Optimal. Leaf size=171 \[ \frac {\log \left (2 \sqrt [3]{1-x^3}+2^{2/3} x-2^{2/3}\right )}{3\ 2^{2/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{\sqrt [3]{1-x^3}-2^{2/3} x+2^{2/3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (-2 \left (1-x^3\right )^{2/3}+\left (2^{2/3} x-2^{2/3}\right ) \sqrt [3]{1-x^3}-\sqrt [3]{2} x^2+2 \sqrt [3]{2} x-\sqrt [3]{2}\right )}{6\ 2^{2/3}} \]
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Rubi [F] time = 0.28, 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+4 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+4 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx &=\int \left (\frac {1-\frac {1}{\sqrt {3}}}{\left (4-2 \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}}+\frac {1+\frac {1}{\sqrt {3}}}{\left (4+2 \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}}\right ) \, dx\\ &=\frac {1}{3} \left (3-\sqrt {3}\right ) \int \frac {1}{\left (4-2 \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}} \, dx+\frac {1}{3} \left (3+\sqrt {3}\right ) \int \frac {1}{\left (4+2 \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.19, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x}{\left (1+4 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 = 1.15, size = 171, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{2^{2/3}-2^{2/3} x+\sqrt [3]{1-x^3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (-2^{2/3}+2^{2/3} x+2 \sqrt [3]{1-x^3}\right )}{3\ 2^{2/3}}-\frac {\log \left (-\sqrt [3]{2}+2 \sqrt [3]{2} x-\sqrt [3]{2} x^2+\left (-2^{2/3}+2^{2/3} x\right ) \sqrt [3]{1-x^3}-2 \left (1-x^3\right )^{2/3}\right )}{6\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 9.96, size = 316, normalized size = 1.85 \begin {gather*} \frac {1}{18} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (2 \, x^{4} + 7 \, x^{3} + 7 \, x + 2\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} \sqrt {3} {\left (91 \, x^{6} - 42 \, x^{5} + 105 \, x^{4} - 92 \, x^{3} + 105 \, x^{2} - 42 \, x + 91\right )} - 12 \, \sqrt {3} {\left (19 \, x^{5} - 29 \, x^{4} + 28 \, x^{3} - 28 \, x^{2} + 29 \, x - 19\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (53 \, x^{6} - 174 \, x^{5} + 111 \, x^{4} - 196 \, x^{3} + 111 \, x^{2} - 174 \, x + 53\right )}}\right ) - \frac {1}{72} \cdot 4^{\frac {2}{3}} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (2 \, x^{2} - x + 2\right )} + 4^{\frac {1}{3}} {\left (19 \, x^{4} - 10 \, x^{3} + 18 \, x^{2} - 10 \, x + 19\right )} - 6 \, {\left (5 \, x^{3} - 3 \, x^{2} + 3 \, x - 5\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{4} + 8 \, x^{3} + 18 \, x^{2} + 8 \, x + 1}\right ) + \frac {1}{36} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {2}{3}} {\left (x^{2} + 4 \, x + 1\right )} - 6 \cdot 4^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 12 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2} + 4 \, 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} + 4 \, x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 9.21, size = 1157, normalized size = 6.77
method | result | size |
trager | \(\text {Expression too large to display}\) | \(1157\) |
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} + 4 \, 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 = -0.01 \begin {gather*} \int \frac {x+1}{{\left (1-x^3\right )}^{1/3}\,\left (x^2+4\,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} + 4 x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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