Optimal. Leaf size=20 \[ -\frac{\tanh ^{-1}\left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
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Rubi [A] time = 0.015723, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {3770} \[ -\frac{\tanh ^{-1}\left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin{align*} \int \frac{\csc \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \csc (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\tanh ^{-1}\left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ \end{align*}
Mathematica [B] time = 0.0571344, size = 54, normalized size = 2.7 \[ \frac{\log \left (\sin \left (\frac{a}{2}+\frac{1}{2} b \log \left (c x^n\right )\right )\right )}{b n}-\frac{\log \left (\cos \left (\frac{a}{2}+\frac{1}{2} b \log \left (c x^n\right )\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 33, normalized size = 1.7 \begin{align*} -{\frac{\ln \left ( \csc \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) +\cot \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }{bn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995549, size = 43, normalized size = 2.15 \begin{align*} -\frac{\log \left (\cot \left (b \log \left (c x^{n}\right ) + a\right ) + \csc \left (b \log \left (c x^{n}\right ) + a\right )\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.497139, size = 147, normalized size = 7.35 \begin{align*} -\frac{\log \left (\frac{1}{2} \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \frac{1}{2}\right ) - \log \left (-\frac{1}{2} \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \frac{1}{2}\right )}{2 \, b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.88101, size = 49, normalized size = 2.45 \begin{align*} - \begin{cases} - \log{\left (x \right )} \csc{\left (a \right )} & \text{for}\: b = 0 \\- \log{\left (x \right )} \csc{\left (a + b \log{\left (c \right )} \right )} & \text{for}\: n = 0 \\\frac{\log{\left (\cot{\left (a + b \log{\left (c x^{n} \right )} \right )} + \csc{\left (a + b \log{\left (c x^{n} \right )} \right )} \right )}}{b n} & \text{otherwise} \end{cases} \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 )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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