Optimal. Leaf size=27 \[ \frac{\log (x)}{a}-\frac{b \log \left (a \log \left (c x^n\right )+b\right )}{a^2 n} \]
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Rubi [A] time = 0.023225, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {43} \[ \frac{\log (x)}{a}-\frac{b \log \left (a \log \left (c x^n\right )+b\right )}{a^2 n} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{1}{a x+\frac{b x}{\log \left (c x^n\right )}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{b+a x} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a}-\frac{b}{a (b+a x)}\right ) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\log (x)}{a}-\frac{b \log \left (b+a \log \left (c x^n\right )\right )}{a^2 n}\\ \end{align*}
Mathematica [A] time = 0.0150973, size = 34, normalized size = 1.26 \[ \frac{\log \left (c x^n\right )}{a n}-\frac{b \log \left (a \log \left (c x^n\right )+b\right )}{a^2 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 35, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( c{x}^{n} \right ) }{na}}-{\frac{b\ln \left ( b+a\ln \left ( c{x}^{n} \right ) \right ) }{{a}^{2}n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04503, size = 45, normalized size = 1.67 \begin{align*} \frac{\log \left (x\right )}{a} - \frac{b \log \left (\frac{a \log \left (c\right ) + a \log \left (x^{n}\right ) + b}{a}\right )}{a^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8948, size = 77, normalized size = 2.85 \begin{align*} \frac{a n \log \left (x\right ) - b \log \left (a n \log \left (x\right ) + a \log \left (c\right ) + b\right )}{a^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30061, size = 72, normalized size = 2.67 \begin{align*} \frac{\log \left (x\right )}{a} - \frac{b \log \left (\frac{1}{4} \,{\left (\pi a n{\left (\mathrm{sgn}\left (x\right ) - 1\right )} + \pi a{\left (\mathrm{sgn}\left (c\right ) - 1\right )}\right )}^{2} +{\left (a n \log \left ({\left | x \right |}\right ) + a \log \left ({\left | c \right |}\right ) + b\right )}^{2}\right )}{2 \, a^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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