Optimal. Leaf size=45 \[ \frac{a (e x)^n}{e n}-\frac{b x^{-n} (e x)^n \tanh ^{-1}\left (\cosh \left (c+d x^n\right )\right )}{d e n} \]
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Rubi [A] time = 0.0543707, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {14, 5441, 5437, 3770} \[ \frac{a (e x)^n}{e n}-\frac{b x^{-n} (e x)^n \tanh ^{-1}\left (\cosh \left (c+d x^n\right )\right )}{d e n} \]
Antiderivative was successfully verified.
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Rule 14
Rule 5441
Rule 5437
Rule 3770
Rubi steps
\begin{align*} \int (e x)^{-1+n} \left (a+b \text{csch}\left (c+d x^n\right )\right ) \, dx &=\int \left (a (e x)^{-1+n}+b (e x)^{-1+n} \text{csch}\left (c+d x^n\right )\right ) \, dx\\ &=\frac{a (e x)^n}{e n}+b \int (e x)^{-1+n} \text{csch}\left (c+d x^n\right ) \, dx\\ &=\frac{a (e x)^n}{e n}+\frac{\left (b x^{-n} (e x)^n\right ) \int x^{-1+n} \text{csch}\left (c+d x^n\right ) \, dx}{e}\\ &=\frac{a (e x)^n}{e n}+\frac{\left (b x^{-n} (e x)^n\right ) \operatorname{Subst}\left (\int \text{csch}(c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac{a (e x)^n}{e n}-\frac{b x^{-n} (e x)^n \tanh ^{-1}\left (\cosh \left (c+d x^n\right )\right )}{d e n}\\ \end{align*}
Mathematica [A] time = 0.0628847, size = 45, normalized size = 1. \[ \frac{x^{-n} (e x)^n \left (a \left (c+d x^n\right )+b \log \left (\tanh \left (\frac{1}{2} \left (c+d x^n\right )\right )\right )\right )}{d e n} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.228, size = 155, normalized size = 3.4 \begin{align*}{\frac{ax}{n}{{\rm e}^{{\frac{ \left ( -1+n \right ) \left ( -i\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) +i\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}+i\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}-i\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}+2\,\ln \left ( x \right ) +2\,\ln \left ( e \right ) \right ) }{2}}}}}-2\,{\frac{b{e}^{n}{\it Artanh} \left ({{\rm e}^{c+d{x}^{n}}} \right ){{\rm e}^{-i/2\pi \,{\it csgn} \left ( iex \right ) \left ( -1+n \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ix \right ) \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ie \right ) \right ) }}}{ned}} \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 = 2.01812, size = 637, normalized size = 14.16 \begin{align*} \frac{a d \cosh \left ({\left (n - 1\right )} \log \left (e\right )\right ) \cosh \left (n \log \left (x\right )\right ) + a d \cosh \left (n \log \left (x\right )\right ) \sinh \left ({\left (n - 1\right )} \log \left (e\right )\right ) -{\left (b \cosh \left ({\left (n - 1\right )} \log \left (e\right )\right ) + b \sinh \left ({\left (n - 1\right )} \log \left (e\right )\right )\right )} \log \left (\cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + 1\right ) +{\left (b \cosh \left ({\left (n - 1\right )} \log \left (e\right )\right ) + b \sinh \left ({\left (n - 1\right )} \log \left (e\right )\right )\right )} \log \left (\cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) - 1\right ) +{\left (a d \cosh \left ({\left (n - 1\right )} \log \left (e\right )\right ) + a d \sinh \left ({\left (n - 1\right )} \log \left (e\right )\right )\right )} \sinh \left (n \log \left (x\right )\right )}{d n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{n - 1} \left (a + b \operatorname{csch}{\left (c + d x^{n} \right )}\right )\, dx \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{\left (b \operatorname{csch}\left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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