Optimal. Leaf size=45 \[ \frac {x \left (1+x^3\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-x^3\right )}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}} \]
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Rubi [A]
time = 0.01, antiderivative size = 45, 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, 251}
\begin {gather*} \frac {x \left (x^3+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-x^3\right )}{(x+1)^{2/3} \left (x^2-x+1\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 727
Rubi steps
\begin {align*} \int \frac {1}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}} \, dx &=\frac {\left (1+x^3\right )^{2/3} \int \frac {1}{\left (1+x^3\right )^{2/3}} \, dx}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}}\\ &=\frac {x \left (1+x^3\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-x^3\right )}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.06, size = 143, normalized size = 3.18 \begin {gather*} \frac {3 \left (-i+\sqrt {3}+2 i x\right ) \sqrt [3]{1+x} \left (-\frac {3 i+\sqrt {3}+\left (-3 i+\sqrt {3}\right ) x}{-3 i+\sqrt {3}+\left (3 i+\sqrt {3}\right ) x}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {4 i \sqrt {3} (1+x)}{\left (3 i+\sqrt {3}\right ) \left (-i+\sqrt {3}+2 i x\right )}\right )}{\left (-3 i+\sqrt {3}\right ) \left (1-x+x^2\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.35, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (x +1\right )^{\frac {2}{3}} \left (x^{2}-x +1\right )^{\frac {2}{3}}}\, dx\]
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (x + 1\right )^{\frac {2}{3}} \left (x^{2} - x + 1\right )^{\frac {2}{3}}}\, 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 = -0.02 \begin {gather*} \int \frac {1}{{\left (x+1\right )}^{2/3}\,{\left (x^2-x+1\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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