Optimal. Leaf size=195 \[ -\frac {2 i (e x)^{m+1} \, _2F_1\left (1,-\frac {i (m+1)}{2 b d n};1-\frac {i (m+1)}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d e n}+\frac {i (e x)^{m+1} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d e n \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac {(e x)^{m+1} (-b d n+i (m+1))}{b d e (m+1) n} \]
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Rubi [F] time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (e x)^m \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (e x)^m \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int (e x)^m \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}
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Mathematica [B] time = 16.57, size = 547, normalized size = 2.81 \[ -\frac {(m+1) x^{-m} (e x)^m \csc \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \left (\frac {x^{m+1} \sin (b d n \log (x)) \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{m+1}-\frac {i \sin \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \exp \left (-\frac {(2 m+1) \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}{b n}\right ) \left (-(2 i b d n+m+1) \exp \left (\frac {2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right ) \, _2F_1\left (1,-\frac {i (m+1)}{2 b d n};1-\frac {i (m+1)}{2 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-(m+1) \exp \left (\frac {a (2 i b d n+2 m+1)}{b n}+\frac {(2 i b d n+2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )}{n}+\log (x) (2 i b d n+m+1)\right ) \, _2F_1\left (1,-\frac {i (m+2 i b d n+1)}{2 b d n};-\frac {i (m+4 i b d n+1)}{2 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+i (2 i b d n+m+1) \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \exp \left (\frac {2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right )\right )}{(m+1) (2 i b d n+m+1)}\right )}{b d n}+\frac {x (e x)^m \sin (b d n \log (x)) \csc \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \csc \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+b d n \log (x)\right )}{b d n}-\frac {x (e x)^m}{m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.16, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{m} \cot \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.45, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \left (\cot ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {cot}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2\,{\left (e\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \cot ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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