Optimal. Leaf size=133 \[ -\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 a^{4/3}}+\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{4/3}}+\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3} a^{4/3}}-\frac{1}{2 a x^2} \]
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Rubi [A] time = 0.104705, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {275, 325, 292, 31, 634, 617, 204, 628} \[ -\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 a^{4/3}}+\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{4/3}}+\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3} a^{4/3}}-\frac{1}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 275
Rule 325
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b x^6\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^3\right )} \, dx,x,x^2\right )\\ &=-\frac{1}{2 a x^2}-\frac{b \operatorname{Subst}\left (\int \frac{x}{a+b x^3} \, dx,x,x^2\right )}{2 a}\\ &=-\frac{1}{2 a x^2}+\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,x^2\right )}{6 a^{4/3}}-\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{6 a^{4/3}}\\ &=-\frac{1}{2 a x^2}+\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{4/3}}-\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{12 a^{4/3}}-\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{4 a}\\ &=-\frac{1}{2 a x^2}+\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{4/3}}-\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 a^{4/3}}-\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}\right )}{2 a^{4/3}}\\ &=-\frac{1}{2 a x^2}+\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3} a^{4/3}}+\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{4/3}}-\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 a^{4/3}}\\ \end{align*}
Mathematica [A] time = 0.0287991, size = 203, normalized size = 1.53 \[ \frac{2 \sqrt [3]{b} x^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-\sqrt [3]{b} x^2 \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-\sqrt [3]{b} x^2 \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+2 \sqrt{3} \sqrt [3]{b} x^2 \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt{3} \sqrt [3]{b} x^2 \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )-6 \sqrt [3]{a}}{12 a^{4/3} x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 105, normalized size = 0.8 \begin{align*}{\frac{1}{6\,a}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{1}{12\,a}\ln \left ({x}^{4}-\sqrt [3]{{\frac{a}{b}}}{x}^{2}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{\sqrt{3}}{6\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{{x}^{2}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{1}{2\,a{x}^{2}}} \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 = 1.6911, size = 282, normalized size = 2.12 \begin{align*} -\frac{2 \, \sqrt{3} x^{2} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3} x^{2} \left (\frac{b}{a}\right )^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) + x^{2} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{4} - a x^{2} \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 2 \, x^{2} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) + 6}{12 \, a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.416471, size = 34, normalized size = 0.26 \begin{align*} \operatorname{RootSum}{\left (216 t^{3} a^{4} - b, \left ( t \mapsto t \log{\left (\frac{36 t^{2} a^{3}}{b} + x^{2} \right )} \right )\right )} - \frac{1}{2 a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22149, size = 171, normalized size = 1.29 \begin{align*} \frac{b \left (-\frac{a}{b}\right )^{\frac{2}{3}} \log \left ({\left | x^{2} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{6 \, a^{2}} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{2} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{6 \, a^{2} b} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}} \log \left (x^{4} + x^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{12 \, a^{2} b} - \frac{1}{2 \, a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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