Optimal. Leaf size=45 \[ \frac{b \sinh (a) \text{Chi}\left (b x^n\right )}{n}+\frac{b \cosh (a) \text{Shi}\left (b x^n\right )}{n}-\frac{x^{-n} \cosh \left (a+b x^n\right )}{n} \]
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Rubi [A] time = 0.0941003, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {5321, 3297, 3303, 3298, 3301} \[ \frac{b \sinh (a) \text{Chi}\left (b x^n\right )}{n}+\frac{b \cosh (a) \text{Shi}\left (b x^n\right )}{n}-\frac{x^{-n} \cosh \left (a+b x^n\right )}{n} \]
Antiderivative was successfully verified.
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Rule 5321
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int x^{-1-n} \cosh \left (a+b x^n\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh (a+b x)}{x^2} \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-n} \cosh \left (a+b x^n\right )}{n}+\frac{b \operatorname{Subst}\left (\int \frac{\sinh (a+b x)}{x} \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-n} \cosh \left (a+b x^n\right )}{n}+\frac{(b \cosh (a)) \operatorname{Subst}\left (\int \frac{\sinh (b x)}{x} \, dx,x,x^n\right )}{n}+\frac{(b \sinh (a)) \operatorname{Subst}\left (\int \frac{\cosh (b x)}{x} \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-n} \cosh \left (a+b x^n\right )}{n}+\frac{b \text{Chi}\left (b x^n\right ) \sinh (a)}{n}+\frac{b \cosh (a) \text{Shi}\left (b x^n\right )}{n}\\ \end{align*}
Mathematica [A] time = 0.0574632, size = 46, normalized size = 1.02 \[ \frac{x^{-n} \left (b \sinh (a) x^n \text{Chi}\left (b x^n\right )+b \cosh (a) x^n \text{Shi}\left (b x^n\right )-\cosh \left (a+b x^n\right )\right )}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 74, normalized size = 1.6 \begin{align*} -{\frac{{{\rm e}^{-a-b{x}^{n}}}}{2\,n{x}^{n}}}+{\frac{b{{\rm e}^{-a}}{\it Ei} \left ( 1,b{x}^{n} \right ) }{2\,n}}-{\frac{{{\rm e}^{a+b{x}^{n}}}}{2\,n{x}^{n}}}-{\frac{{{\rm e}^{a}}b{\it Ei} \left ( 1,-b{x}^{n} \right ) }{2\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.23315, size = 46, normalized size = 1.02 \begin{align*} -\frac{b e^{\left (-a\right )} \Gamma \left (-1, b x^{n}\right )}{2 \, n} + \frac{b e^{a} \Gamma \left (-1, -b x^{n}\right )}{2 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.87735, size = 462, normalized size = 10.27 \begin{align*} \frac{{\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 ) -{\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 ) - 2 \, \cosh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )}{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} \cosh \left (b x^{n} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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