Optimal. Leaf size=102 \[ \frac {\sqrt [3]{1+x^3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1+x} \sqrt [3]{1-x+x^2}}-\frac {\sqrt [3]{1+x^3} \log \left (-x+\sqrt [3]{1+x^3}\right )}{2 \sqrt [3]{1+x} \sqrt [3]{1-x+x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {727, 245}
\begin {gather*} \frac {\sqrt [3]{x^3+1} \text {ArcTan}\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 245
Rule 727
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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 20.06, size = 132, normalized size = 1.29 \begin {gather*} \frac {3 \sqrt [3]{\frac {i+\sqrt {3}-2 i x}{3 i+\sqrt {3}}} \sqrt [3]{\frac {-i+\sqrt {3}+2 i x}{-3 i+\sqrt {3}}} (1+x)^{2/3} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {2 i (1+x)}{3 i+\sqrt {3}},-\frac {2 i (1+x)}{-3 i+\sqrt {3}}\right )}{2 \sqrt [3]{1-x+x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.36, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (x +1\right )^{\frac {1}{3}} \left (x^{2}-x +1\right )^{\frac {1}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.19, size = 115, normalized size = 1.13 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{x + 1} \sqrt [3]{x^{2} - x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (x+1\right )}^{1/3}\,{\left (x^2-x+1\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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