(a+b x) \cosh (a+b x)}{b^2}+\frac{\log (a+b x)}{2 b^2}+\frac{\cosh (2 a+2 b x)}{4 b^2}+\frac{x (a+b x) \text{Shi}(a+b x)^2}{2 b}-\frac{x \text{Shi}(a+b x) \cosh (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6538
Rule 6542
Rule 5617
Rule 6741
Rule 6742
Rule 2638
Rule 3298
Rule 6546
Rule 3312
Rule 3301
Rule 6534
Rule 6540
Rule 5448
Rule 12
Rubi steps
\begin{align*} \int x \text{Shi}(a+b x)^2 \, dx &=\frac{x (a+b x) \text{Shi}(a+b x)^2}{2 b}-\frac{a \int \text{Shi}(a+b x)^2 \, dx}{2 b}-\int x \sinh (a+b x) \text{Shi}(a+b x) \, dx\\ &=-\frac{x \cosh (a+b x) \text{Shi}(a+b x)}{b}-\frac{a (a+b x) \text{Shi}(a+b x)^2}{2 b^2}+\frac{x (a+b x) \text{Shi}(a+b x)^2}{2 b}+\frac{\int \cosh (a+b x) \text{Shi}(a+b x) \, dx}{b}+\frac{a \int \sinh (a+b x) \text{Shi}(a+b x) \, dx}{b}+\int \frac{x \cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx\\ &=\frac{a \cosh (a+b x) \text{Shi}(a+b x)}{b^2}-\frac{x \cosh (a+b x) \text{Shi}(a+b x)}{b}+\frac{\sinh (a+b x) \text{Shi}(a+b x)}{b^2}-\frac{a (a+b x) \text{Shi}(a+b x)^2}{2 b^2}+\frac{x (a+b x) \text{Shi}(a+b x)^2}{2 b}+\frac{1}{2} \int \frac{x \sinh (2 (a+b x))}{a+b x} \, dx-\frac{\int \frac{\sinh ^2(a+b x)}{a+b x} \, dx}{b}-\frac{a \int \frac{\cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx}{b}\\ &=\frac{a \cosh (a+b x) \text{Shi}(a+b x)}{b^2}-\frac{x \cosh (a+b x) \text{Shi}(a+b x)}{b}+\frac{\sinh (a+b x) \text{Shi}(a+b x)}{b^2}-\frac{a (a+b x) \text{Shi}(a+b x)^2}{2 b^2}+\frac{x (a+b x) \text{Shi}(a+b x)^2}{2 b}+\frac{1}{2} \int \frac{x \sinh (2 a+2 b x)}{a+b x} \, dx+\frac{\int \left (\frac{1}{2 (a+b x)}-\frac{\cosh (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{b}-\frac{a \int \frac{\sinh (2 a+2 b x)}{2 (a+b x)} \, dx}{b}\\ &=\frac{\log (a+b x)}{2 b^2}+\frac{a \cosh (a+b x) \text{Shi}(a+b x)}{b^2}-\frac{x \cosh (a+b x) \text{Shi}(a+b x)}{b}+\frac{\sinh (a+b x) \text{Shi}(a+b x)}{b^2}-\frac{a (a+b x) \text{Shi}(a+b x)^2}{2 b^2}+\frac{x (a+b x) \text{Shi}(a+b x)^2}{2 b}+\frac{1}{2} \int \left (\frac{\sinh (2 a+2 b x)}{b}+\frac{a \sinh (2 a+2 b x)}{b (-a-b x)}\right ) \, dx-\frac{\int \frac{\cosh (2 a+2 b x)}{a+b x} \, dx}{2 b}-\frac{a \int \frac{\sinh (2 a+2 b x)}{a+b x} \, dx}{2 b}\\ &=-\frac{\text{Chi}(2 a+2 b x)}{2 b^2}+\frac{\log (a+b x)}{2 b^2}+\frac{a \cosh (a+b x) \text{Shi}(a+b x)}{b^2}-\frac{x \cosh (a+b x) \text{Shi}(a+b x)}{b}+\frac{\sinh (a+b x) \text{Shi}(a+b x)}{b^2}-\frac{a (a+b x) \text{Shi}(a+b x)^2}{2 b^2}+\frac{x (a+b x) \text{Shi}(a+b x)^2}{2 b}-\frac{a \text{Shi}(2 a+2 b x)}{2 b^2}+\frac{\int \sinh (2 a+2 b x) \, dx}{2 b}+\frac{a \int \frac{\sinh (2 a+2 b x)}{-a-b x} \, dx}{2 b}\\ &=\frac{\cosh (2 a+2 b x)}{4 b^2}-\frac{\text{Chi}(2 a+2 b x)}{2 b^2}+\frac{\log (a+b x)}{2 b^2}+\frac{a \cosh (a+b x) \text{Shi}(a+b x)}{b^2}-\frac{x \cosh (a+b x) \text{Shi}(a+b x)}{b}+\frac{\sinh (a+b x) \text{Shi}(a+b x)}{b^2}-\frac{a (a+b x) \text{Shi}(a+b x)^2}{2 b^2}+\frac{x (a+b x) \text{Shi}(a+b x)^2}{2 b}-\frac{a \text{Shi}(2 a+2 b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.283587, size = 95, normalized size = 0.62 \[ \frac{-2 \left (a^2-b^2 x^2\right ) \text{Shi}(a+b x)^2-2 \text{Chi}(2 (a+b x))-4 a \text{Shi}(2 (a+b x))+4 \text{Shi}(a+b x) (\sinh (a+b x)+(a-b x) \cosh (a+b x))+2 \log (a+b x)+\cosh (2 (a+b x))}{4 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.056, size = 135, normalized size = 0.9 \begin{align*}{\frac{{x}^{2} \left ({\it Shi} \left ( bx+a \right ) \right ) ^{2}}{2}}-{\frac{ \left ({\it Shi} \left ( bx+a \right ) \right ) ^{2}{a}^{2}}{2\,{b}^{2}}}-{\frac{x\cosh \left ( bx+a \right ){\it Shi} \left ( bx+a \right ) }{b}}+{\frac{\cosh \left ( bx+a \right ) a{\it Shi} \left ( bx+a \right ) }{{b}^{2}}}+{\frac{{\it Shi} \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{{b}^{2}}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{2\,{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) }{2\,{b}^{2}}}-{\frac{{\it Chi} \left ( 2\,bx+2\,a \right ) }{2\,{b}^{2}}}-{\frac{a{\it Shi} \left ( 2\,bx+2\,a \right ) }{{b}^{2}}} \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{\rm Shi}\left (b x + a\right )^{2}\,{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 \operatorname{Shi}\left (b x + a\right )^{2}, 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 \operatorname{Shi}^{2}{\left (a + 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{\rm Shi}\left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]