Optimal. Leaf size=67 \[ \frac{b \sinh (2 a) \text{Chi}\left (2 b x^n\right )}{n}+\frac{b \cosh (2 a) \text{Shi}\left (2 b x^n\right )}{n}-\frac{x^{-n} \cosh \left (2 \left (a+b x^n\right )\right )}{2 n}+\frac{x^{-n}}{2 n} \]
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Rubi [A] time = 0.122997, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5362, 5321, 3297, 3303, 3298, 3301} \[ \frac{b \sinh (2 a) \text{Chi}\left (2 b x^n\right )}{n}+\frac{b \cosh (2 a) \text{Shi}\left (2 b x^n\right )}{n}-\frac{x^{-n} \cosh \left (2 \left (a+b x^n\right )\right )}{2 n}+\frac{x^{-n}}{2 n} \]
Antiderivative was successfully verified.
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Rule 5362
Rule 5321
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int x^{-1-n} \sinh ^2\left (a+b x^n\right ) \, dx &=\int \left (-\frac{1}{2} x^{-1-n}+\frac{1}{2} x^{-1-n} \cosh \left (2 a+2 b x^n\right )\right ) \, dx\\ &=\frac{x^{-n}}{2 n}+\frac{1}{2} \int x^{-1-n} \cosh \left (2 a+2 b x^n\right ) \, dx\\ &=\frac{x^{-n}}{2 n}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 a+2 b x)}{x^2} \, dx,x,x^n\right )}{2 n}\\ &=\frac{x^{-n}}{2 n}-\frac{x^{-n} \cosh \left (2 \left (a+b x^n\right )\right )}{2 n}+\frac{b \operatorname{Subst}\left (\int \frac{\sinh (2 a+2 b x)}{x} \, dx,x,x^n\right )}{n}\\ &=\frac{x^{-n}}{2 n}-\frac{x^{-n} \cosh \left (2 \left (a+b x^n\right )\right )}{2 n}+\frac{(b \cosh (2 a)) \operatorname{Subst}\left (\int \frac{\sinh (2 b x)}{x} \, dx,x,x^n\right )}{n}+\frac{(b \sinh (2 a)) \operatorname{Subst}\left (\int \frac{\cosh (2 b x)}{x} \, dx,x,x^n\right )}{n}\\ &=\frac{x^{-n}}{2 n}-\frac{x^{-n} \cosh \left (2 \left (a+b x^n\right )\right )}{2 n}+\frac{b \text{Chi}\left (2 b x^n\right ) \sinh (2 a)}{n}+\frac{b \cosh (2 a) \text{Shi}\left (2 b x^n\right )}{n}\\ \end{align*}
Mathematica [A] time = 0.130863, size = 54, normalized size = 0.81 \[ \frac{x^{-n} \left (b \sinh (2 a) x^n \text{Chi}\left (2 b x^n\right )+b \cosh (2 a) x^n \text{Shi}\left (2 b x^n\right )-\sinh ^2\left (a+b x^n\right )\right )}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 90, normalized size = 1.3 \begin{align*}{\frac{1}{2\,n{x}^{n}}}-{\frac{{{\rm e}^{-2\,a-2\,b{x}^{n}}}}{4\,n{x}^{n}}}+{\frac{b{{\rm e}^{-2\,a}}{\it Ei} \left ( 1,2\,b{x}^{n} \right ) }{2\,n}}-{\frac{{{\rm e}^{2\,a+2\,b{x}^{n}}}}{4\,n{x}^{n}}}-{\frac{b{{\rm e}^{2\,a}}{\it Ei} \left ( 1,-2\,b{x}^{n} \right ) }{2\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.89354, size = 567, normalized size = 8.46 \begin{align*} \frac{{\left ({\left (b \cosh \left (2 \, a\right ) + b \sinh \left (2 \, a\right )\right )} \cosh \left (n \log \left (x\right )\right ) +{\left (b \cosh \left (2 \, a\right ) + b \sinh \left (2 \, a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )}{\rm Ei}\left (2 \, b \cosh \left (n \log \left (x\right )\right ) + 2 \, b \sinh \left (n \log \left (x\right )\right )\right ) -{\left ({\left (b \cosh \left (2 \, a\right ) - b \sinh \left (2 \, a\right )\right )} \cosh \left (n \log \left (x\right )\right ) +{\left (b \cosh \left (2 \, a\right ) - b \sinh \left (2 \, a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )}{\rm Ei}\left (-2 \, b \cosh \left (n \log \left (x\right )\right ) - 2 \, b \sinh \left (n \log \left (x\right )\right )\right ) - \cosh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )^{2} - \sinh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )^{2} + 1}{2 \,{\left (n \cosh \left (n \log \left (x\right )\right ) + n \sinh \left (n \log \left (x\right )\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{-n - 1} \sinh \left (b x^{n} + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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