Optimal. Leaf size=102 \[ \frac{\sqrt [3]{x^3+1} \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{x+1} \sqrt [3]{x^2-x+1}}-\frac{\sqrt [3]{x^3+1} \log \left (\sqrt [3]{x^3+1}-x\right )}{2 \sqrt [3]{x+1} \sqrt [3]{x^2-x+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0171366, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {713, 239} \[ \frac{\sqrt [3]{x^3+1} \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{x+1} \sqrt [3]{x^2-x+1}}-\frac{\sqrt [3]{x^3+1} \log \left (\sqrt [3]{x^3+1}-x\right )}{2 \sqrt [3]{x+1} \sqrt [3]{x^2-x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 713
Rule 239
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [3]{1+x} \sqrt [3]{1-x+x^2}} \, dx &=\frac{\sqrt [3]{1+x^3} \int \frac{1}{\sqrt [3]{1+x^3}} \, dx}{\sqrt [3]{1+x} \sqrt [3]{1-x+x^2}}\\ &=\frac{\sqrt [3]{1+x^3} \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{1+x^3}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{1+x} \sqrt [3]{1-x+x^2}}-\frac{\sqrt [3]{1+x^3} \log \left (-x+\sqrt [3]{1+x^3}\right )}{2 \sqrt [3]{1+x} \sqrt [3]{1-x+x^2}}\\ \end{align*}
Mathematica [C] time = 0.0704854, size = 132, normalized size = 1.29 \[ \frac{3 \sqrt [3]{\frac{-2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt [3]{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} (x+1)^{2/3} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{2 i (x+1)}{3 i+\sqrt{3}},-\frac{2 i (x+1)}{-3 i+\sqrt{3}}\right )}{2 \sqrt [3]{x^2-x+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 1.582, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [3]{1+x}}}{\frac{1}{\sqrt [3]{{x}^{2}-x+1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{2} - x + 1\right )}^{\frac{1}{3}}{\left (x + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 3.30881, size = 333, normalized size = 3.26 \begin{align*} \frac{1}{3} \, \sqrt{3} \arctan \left (-\frac{4 \, \sqrt{3}{\left (x^{2} - x + 1\right )}^{\frac{1}{3}}{\left (x + 1\right )}^{\frac{1}{3}} x^{2} - 2 \, \sqrt{3}{\left (x^{2} - x + 1\right )}^{\frac{2}{3}}{\left (x + 1\right )}^{\frac{2}{3}} x + \sqrt{3}{\left (x^{3} + 1\right )}}{9 \, x^{3} + 1}\right ) - \frac{1}{6} \, \log \left (3 \,{\left (x^{2} - x + 1\right )}^{\frac{1}{3}}{\left (x + 1\right )}^{\frac{1}{3}} x^{2} - 3 \,{\left (x^{2} - x + 1\right )}^{\frac{2}{3}}{\left (x + 1\right )}^{\frac{2}{3}} x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{x + 1} \sqrt [3]{x^{2} - x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{2} - x + 1\right )}^{\frac{1}{3}}{\left (x + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]