Optimal. Leaf size=83 \[ \frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a}}-\frac{\log (x)}{2 \sqrt [3]{a}} \]
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Rubi [A] time = 0.0506088, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {266, 55, 617, 204, 31} \[ \frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a}}-\frac{\log (x)}{2 \sqrt [3]{a}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt [3]{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x \sqrt [3]{a+b x}} \, dx,x,x^3\right )\\ &=-\frac{\log (x)}{2 \sqrt [3]{a}}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}\\ &=-\frac{\log (x)}{2 \sqrt [3]{a}}+\frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{\sqrt [3]{a}}\\ &=\frac{\tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{a}}-\frac{\log (x)}{2 \sqrt [3]{a}}+\frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}\\ \end{align*}
Mathematica [A] time = 0.0176008, size = 70, normalized size = 0.84 \[ \frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )-3 \log (x)}{6 \sqrt [3]{a}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57631, size = 690, normalized size = 8.31 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}} a \sqrt{-\frac{1}{a^{\frac{2}{3}}}} \log \left (\frac{2 \, b x^{3} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} a^{\frac{2}{3}} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} a - a^{\frac{4}{3}}\right )} \sqrt{-\frac{1}{a^{\frac{2}{3}}}} - 3 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + 3 \, a}{x^{3}}\right ) - a^{\frac{2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right ) + 2 \, a^{\frac{2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}}\right )}{6 \, a}, \frac{6 \, \sqrt{\frac{1}{3}} a^{\frac{2}{3}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{a^{\frac{1}{3}}}\right ) - a^{\frac{2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right ) + 2 \, a^{\frac{2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}}\right )}{6 \, a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.36052, size = 37, normalized size = 0.45 \begin{align*} - \frac{\Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{3}}} \right )}}{3 \sqrt [3]{b} x \Gamma \left (\frac{4}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.26144, size = 117, normalized size = 1.41 \begin{align*} \frac{\sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{3 \, a^{\frac{1}{3}}} - \frac{\log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{6 \, a^{\frac{1}{3}}} + \frac{\log \left ({\left |{\left (b x^{3} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{3 \, a^{\frac{1}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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