Optimal. Leaf size=101 \[ \frac{2 i (e x)^{m+1} \text{Hypergeometric2F1}\left (1,-\frac{i (m+1)}{2 b d n},1-\frac{i (m+1)}{2 b d n},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (m+1)}-\frac{i (e x)^{m+1}}{e (m+1)} \]
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Rubi [F] time = 0.0478934, 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 = {} \[ \int (e x)^m \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int (e x)^m \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int (e x)^m \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end{align*}
Mathematica [A] time = 14.5072, size = 186, normalized size = 1.84 \[ \frac{i x (e x)^m \left (\text{Hypergeometric2F1}\left (1,-\frac{i (m+1)}{2 b d n},1-\frac{i (m+1)}{2 b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-\frac{(m+1) e^{2 i a d} \left (c x^n\right )^{2 i b d} \text{Hypergeometric2F1}\left (1,-\frac{i (2 i b d n+m+1)}{2 b d n},-\frac{i (4 i b d n+m+1)}{2 b d n},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{2 i b d n+m+1}\right )}{m+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.721, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m}\tan \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*} \int \left (e x\right )^{m} \tan \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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (e x\right )^{m} \tan \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} \tan{\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(-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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