Optimal. Leaf size=210 \[ \frac{(e x)^{m+1} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (-\frac{i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p F_1\left (-\frac{i (m+1)}{2 b d n};p,-p;1-\frac{i (m+1)}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (m+1)} \]
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Rubi [F] time = 0.10399, 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 = {} \[ \int (e x)^m \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int (e x)^m \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int (e x)^m \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end{align*}
Mathematica [A] time = 1.12706, size = 205, normalized size = 0.98 \[ \frac{x (e x)^m \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (\frac{i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p F_1\left (-\frac{i (m+1)}{2 b d n};p,-p;1-\frac{i (m+1)}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{m+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.213, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( \cot \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} \cot \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} \cot \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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