Optimal. Leaf size=113 \[ -\frac{3 b \cosh (a) \text{Chi}\left (b x^n\right )}{4 n}+\frac{3 b \cosh (3 a) \text{Chi}\left (3 b x^n\right )}{4 n}-\frac{3 b \sinh (a) \text{Shi}\left (b x^n\right )}{4 n}+\frac{3 b \sinh (3 a) \text{Shi}\left (3 b x^n\right )}{4 n}+\frac{3 x^{-n} \sinh \left (a+b x^n\right )}{4 n}-\frac{x^{-n} \sinh \left (3 \left (a+b x^n\right )\right )}{4 n} \]
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Rubi [A] time = 0.216987, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5362, 5320, 3297, 3303, 3298, 3301} \[ -\frac{3 b \cosh (a) \text{Chi}\left (b x^n\right )}{4 n}+\frac{3 b \cosh (3 a) \text{Chi}\left (3 b x^n\right )}{4 n}-\frac{3 b \sinh (a) \text{Shi}\left (b x^n\right )}{4 n}+\frac{3 b \sinh (3 a) \text{Shi}\left (3 b x^n\right )}{4 n}+\frac{3 x^{-n} \sinh \left (a+b x^n\right )}{4 n}-\frac{x^{-n} \sinh \left (3 \left (a+b x^n\right )\right )}{4 n} \]
Antiderivative was successfully verified.
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Rule 5362
Rule 5320
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int x^{-1-n} \sinh ^3\left (a+b x^n\right ) \, dx &=\int \left (-\frac{3}{4} x^{-1-n} \sinh \left (a+b x^n\right )+\frac{1}{4} x^{-1-n} \sinh \left (3 a+3 b x^n\right )\right ) \, dx\\ &=\frac{1}{4} \int x^{-1-n} \sinh \left (3 a+3 b x^n\right ) \, dx-\frac{3}{4} \int x^{-1-n} \sinh \left (a+b x^n\right ) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sinh (3 a+3 b x)}{x^2} \, dx,x,x^n\right )}{4 n}-\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (a+b x)}{x^2} \, dx,x,x^n\right )}{4 n}\\ &=\frac{3 x^{-n} \sinh \left (a+b x^n\right )}{4 n}-\frac{x^{-n} \sinh \left (3 \left (a+b x^n\right )\right )}{4 n}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{\cosh (a+b x)}{x} \, dx,x,x^n\right )}{4 n}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{\cosh (3 a+3 b x)}{x} \, dx,x,x^n\right )}{4 n}\\ &=\frac{3 x^{-n} \sinh \left (a+b x^n\right )}{4 n}-\frac{x^{-n} \sinh \left (3 \left (a+b x^n\right )\right )}{4 n}-\frac{(3 b \cosh (a)) \operatorname{Subst}\left (\int \frac{\cosh (b x)}{x} \, dx,x,x^n\right )}{4 n}+\frac{(3 b \cosh (3 a)) \operatorname{Subst}\left (\int \frac{\cosh (3 b x)}{x} \, dx,x,x^n\right )}{4 n}-\frac{(3 b \sinh (a)) \operatorname{Subst}\left (\int \frac{\sinh (b x)}{x} \, dx,x,x^n\right )}{4 n}+\frac{(3 b \sinh (3 a)) \operatorname{Subst}\left (\int \frac{\sinh (3 b x)}{x} \, dx,x,x^n\right )}{4 n}\\ &=-\frac{3 b \cosh (a) \text{Chi}\left (b x^n\right )}{4 n}+\frac{3 b \cosh (3 a) \text{Chi}\left (3 b x^n\right )}{4 n}+\frac{3 x^{-n} \sinh \left (a+b x^n\right )}{4 n}-\frac{x^{-n} \sinh \left (3 \left (a+b x^n\right )\right )}{4 n}-\frac{3 b \sinh (a) \text{Shi}\left (b x^n\right )}{4 n}+\frac{3 b \sinh (3 a) \text{Shi}\left (3 b x^n\right )}{4 n}\\ \end{align*}
Mathematica [A] time = 0.200762, size = 95, normalized size = 0.84 \[ -\frac{x^{-n} \left (3 b \cosh (a) x^n \text{Chi}\left (b x^n\right )-3 b \cosh (3 a) x^n \text{Chi}\left (3 b x^n\right )+3 b \sinh (a) x^n \text{Shi}\left (b x^n\right )-3 b \sinh (3 a) x^n \text{Shi}\left (3 b x^n\right )-3 \sinh \left (a+b x^n\right )+\sinh \left (3 \left (a+b x^n\right )\right )\right )}{4 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.087, size = 152, normalized size = 1.4 \begin{align*}{\frac{{{\rm e}^{-3\,a-3\,b{x}^{n}}}}{8\,n{x}^{n}}}-{\frac{3\,b{{\rm e}^{-3\,a}}{\it Ei} \left ( 1,3\,b{x}^{n} \right ) }{8\,n}}-{\frac{3\,{{\rm e}^{-a-b{x}^{n}}}}{8\,n{x}^{n}}}+{\frac{3\,b{{\rm e}^{-a}}{\it Ei} \left ( 1,b{x}^{n} \right ) }{8\,n}}-{\frac{{{\rm e}^{3\,a+3\,b{x}^{n}}}}{8\,n{x}^{n}}}-{\frac{3\,b{{\rm e}^{3\,a}}{\it Ei} \left ( 1,-3\,b{x}^{n} \right ) }{8\,n}}+{\frac{3\,{{\rm e}^{a+b{x}^{n}}}}{8\,n{x}^{n}}}+{\frac{3\,{{\rm e}^{a}}b{\it Ei} \left ( 1,-b{x}^{n} \right ) }{8\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.3355, size = 95, normalized size = 0.84 \begin{align*} \frac{3 \, b e^{\left (-3 \, a\right )} \Gamma \left (-1, 3 \, b x^{n}\right )}{8 \, n} - \frac{3 \, b e^{\left (-a\right )} \Gamma \left (-1, b x^{n}\right )}{8 \, n} - \frac{3 \, b e^{a} \Gamma \left (-1, -b x^{n}\right )}{8 \, n} + \frac{3 \, b e^{\left (3 \, a\right )} \Gamma \left (-1, -3 \, b x^{n}\right )}{8 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.89298, size = 986, normalized size = 8.73 \begin{align*} -\frac{2 \, \sinh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )^{3} - 3 \,{\left ({\left (b \cosh \left (3 \, a\right ) + b \sinh \left (3 \, a\right )\right )} \cosh \left (n \log \left (x\right )\right ) +{\left (b \cosh \left (3 \, a\right ) + b \sinh \left (3 \, a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )}{\rm Ei}\left (3 \, b \cosh \left (n \log \left (x\right )\right ) + 3 \, b \sinh \left (n \log \left (x\right )\right )\right ) + 3 \,{\left ({\left (b \cosh \left (a\right ) + b \sinh \left (a\right )\right )} \cosh \left (n \log \left (x\right )\right ) +{\left (b \cosh \left (a\right ) + b \sinh \left (a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )}{\rm Ei}\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right )\right ) + 3 \,{\left ({\left (b \cosh \left (a\right ) - b \sinh \left (a\right )\right )} \cosh \left (n \log \left (x\right )\right ) +{\left (b \cosh \left (a\right ) - b \sinh \left (a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )}{\rm Ei}\left (-b \cosh \left (n \log \left (x\right )\right ) - b \sinh \left (n \log \left (x\right )\right )\right ) - 3 \,{\left ({\left (b \cosh \left (3 \, a\right ) - b \sinh \left (3 \, a\right )\right )} \cosh \left (n \log \left (x\right )\right ) +{\left (b \cosh \left (3 \, a\right ) - b \sinh \left (3 \, a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )}{\rm Ei}\left (-3 \, b \cosh \left (n \log \left (x\right )\right ) - 3 \, b \sinh \left (n \log \left (x\right )\right )\right ) + 6 \,{\left (\cosh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )^{2} - 1\right )} \sinh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )}{8 \,{\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 )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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