Optimal. Leaf size=97 \[ -\text {RootSum}\left [\text {$\#$1}^9-4 \text {$\#$1}^6+5 \text {$\#$1}^3-1\& ,\frac {-\text {$\#$1}^3 \log \left (\sqrt [3]{x^3-x^2}-\text {$\#$1} x\right )+\text {$\#$1}^3 \log (x)+2 \log \left (\sqrt [3]{x^3-x^2}-\text {$\#$1} x\right )-2 \log (x)}{3 \text {$\#$1}^4-5 \text {$\#$1}}\& \right ] \]
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Rubi [F] time = 0.61, 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-x+x^3\right ) \sqrt [3]{-x^2+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-x+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx &=\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1+x}{\sqrt [3]{-1+x} x^{2/3} \left (-1-x+x^3\right )} \, dx}{\sqrt [3]{-x^2+x^3}}\\ &=\frac {\left (3 \sqrt [3]{-1+x} x^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1+x^3}{\sqrt [3]{-1+x^3} \left (-1-x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}}\\ &=\frac {\left (3 \sqrt [3]{-1+x} x^{2/3}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [3]{-1+x^3} \left (-1-x^3+x^9\right )}+\frac {x^3}{\sqrt [3]{-1+x^3} \left (-1-x^3+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}}\\ &=\frac {\left (3 \sqrt [3]{-1+x} x^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3} \left (-1-x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}}+\frac {\left (3 \sqrt [3]{-1+x} x^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{-1+x^3} \left (-1-x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}}\\ \end {align*}
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Mathematica [B] time = 0.10, size = 481, normalized size = 4.96 \begin {gather*} -\frac {\left (\frac {1}{x-1}+1\right )^{2/3} (x-1) \left (\text {RootSum}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+4 \text {$\#$1}-1\&,\frac {-\frac {2 \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {1}{x-1}+1}\right )}{\text {$\#$1}^{2/3}}+\frac {\log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {1}{x-1}+1}+\left (\frac {1}{x-1}+1\right )^{2/3}\right )}{\text {$\#$1}^{2/3}}+\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {1}{x-1}+1}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )}{\text {$\#$1}^{2/3}}}{3 \text {$\#$1}^2-10 \text {$\#$1}+4}\&\right ]-3 \text {RootSum}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+4 \text {$\#$1}-1\&,\frac {\sqrt [3]{\text {$\#$1}} \log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {1}{x-1}+1}+\left (\frac {1}{x-1}+1\right )^{2/3}\right )-2 \sqrt [3]{\text {$\#$1}} \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {1}{x-1}+1}\right )+2 \sqrt {3} \sqrt [3]{\text {$\#$1}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {1}{x-1}+1}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )}{3 \text {$\#$1}^2-10 \text {$\#$1}+4}\&\right ]+2 \text {RootSum}\left [\text {$\#$1}^3-5 \text {$\#$1}^2+4 \text {$\#$1}-1\&,\frac {-2 \text {$\#$1}^{4/3} \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {1}{x-1}+1}\right )+\text {$\#$1}^{4/3} \log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {1}{x-1}+1}+\left (\frac {1}{x-1}+1\right )^{2/3}\right )+2 \sqrt {3} \text {$\#$1}^{4/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {1}{x-1}+1}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )}{3 \text {$\#$1}^2-10 \text {$\#$1}+4}\&\right ]\right )}{2 \sqrt [3]{(x-1) x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 97, normalized size = 1.00 \begin {gather*} -\text {RootSum}\left [-1+5 \text {$\#$1}^3-4 \text {$\#$1}^6+\text {$\#$1}^9\&,\frac {-2 \log (x)+2 \log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+3 \text {$\#$1}^4}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 163.86, size = 190624, normalized size = 1965.20
method | result | size |
trager | \(\text {Expression too large to display}\) | \(190624\) |
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} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{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 = -0.01 \begin {gather*} \int -\frac {x+1}{{\left (x^3-x^2\right )}^{1/3}\,\left (-x^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]{x^{2} \left (x - 1\right )} \left (x^{3} - x - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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