Integrand size = 13, antiderivative size = 66 \[ \int \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=i x-2 i x \operatorname {Hypergeometric2F1}\left (1,-\frac {i}{2 b d n},1-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {4590, 4592, 470, 371} \[ \int \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=i x-2 i x \operatorname {Hypergeometric2F1}\left (1,-\frac {i}{2 b d n},1-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]
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Rule 371
Rule 470
Rule 4590
Rule 4592
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \cot (d (a+b \log (x))) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {1}{n}} \left (-i-i e^{2 i a d} x^{2 i b d}\right )}{1-e^{2 i a d} x^{2 i b d}} \, dx,x,c x^n\right )}{n} \\ & = i x-\frac {\left (2 i x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {1}{n}}}{1-e^{2 i a d} x^{2 i b d}} \, dx,x,c x^n\right )}{n} \\ & = i x-2 i x \operatorname {Hypergeometric2F1}\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 ) \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(141\) vs. \(2(66)=132\).
Time = 8.06 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.14 \[ \int \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \left (-\frac {e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1-\frac {i}{2 b d n},2-\frac {i}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )}{-i+2 b d n}-i \operatorname {Hypergeometric2F1}\left (1,-\frac {i}{2 b d n},1-\frac {i}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right ) \]
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\[\int \cot \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
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\[ \int \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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\[ \int \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \cot {\left (d \left (a + b \log {\left (c x^{n} \right )}\right ) \right )}\, dx \]
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\[ \int \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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\[ \int \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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Timed out. \[ \int \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \mathrm {cot}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]
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