Optimal. Leaf size=30 \[ -\frac{\cot \left (a d+b d \log \left (c x^n\right )\right )}{b d n}-\log (x) \]
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Rubi [A] time = 0.0300002, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3473, 8} \[ -\frac{\cot \left (a d+b d \log \left (c x^n\right )\right )}{b d n}-\log (x) \]
Antiderivative was successfully verified.
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Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \frac{\cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \cot ^2(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\cot \left (a d+b d \log \left (c x^n\right )\right )}{b d n}-\frac{\operatorname{Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\cot \left (a d+b d \log \left (c x^n\right )\right )}{b d n}-\log (x)\\ \end{align*}
Mathematica [C] time = 0.109699, size = 51, normalized size = 1.7 \[ -\frac{\cot \left (a d+b d \log \left (c x^n\right )\right ) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-\tan ^2\left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.022, size = 63, normalized size = 2.1 \begin{align*} -{\frac{\cot \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }{bdn}}+{\frac{\pi }{2\,bdn}}-{\frac{{\rm arccot} \left (\cot \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right )}{bdn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.22975, size = 435, normalized size = 14.5 \begin{align*} \frac{{\left (b d \cos \left (2 \, b d \log \left (c\right )\right )^{2} + b d \sin \left (2 \, b d \log \left (c\right )\right )^{2}\right )} n \cos \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )^{2} \log \left (x\right ) +{\left (b d \cos \left (2 \, b d \log \left (c\right )\right )^{2} + b d \sin \left (2 \, b d \log \left (c\right )\right )^{2}\right )} n \log \left (x\right ) \sin \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )^{2} + b d n \log \left (x\right ) - 2 \,{\left (b d n \cos \left (2 \, b d \log \left (c\right )\right ) \log \left (x\right ) - \sin \left (2 \, b d \log \left (c\right )\right )\right )} \cos \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right ) + 2 \,{\left (b d n \log \left (x\right ) \sin \left (2 \, b d \log \left (c\right )\right ) + \cos \left (2 \, b d \log \left (c\right )\right )\right )} \sin \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )}{2 \, b d n \cos \left (2 \, b d \log \left (c\right )\right ) \cos \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right ) - 2 \, b d n \sin \left (2 \, b d \log \left (c\right )\right ) \sin \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right ) -{\left (b d \cos \left (2 \, b d \log \left (c\right )\right )^{2} + b d \sin \left (2 \, b d \log \left (c\right )\right )^{2}\right )} n \cos \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )^{2} -{\left (b d \cos \left (2 \, b d \log \left (c\right )\right )^{2} + b d \sin \left (2 \, b d \log \left (c\right )\right )^{2}\right )} n \sin \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )^{2} - b d n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.490541, size = 216, normalized size = 7.2 \begin{align*} -\frac{b d n \log \left (x\right ) \sin \left (2 \, b d n \log \left (x\right ) + 2 \, b d \log \left (c\right ) + 2 \, a d\right ) + \cos \left (2 \, b d n \log \left (x\right ) + 2 \, b d \log \left (c\right ) + 2 \, a d\right ) + 1}{b d n \sin \left (2 \, b d n \log \left (x\right ) + 2 \, b d \log \left (c\right ) + 2 \, a d\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}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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