Optimal. Leaf size=135 \[ \frac{(e x)^{m+1} \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )^{-p} \left (e^{2 a d} \left (c x^n\right )^{2 b d}-1\right )^p F_1\left (\frac{m+1}{2 b d n};-p,p;\frac{m+1}{2 b d n}+1;e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{e (m+1)} \]
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Rubi [F] time = 0.0988439, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (e x)^m \tanh ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int (e x)^m \tanh ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int (e x)^m \tanh ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end{align*}
Mathematica [A] time = 4.99858, size = 174, normalized size = 1.29 \[ \frac{x (e x)^m \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )^{-p} \left (\frac{e^{2 a d} \left (c x^n\right )^{2 b d}-1}{e^{2 a d} \left (c x^n\right )^{2 b d}+1}\right )^p \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )^p F_1\left (\frac{m+1}{2 b d n};-p,p;\frac{m+1}{2 b d n}+1;e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{m+1} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.057, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( \tanh \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (e x\right )^{m} \tanh \left (b d \log \left (c x^{n}\right ) + a d\right )^{p}, x\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 \left (e x\right )^{m} \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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