Optimal. Leaf size=19 \[ -\frac{\cot \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Rubi [A] time = 0.0279832, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3767, 8} \[ -\frac{\cot \left (a+b \log \left (c x^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{\csc ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \csc ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\operatorname{Subst}\left (\int 1 \, dx,x,\cot \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=-\frac{\cot \left (a+b \log \left (c x^n\right )\right )}{b n}\\ \end{align*}
Mathematica [A] time = 0.096024, size = 19, normalized size = 1. \[ -\frac{\cot \left (a+b \log \left (c x^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 20, normalized size = 1.1 \begin{align*} -{\frac{\cot \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{bn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.20872, size = 227, normalized size = 11.95 \begin{align*} \frac{2 \,{\left (\cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) \sin \left (2 \, b \log \left (c\right )\right ) + \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )\right )}}{2 \, b n \cos \left (2 \, b \log \left (c\right )\right ) \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) -{\left (b \cos \left (2 \, b \log \left (c\right )\right )^{2} + b \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} n \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )^{2} - 2 \, b n \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) -{\left (b \cos \left (2 \, b \log \left (c\right )\right )^{2} + b \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} n \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )^{2} - b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.467328, size = 95, normalized size = 5. \begin{align*} -\frac{\cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b n \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\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{\csc ^{2}{\left (a + b \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (b \log \left (c x^{n}\right ) + a\right )^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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