Optimal. Leaf size=70 \[ \frac {2 i \, _2F_1\left (1,\frac {i}{2 b d n};1+\frac {i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{x}-\frac {i}{x} \]
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Rubi [F] time = 0.03, 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 = {} \[ \int \frac {\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx &=\int \frac {\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx\\ \end {align*}
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Mathematica [B] time = 4.59, size = 217, normalized size = 3.10 \[ \frac {-\frac {e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1+\frac {i}{2 b d n};2+\frac {i}{2 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b d n+i}+i \, _2F_1\left (1,\frac {i}{2 b d n};1+\frac {i}{2 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\cot \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+\sin (b d n \log (x)) \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \csc \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )}{x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cot \left (b d \log \left (c x^{n}\right ) + a d\right )}{x^{2}}, x\right ) \]
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 = 1.45, size = 0, normalized size = 0.00 \[ \int \frac {\cot \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}}\,{d x} \]
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 \[ \int \frac {\mathrm {cot}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^2} \,d x \]
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 \[ \int \frac {\cot {\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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