Optimal. Leaf size=231 \[ \frac {\log \left (5 \sqrt [3]{1-x^3}+\sqrt [3]{2} 5^{2/3} x-\sqrt [3]{2} 5^{2/3}\right )}{\sqrt [3]{2} 5^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {5 \sqrt {3} \sqrt [3]{1-x^3}}{5 \sqrt [3]{1-x^3}-2 \sqrt [3]{2} 5^{2/3} x+2 \sqrt [3]{2} 5^{2/3}}\right )}{\sqrt [3]{2} 5^{2/3}}-\frac {\log \left (5 \left (1-x^3\right )^{2/3}+\left (\sqrt [3]{2} 5^{2/3}-\sqrt [3]{2} 5^{2/3} x\right ) \sqrt [3]{1-x^3}+2^{2/3} \sqrt [3]{5} x^2-2\ 2^{2/3} \sqrt [3]{5} x+2^{2/3} \sqrt [3]{5}\right )}{2 \sqrt [3]{2} 5^{2/3}} \]
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Rubi [F] time = 0.31, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1+x}{\left (1+3 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1+x}{\left (1+3 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx &=\int \left (\frac {1-\frac {1}{\sqrt {5}}}{\left (3-\sqrt {5}+2 x\right ) \sqrt [3]{1-x^3}}+\frac {1+\frac {1}{\sqrt {5}}}{\left (3+\sqrt {5}+2 x\right ) \sqrt [3]{1-x^3}}\right ) \, dx\\ &=\frac {1}{5} \left (5-\sqrt {5}\right ) \int \frac {1}{\left (3-\sqrt {5}+2 x\right ) \sqrt [3]{1-x^3}} \, dx+\frac {1}{5} \left (5+\sqrt {5}\right ) \int \frac {1}{\left (3+\sqrt {5}+2 x\right ) \sqrt [3]{1-x^3}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x}{\left (1+3 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.23, size = 231, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {5 \sqrt {3} \sqrt [3]{1-x^3}}{2 \sqrt [3]{2} 5^{2/3}-2 \sqrt [3]{2} 5^{2/3} x+5 \sqrt [3]{1-x^3}}\right )}{\sqrt [3]{2} 5^{2/3}}+\frac {\log \left (-\sqrt [3]{2} 5^{2/3}+\sqrt [3]{2} 5^{2/3} x+5 \sqrt [3]{1-x^3}\right )}{\sqrt [3]{2} 5^{2/3}}-\frac {\log \left (2^{2/3} \sqrt [3]{5}-2\ 2^{2/3} \sqrt [3]{5} x+2^{2/3} \sqrt [3]{5} x^2+\left (\sqrt [3]{2} 5^{2/3}-\sqrt [3]{2} 5^{2/3} x\right ) \sqrt [3]{1-x^3}+5 \left (1-x^3\right )^{2/3}\right )}{2 \sqrt [3]{2} 5^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 12.66, size = 324, normalized size = 1.40 \begin {gather*} \frac {1}{30} \cdot 50^{\frac {1}{6}} \sqrt {3} \sqrt {2} \arctan \left (\frac {50^{\frac {1}{6}} \sqrt {3} {\left (2 \cdot 50^{\frac {2}{3}} \sqrt {2} {\left (3 \, x^{4} + 8 \, x^{3} + 3 \, x^{2} + 8 \, x + 3\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 50^{\frac {1}{3}} \sqrt {2} {\left (41 \, x^{6} - 11 \, x^{5} + 50 \, x^{4} - 35 \, x^{3} + 50 \, x^{2} - 11 \, x + 41\right )} - 20 \, \sqrt {2} {\left (11 \, x^{5} - 15 \, x^{4} + 15 \, x^{3} - 15 \, x^{2} + 15 \, x - 11\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{30 \, {\left (19 \, x^{6} - 69 \, x^{5} + 30 \, x^{4} - 85 \, x^{3} + 30 \, x^{2} - 69 \, x + 19\right )}}\right ) - \frac {1}{300} \cdot 50^{\frac {2}{3}} \log \left (\frac {50^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (3 \, x^{2} - x + 3\right )} + 50^{\frac {1}{3}} {\left (11 \, x^{4} - 4 \, x^{3} + 11 \, x^{2} - 4 \, x + 11\right )} - 20 \, {\left (2 \, x^{3} - x^{2} + x - 2\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{4} + 6 \, x^{3} + 11 \, x^{2} + 6 \, x + 1}\right ) + \frac {1}{150} \cdot 50^{\frac {2}{3}} \log \left (\frac {50^{\frac {2}{3}} {\left (x^{2} + 3 \, x + 1\right )} - 10 \cdot 50^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 50 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2} + 3 \, x + 1}\right ) \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*} \int \frac {x + 1}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + 3 \, x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 10.68, size = 1175, normalized size = 5.09
method | result | size |
trager | \(\text {Expression too large to display}\) | \(1175\) |
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*} \int \frac {x + 1}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + 3 \, x + 1\right )}}\,{d x} \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 = -0.00 \begin {gather*} \int \frac {x+1}{{\left (1-x^3\right )}^{1/3}\,\left (x^2+3\,x+1\right )} \,d x \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 {x + 1}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 3 x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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