Optimal. Leaf size=158 \[ -\frac {2 i x^2 \, _2F_1\left (1,-\frac {i}{b d n};1-\frac {i}{b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n}+\frac {i x^2 \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac {x^2 (-b d n+2 i)}{2 b d n} \]
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Rubi [F] time = 0.04, 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 = {} \[ \int x \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int x \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}
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Mathematica [A] time = 5.20, size = 175, normalized size = 1.11 \[ -\frac {x^2 \left (2 e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac {i}{b d n};2-\frac {i}{b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(b d n-i) \left (2 i \, _2F_1\left (1,-\frac {i}{b d n};1-\frac {i}{b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )\right )}{2 b d n (b d n-i)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \cot \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}, x\right ) \]
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 = 0.27, size = 0, normalized size = 0.00 \[ \int x \left (\cot ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\, dx \]
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 \[ \int x\,{\mathrm {cot}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2 \,d x \]
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 \[ \int x \cot ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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