Optimal. Leaf size=118 \[ \frac{a^3 \text{Shi}(a+b x)}{3 b^3}-\frac{a^2 \cosh (a+b x)}{3 b^3}-\frac{a \sinh (a+b x)}{3 b^3}+\frac{2 x \sinh (a+b x)}{3 b^2}+\frac{a x \cosh (a+b x)}{3 b^2}-\frac{2 \cosh (a+b x)}{3 b^3}+\frac{1}{3} x^3 \text{Shi}(a+b x)-\frac{x^2 \cosh (a+b x)}{3 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.282966, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6532, 6742, 2638, 3296, 2637, 3298} \[ \frac{a^3 \text{Shi}(a+b x)}{3 b^3}-\frac{a^2 \cosh (a+b x)}{3 b^3}-\frac{a \sinh (a+b x)}{3 b^3}+\frac{2 x \sinh (a+b x)}{3 b^2}+\frac{a x \cosh (a+b x)}{3 b^2}-\frac{2 \cosh (a+b x)}{3 b^3}+\frac{1}{3} x^3 \text{Shi}(a+b x)-\frac{x^2 \cosh (a+b x)}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6532
Rule 6742
Rule 2638
Rule 3296
Rule 2637
Rule 3298
Rubi steps
\begin{align*} \int x^2 \text{Shi}(a+b x) \, dx &=\frac{1}{3} x^3 \text{Shi}(a+b x)-\frac{1}{3} b \int \frac{x^3 \sinh (a+b x)}{a+b x} \, dx\\ &=\frac{1}{3} x^3 \text{Shi}(a+b x)-\frac{1}{3} b \int \left (\frac{a^2 \sinh (a+b x)}{b^3}-\frac{a x \sinh (a+b x)}{b^2}+\frac{x^2 \sinh (a+b x)}{b}-\frac{a^3 \sinh (a+b x)}{b^3 (a+b x)}\right ) \, dx\\ &=\frac{1}{3} x^3 \text{Shi}(a+b x)-\frac{1}{3} \int x^2 \sinh (a+b x) \, dx-\frac{a^2 \int \sinh (a+b x) \, dx}{3 b^2}+\frac{a^3 \int \frac{\sinh (a+b x)}{a+b x} \, dx}{3 b^2}+\frac{a \int x \sinh (a+b x) \, dx}{3 b}\\ &=-\frac{a^2 \cosh (a+b x)}{3 b^3}+\frac{a x \cosh (a+b x)}{3 b^2}-\frac{x^2 \cosh (a+b x)}{3 b}+\frac{a^3 \text{Shi}(a+b x)}{3 b^3}+\frac{1}{3} x^3 \text{Shi}(a+b x)-\frac{a \int \cosh (a+b x) \, dx}{3 b^2}+\frac{2 \int x \cosh (a+b x) \, dx}{3 b}\\ &=-\frac{a^2 \cosh (a+b x)}{3 b^3}+\frac{a x \cosh (a+b x)}{3 b^2}-\frac{x^2 \cosh (a+b x)}{3 b}-\frac{a \sinh (a+b x)}{3 b^3}+\frac{2 x \sinh (a+b x)}{3 b^2}+\frac{a^3 \text{Shi}(a+b x)}{3 b^3}+\frac{1}{3} x^3 \text{Shi}(a+b x)-\frac{2 \int \sinh (a+b x) \, dx}{3 b^2}\\ &=-\frac{2 \cosh (a+b x)}{3 b^3}-\frac{a^2 \cosh (a+b x)}{3 b^3}+\frac{a x \cosh (a+b x)}{3 b^2}-\frac{x^2 \cosh (a+b x)}{3 b}-\frac{a \sinh (a+b x)}{3 b^3}+\frac{2 x \sinh (a+b x)}{3 b^2}+\frac{a^3 \text{Shi}(a+b x)}{3 b^3}+\frac{1}{3} x^3 \text{Shi}(a+b x)\\ \end{align*}
Mathematica [A] time = 0.185289, size = 64, normalized size = 0.54 \[ -\frac{-\left (a^3+b^3 x^3\right ) \text{Shi}(a+b x)+\left (a^2-a b x+b^2 x^2+2\right ) \cosh (a+b x)+(a-2 b x) \sinh (a+b x)}{3 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.047, size = 101, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{3}} \left ({\frac{{b}^{3}{x}^{3}{\it Shi} \left ( bx+a \right ) }{3}}-{\frac{ \left ( bx+a \right ) ^{2}\cosh \left ( bx+a \right ) }{3}}+{\frac{ \left ( 2\,bx+2\,a \right ) \sinh \left ( bx+a \right ) }{3}}-{\frac{2\,\cosh \left ( bx+a \right ) }{3}}+a \left ( \left ( bx+a \right ) \cosh \left ( bx+a \right ) -\sinh \left ( bx+a \right ) \right ) -\cosh \left ( bx+a \right ){a}^{2}+{\frac{{a}^{3}{\it Shi} \left ( bx+a \right ) }{3}} \right ) } \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^{2}{\rm Shi}\left (b x + a\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^{2} \operatorname{Shi}\left (b x + a\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^{2} \operatorname{Shi}{\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^{2}{\rm Shi}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]