Integrand size = 17, antiderivative size = 74 \[ \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {i x^3}{3}-\frac {2}{3} i x^3 \operatorname {Hypergeometric2F1}\left (1,-\frac {3 i}{2 b d n},1-\frac {3 i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {4594, 4592, 470, 371} \[ \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {i x^3}{3}-\frac {2}{3} i x^3 \operatorname {Hypergeometric2F1}\left (1,-\frac {3 i}{2 b d n},1-\frac {3 i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]
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Rule 371
Rule 470
Rule 4592
Rule 4594
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int x^{-1+\frac {3}{n}} \cot (d (a+b \log (x))) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {3}{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} \\ & = \frac {i x^3}{3}-\frac {\left (2 i x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {3}{n}}}{1-e^{2 i a d} x^{2 i b d}} \, dx,x,c x^n\right )}{n} \\ & = \frac {i x^3}{3}-\frac {2}{3} i x^3 \operatorname {Hypergeometric2F1}\left (1,-\frac {3 i}{2 b d n},1-\frac {3 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. \(229\) vs. \(2(74)=148\).
Time = 4.38 (sec) , antiderivative size = 229, normalized size of antiderivative = 3.09 \[ \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {x^3 \left (3 e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1-\frac {3 i}{2 b d n},2-\frac {3 i}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(-3 i+2 b d n) \left (\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\cot \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )+i \operatorname {Hypergeometric2F1}\left (1,-\frac {3 i}{2 b d n},1-\frac {3 i}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+\csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \csc \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right ) \sin (b d n \log (x))\right )\right )}{-9 i+6 b d n} \]
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\[\int x^{2} \cot \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
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\[ \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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\[ \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^{2} \cot {\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
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\[ \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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\[ \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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Timed out. \[ \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^2\,\mathrm {cot}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]
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