Optimal. Leaf size=53 \[ -\frac{C \log \left (\sqrt [3]{-\frac{a}{b}}+x\right )}{b}-\frac{2 C \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0931049, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {1867, 31, 617, 204} \[ -\frac{C \log \left (\sqrt [3]{-\frac{a}{b}}+x\right )}{b}-\frac{2 C \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1867
Rule 31
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x \left (-2 \sqrt [3]{-\frac{a}{b}} C+C x\right )}{a-b x^3} \, dx &=-\frac{C \int \frac{1}{\sqrt [3]{-\frac{a}{b}}+x} \, dx}{b}+\frac{\left (\sqrt [3]{-\frac{a}{b}} C\right ) \int \frac{1}{\left (-\frac{a}{b}\right )^{2/3}-\sqrt [3]{-\frac{a}{b}} x+x^2} \, dx}{b}\\ &=-\frac{C \log \left (\sqrt [3]{-\frac{a}{b}}+x\right )}{b}+\frac{(2 C) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}\right )}{b}\\ &=-\frac{2 C \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} b}-\frac{C \log \left (\sqrt [3]{-\frac{a}{b}}+x\right )}{b}\\ \end{align*}
Mathematica [B] time = 0.0692506, size = 149, normalized size = 2.81 \[ -\frac{C \left (\sqrt [3]{b} \sqrt [3]{-\frac{a}{b}} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\sqrt [3]{a} \log \left (a-b x^3\right )-2 \sqrt [3]{b} \sqrt [3]{-\frac{a}{b}} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )-2 \sqrt{3} \sqrt [3]{b} \sqrt [3]{-\frac{a}{b}} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )\right )}{3 \sqrt [3]{a} b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.006, size = 135, normalized size = 2.6 \begin{align*}{\frac{2\,C}{3\,b}\sqrt [3]{-{\frac{a}{b}}}\ln \left ( x-\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{C}{3\,b}\sqrt [3]{-{\frac{a}{b}}}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,C\sqrt{3}}{3\,b}\sqrt [3]{-{\frac{a}{b}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{C\ln \left ( b{x}^{3}-a \right ) }{3\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.10121, size = 142, normalized size = 2.68 \begin{align*} -\frac{2 \, \sqrt{3} C \arctan \left (\frac{2 \, \sqrt{3} b x \left (-\frac{a}{b}\right )^{\frac{2}{3}} + \sqrt{3} a}{3 \, a}\right ) + 3 \, C \log \left (x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 0.435654, size = 110, normalized size = 2.08 \begin{align*} - \frac{C \left (\log{\left (- \frac{a}{b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} + x \right )} - \frac{\sqrt{3} i \log{\left (\frac{a}{2 b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} - \frac{\sqrt{3} i a}{2 b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} + x \right )}}{3} + \frac{\sqrt{3} i \log{\left (\frac{a}{2 b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} + \frac{\sqrt{3} i a}{2 b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} + x \right )}}{3}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.09418, size = 223, normalized size = 4.21 \begin{align*} -\frac{{\left (C b \left (\frac{a}{b}\right )^{\frac{2}{3}} - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} C \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b} + \frac{\sqrt{3}{\left (a b^{2} + \sqrt{3} \sqrt{a^{2} b^{4}} i\right )} C \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{3}} - \frac{{\left (3 \, a b^{2} + \sqrt{3} \sqrt{a^{2} b^{4}} i\right )} C \log \left (x^{2} + x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]