Optimal. Leaf size=88 \[ 3 \sqrt [3]{x-a}+\frac{1}{2} \sqrt [3]{a} \log (x)-\frac{3}{2} \sqrt [3]{a} \log \left (\sqrt [3]{x-a}+\sqrt [3]{a}\right )+\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{x-a}}{\sqrt{3} \sqrt [3]{a}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0386913, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {50, 58, 617, 204, 31} \[ 3 \sqrt [3]{x-a}+\frac{1}{2} \sqrt [3]{a} \log (x)-\frac{3}{2} \sqrt [3]{a} \log \left (\sqrt [3]{x-a}+\sqrt [3]{a}\right )+\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{x-a}}{\sqrt{3} \sqrt [3]{a}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 58
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{-a+x}}{x} \, dx &=3 \sqrt [3]{-a+x}-a \int \frac{1}{x (-a+x)^{2/3}} \, dx\\ &=3 \sqrt [3]{-a+x}+\frac{1}{2} \sqrt [3]{a} \log (x)-\frac{1}{2} \left (3 \sqrt [3]{a}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+x} \, dx,x,\sqrt [3]{-a+x}\right )-\frac{1}{2} \left (3 a^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{-a+x}\right )\\ &=3 \sqrt [3]{-a+x}+\frac{1}{2} \sqrt [3]{a} \log (x)-\frac{3}{2} \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{-a+x}\right )-\left (3 \sqrt [3]{a}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{-a+x}}{\sqrt [3]{a}}\right )\\ &=3 \sqrt [3]{-a+x}+\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{-a+x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )+\frac{1}{2} \sqrt [3]{a} \log (x)-\frac{3}{2} \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{-a+x}\right )\\ \end{align*}
Mathematica [A] time = 0.039665, size = 112, normalized size = 1.27 \[ \frac{1}{2} \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{x-a}+(x-a)^{2/3}\right )+3 \sqrt [3]{x-a}-\sqrt [3]{a} \log \left (\sqrt [3]{x-a}+\sqrt [3]{a}\right )+\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{x-a}}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 85, normalized size = 1. \begin{align*} 3\,\sqrt [3]{-a+x}-\sqrt [3]{a}\ln \left ( \sqrt [3]{a}+\sqrt [3]{-a+x} \right ) +{\frac{1}{2}\sqrt [3]{a}\ln \left ( \left ( -a+x \right ) ^{{\frac{2}{3}}}-\sqrt [3]{a}\sqrt [3]{-a+x}+{a}^{{\frac{2}{3}}} \right ) }-\sqrt [3]{a}\sqrt{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{-a+x}}{\sqrt [3]{a}}}-1 \right ) } \right ) \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 = 1.60695, size = 309, normalized size = 3.51 \begin{align*} \sqrt{3} \left (-a\right )^{\frac{1}{3}} \arctan \left (-\frac{\sqrt{3} a - 2 \, \sqrt{3} \left (-a\right )^{\frac{2}{3}}{\left (-a + x\right )}^{\frac{1}{3}}}{3 \, a}\right ) - \frac{1}{2} \, \left (-a\right )^{\frac{1}{3}} \log \left (\left (-a\right )^{\frac{2}{3}} + \left (-a\right )^{\frac{1}{3}}{\left (-a + x\right )}^{\frac{1}{3}} +{\left (-a + x\right )}^{\frac{2}{3}}\right ) + \left (-a\right )^{\frac{1}{3}} \log \left (-\left (-a\right )^{\frac{1}{3}} +{\left (-a + x\right )}^{\frac{1}{3}}\right ) + 3 \,{\left (-a + x\right )}^{\frac{1}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 1.74508, size = 153, normalized size = 1.74 \begin{align*} \frac{4 \sqrt [3]{a} e^{- \frac{i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{- a + x} e^{\frac{i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{4}{3}\right )}{3 \Gamma \left (\frac{7}{3}\right )} - \frac{4 \sqrt [3]{a} \log{\left (1 - \frac{\sqrt [3]{- a + x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{4}{3}\right )}{3 \Gamma \left (\frac{7}{3}\right )} + \frac{4 \sqrt [3]{a} e^{\frac{i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{- a + x} e^{\frac{5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{4}{3}\right )}{3 \Gamma \left (\frac{7}{3}\right )} + \frac{4 \sqrt [3]{- a + x} \Gamma \left (\frac{4}{3}\right )}{\Gamma \left (\frac{7}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.83854, size = 139, normalized size = 1.58 \begin{align*} -\sqrt{3} \left (-a\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (\left (-a\right )^{\frac{1}{3}} + 2 \,{\left (-a + x\right )}^{\frac{1}{3}}\right )}}{3 \, \left (-a\right )^{\frac{1}{3}}}\right ) - \frac{1}{2} \, \left (-a\right )^{\frac{1}{3}} \log \left (\left (-a\right )^{\frac{2}{3}} + \left (-a\right )^{\frac{1}{3}}{\left (-a + x\right )}^{\frac{1}{3}} +{\left (-a + x\right )}^{\frac{2}{3}}\right ) + \left (-a\right )^{\frac{1}{3}} \log \left ({\left | -\left (-a\right )^{\frac{1}{3}} +{\left (-a + x\right )}^{\frac{1}{3}} \right |}\right ) + 3 \,{\left (-a + x\right )}^{\frac{1}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]