Optimal. Leaf size=149 \[ \frac{\sqrt [3]{b} \log \left (a^{2/3} \log ^2\left (c x^n\right )-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3}\right )}{6 a^{4/3} n}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a} \log \left (c x^n\right )+\sqrt [3]{b}\right )}{3 a^{4/3} n}+\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} \log \left (c x^n\right )}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{4/3} n}+\frac{\log (x)}{a} \]
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Rubi [A] time = 0.109425, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {321, 200, 31, 634, 617, 204, 628} \[ \frac{\sqrt [3]{b} \log \left (a^{2/3} \log ^2\left (c x^n\right )-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3}\right )}{6 a^{4/3} n}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a} \log \left (c x^n\right )+\sqrt [3]{b}\right )}{3 a^{4/3} n}+\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} \log \left (c x^n\right )}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{4/3} n}+\frac{\log (x)}{a} \]
Antiderivative was successfully verified.
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Rule 321
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{a x+\frac{b x}{\log ^3\left (c x^n\right )}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{b+a x^3} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\log (x)}{a}-\frac{b \operatorname{Subst}\left (\int \frac{1}{b+a x^3} \, dx,x,\log \left (c x^n\right )\right )}{a n}\\ &=\frac{\log (x)}{a}-\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx,x,\log \left (c x^n\right )\right )}{3 a n}-\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,\log \left (c x^n\right )\right )}{3 a n}\\ &=\frac{\log (x)}{a}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \log \left (c x^n\right )\right )}{3 a^{4/3} n}+\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,\log \left (c x^n\right )\right )}{6 a^{4/3} n}-\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,\log \left (c x^n\right )\right )}{2 a n}\\ &=\frac{\log (x)}{a}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \log \left (c x^n\right )\right )}{3 a^{4/3} n}+\frac{\sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+a^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{4/3} n}-\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{a} \log \left (c x^n\right )}{\sqrt [3]{b}}\right )}{a^{4/3} n}\\ &=\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} \log \left (c x^n\right )}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{\sqrt{3} a^{4/3} n}+\frac{\log (x)}{a}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \log \left (c x^n\right )\right )}{3 a^{4/3} n}+\frac{\sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+a^{2/3} \log ^2\left (c x^n\right )\right )}{6 a^{4/3} n}\\ \end{align*}
Mathematica [A] time = 0.0502176, size = 132, normalized size = 0.89 \[ \frac{\sqrt [3]{b} \left (\log \left (a^{2/3} \log ^2\left (c x^n\right )-\sqrt [3]{a} \sqrt [3]{b} \log \left (c x^n\right )+b^{2/3}\right )-2 \log \left (\sqrt [3]{a} \log \left (c x^n\right )+\sqrt [3]{b}\right )\right )+2 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} \log \left (c x^n\right )}{\sqrt [3]{b}}}{\sqrt{3}}\right )+6 \sqrt [3]{a} \log \left (c x^n\right )}{6 a^{4/3} n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 136, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( c{x}^{n} \right ) }{na}}-{\frac{b}{3\,n{a}^{2}}\ln \left ( \ln \left ( c{x}^{n} \right ) +\sqrt [3]{{\frac{b}{a}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}+{\frac{b}{6\,n{a}^{2}}\ln \left ( \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2}-\sqrt [3]{{\frac{b}{a}}}\ln \left ( c{x}^{n} \right ) + \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b\sqrt{3}}{3\,n{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\ln \left ( c{x}^{n} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -b \int \frac{1}{3 \, a^{2} x \log \left (c\right )^{2} \log \left (x^{n}\right ) + 3 \, a^{2} x \log \left (c\right ) \log \left (x^{n}\right )^{2} + a^{2} x \log \left (x^{n}\right )^{3} +{\left (a^{2} \log \left (c\right )^{3} + a b\right )} x}\,{d x} + \frac{\log \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9596, size = 406, normalized size = 2.72 \begin{align*} \frac{6 \, n \log \left (x\right ) + 2 \, \sqrt{3} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \,{\left (\sqrt{3} a n \log \left (x\right ) + \sqrt{3} a \log \left (c\right )\right )} \left (-\frac{b}{a}\right )^{\frac{2}{3}} - \sqrt{3} b}{3 \, b}\right ) - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (n^{2} \log \left (x\right )^{2} + 2 \, n \log \left (c\right ) \log \left (x\right ) + \log \left (c\right )^{2} +{\left (n \log \left (x\right ) + \log \left (c\right )\right )} \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right ) + 2 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (n \log \left (x\right ) - \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \log \left (c\right )\right )}{6 \, a 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 [B] time = 1.39006, size = 431, normalized size = 2.89 \begin{align*} \frac{1}{3} \, \sqrt{3} \left (-\frac{b}{a^{4} n^{3}}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3} \pi a n^{3}{\left (\mathrm{sgn}\left (c\right ) - 1\right )} - 2 \, a n^{4} \log \left (x\right ) - 2 \, a n^{3} \log \left ({\left | c \right |}\right ) + 2 \, \left (-a^{2} b\right )^{\frac{1}{3}} n^{3}}{2 \, \sqrt{3} a n^{4} \log \left (x\right ) + \pi a n^{3}{\left (\mathrm{sgn}\left (c\right ) - 1\right )} + 2 \, \sqrt{3} a n^{3} \log \left ({\left | c \right |}\right ) + 2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} n^{3}}\right ) + \frac{1}{6} \, \left (-\frac{b}{a^{4} n^{3}}\right )^{\frac{1}{3}} \log \left (\frac{1}{4} \,{\left (\pi a^{2} n^{4}{\left (\mathrm{sgn}\left (x\right ) - 1\right )} + \pi a^{2} n^{3}{\left (\mathrm{sgn}\left (c\right ) - 1\right )}\right )}^{2} +{\left (a^{2} n^{4} \log \left ({\left | x \right |}\right ) + a^{2} n^{3} \log \left ({\left | c \right |}\right ) - \left (-a^{2} b\right )^{\frac{1}{3}} a n^{3}\right )}^{2}\right ) - \frac{1}{6} \, \left (-\frac{b}{a^{4} n^{3}}\right )^{\frac{1}{3}} \log \left (\frac{1}{4} \,{\left (\sqrt{3} \pi a n^{3}{\left (\mathrm{sgn}\left (c\right ) - 1\right )} - 2 \, a n^{4} \log \left (x\right ) - 2 \, a n^{3} \log \left ({\left | c \right |}\right ) + 2 \, \left (-a^{2} b\right )^{\frac{1}{3}} n^{3}\right )}^{2} + \frac{1}{4} \,{\left (2 \, \sqrt{3} a n^{4} \log \left (x\right ) + \pi a n^{3}{\left (\mathrm{sgn}\left (c\right ) - 1\right )} + 2 \, \sqrt{3} a n^{3} \log \left ({\left | c \right |}\right ) + 2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} n^{3}\right )}^{2}\right ) + \frac{\log \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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