Optimal. Leaf size=73 \[ \frac{3 \log \left (a-\sqrt [3]{a^3-b^3 x}\right )}{2 a}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a^3-b^3 x}+a}{\sqrt{3} a}\right )}{a}-\frac{\log (x)}{2 a} \]
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Rubi [A] time = 0.030922, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {55, 617, 204, 31} \[ \frac{3 \log \left (a-\sqrt [3]{a^3-b^3 x}\right )}{2 a}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a^3-b^3 x}+a}{\sqrt{3} a}\right )}{a}-\frac{\log (x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt [3]{a^3-b^3 x}} \, dx &=-\frac{\log (x)}{2 a}+\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{a^2+a x+x^2} \, dx,x,\sqrt [3]{a^3-b^3 x}\right )-\frac{3 \operatorname{Subst}\left (\int \frac{1}{a-x} \, dx,x,\sqrt [3]{a^3-b^3 x}\right )}{2 a}\\ &=-\frac{\log (x)}{2 a}+\frac{3 \log \left (a-\sqrt [3]{a^3-b^3 x}\right )}{2 a}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a^3-b^3 x}}{a}\right )}{a}\\ &=\frac{\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a^3-b^3 x}}{a}}{\sqrt{3}}\right )}{a}-\frac{\log (x)}{2 a}+\frac{3 \log \left (a-\sqrt [3]{a^3-b^3 x}\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0618446, size = 68, normalized size = 0.93 \[ \frac{3 \log \left (a-\sqrt [3]{a^3-b^3 x}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a^3-b^3 x}+a}{\sqrt{3} a}\right )-\log (x)}{2 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 91, normalized size = 1.3 \begin{align*}{\frac{1}{a}\ln \left ( -a+\sqrt [3]{-{b}^{3}x+{a}^{3}} \right ) }-{\frac{1}{2\,a}\ln \left ( \left ( -{b}^{3}x+{a}^{3} \right ) ^{{\frac{2}{3}}}+a\sqrt [3]{-{b}^{3}x+{a}^{3}}+{a}^{2} \right ) }+{\frac{\sqrt{3}}{a}\arctan \left ({\frac{\sqrt{3}}{3\,a} \left ( a+2\,\sqrt [3]{-{b}^{3}x+{a}^{3}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52501, size = 122, normalized size = 1.67 \begin{align*} \frac{\sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (a + 2 \,{\left (-b^{3} x + a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right )}{a} - \frac{\log \left (a^{2} +{\left (-b^{3} x + a^{3}\right )}^{\frac{1}{3}} a +{\left (-b^{3} x + a^{3}\right )}^{\frac{2}{3}}\right )}{2 \, a} + \frac{\log \left (-a +{\left (-b^{3} x + a^{3}\right )}^{\frac{1}{3}}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63676, size = 232, normalized size = 3.18 \begin{align*} \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3} a + 2 \, \sqrt{3}{\left (-b^{3} x + a^{3}\right )}^{\frac{1}{3}}}{3 \, a}\right ) - \log \left (a^{2} +{\left (-b^{3} x + a^{3}\right )}^{\frac{1}{3}} a +{\left (-b^{3} x + a^{3}\right )}^{\frac{2}{3}}\right ) + 2 \, \log \left (-a +{\left (-b^{3} x + a^{3}\right )}^{\frac{1}{3}}\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.40131, size = 136, normalized size = 1.86 \begin{align*} - \frac{e^{- \frac{2 i \pi }{3}} \log{\left (- \frac{a e^{\frac{i \pi }{3}}}{b \sqrt [3]{- \frac{a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a \Gamma \left (\frac{2}{3}\right )} + \frac{e^{- \frac{i \pi }{3}} \log{\left (- \frac{a e^{i \pi }}{b \sqrt [3]{- \frac{a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a \Gamma \left (\frac{2}{3}\right )} - \frac{\log{\left (- \frac{a e^{\frac{5 i \pi }{3}}}{b \sqrt [3]{- \frac{a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a \Gamma \left (\frac{2}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23209, size = 123, normalized size = 1.68 \begin{align*} \frac{\sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (a + 2 \,{\left (-b^{3} x + a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right )}{a} - \frac{\log \left (a^{2} +{\left (-b^{3} x + a^{3}\right )}^{\frac{1}{3}} a +{\left (-b^{3} x + a^{3}\right )}^{\frac{2}{3}}\right )}{2 \, a} + \frac{\log \left ({\left | -a +{\left (-b^{3} x + a^{3}\right )}^{\frac{1}{3}} \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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