Optimal. Leaf size=155 \[ -\frac{2 i \text{Hypergeometric2F1}\left (1,\frac{i}{b d n},1+\frac{i}{b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x^2}+\frac{i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x^2 \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac{1+\frac{2 i}{b d n}}{2 x^2} \]
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Rubi [F] time = 0.0533164, 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 = {} \[ \int \frac{\cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx &=\int \frac{\cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx\\ \end{align*}
Mathematica [A] time = 4.02793, size = 175, normalized size = 1.13 \[ \frac{2 e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (1,1+\frac{i}{b d n},2+\frac{i}{b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(b d n+i) \left (-2 i \text{Hypergeometric2F1}\left (1,\frac{i}{b d n},1+\frac{i}{b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )}{2 b d n x^2 (b d n+i)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.863, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cot \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{2}}{{x}^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cot \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}}{x^{3}}, 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 \frac{\cot ^{2}{\left (a d + b d \log{\left (c x^{n} \right )} \right )}}{x^{3}}\, 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 \frac{\cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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