Optimal. Leaf size=88 \[ \frac{(e x)^{m+1}}{e (m+1)}-\frac{2 (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2 b d n};\frac{m+1}{2 b d n}+1;-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{e (m+1)} \]
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Rubi [F] time = 0.0464648, 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 = {} \[ \int (e x)^m \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int (e x)^m \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int (e x)^m \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end{align*}
Mathematica [A] time = 13.584, size = 160, normalized size = 1.82 \[ \frac{x (e x)^m \left (\frac{(m+1) e^{2 a d} \left (c x^n\right )^{2 b d} \, _2F_1\left (1,\frac{m+2 b d n+1}{2 b d n};\frac{m+4 b d n+1}{2 b d n};-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{2 b d n+m+1}-\, _2F_1\left (1,\frac{m+1}{2 b d n};\frac{m+1}{2 b d n}+1;-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{m+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.214, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m}\tanh \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \, 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*} \frac{e^{m} x x^{m}}{m + 1} - 2 \, e^{m} \int \frac{x^{m}}{c^{2 \, b d} e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + 1}\,{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 ), x\right ) \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 )^{m} \tanh{\left (a d + b d \log{\left (c 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 (e x\right )^{m} \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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