Optimal. Leaf size=39 \[ 2 \tan ^{-1}\left (\frac {\sqrt [4]{x^2-x}}{x}\right )-2 \tanh ^{-1}\left (\frac {\left (x^2-x\right )^{3/4}}{x-1}\right ) \]
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Rubi [F] time = 1.15, 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 {-3+2 x}{\sqrt [4]{-x+x^2} \left (1-x+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-3+2 x}{\sqrt [4]{-x+x^2} \left (1-x+x^3\right )} \, dx &=\frac {\left (\sqrt [4]{-1+x} \sqrt [4]{x}\right ) \int \frac {-3+2 x}{\sqrt [4]{-1+x} \sqrt [4]{x} \left (1-x+x^3\right )} \, dx}{\sqrt [4]{-x+x^2}}\\ &=\frac {\left (4 \sqrt [4]{-1+x} \sqrt [4]{x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-3+2 x^4\right )}{\sqrt [4]{-1+x^4} \left (1-x^4+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^2}}\\ &=\frac {\left (4 \sqrt [4]{-1+x} \sqrt [4]{x}\right ) \operatorname {Subst}\left (\int \left (-\frac {3 x^2}{\sqrt [4]{-1+x^4} \left (1-x^4+x^{12}\right )}+\frac {2 x^6}{\sqrt [4]{-1+x^4} \left (1-x^4+x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^2}}\\ &=\frac {\left (8 \sqrt [4]{-1+x} \sqrt [4]{x}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt [4]{-1+x^4} \left (1-x^4+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^2}}-\frac {\left (12 \sqrt [4]{-1+x} \sqrt [4]{x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{-1+x^4} \left (1-x^4+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^2}}\\ \end {align*}
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Mathematica [F] time = 0.54, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-3+2 x}{\sqrt [4]{-x+x^2} \left (1-x+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.10, size = 39, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\frac {\sqrt [4]{-x+x^2}}{x}\right )-2 \tanh ^{-1}\left (\frac {\left (-x+x^2\right )^{3/4}}{-1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 7.78, size = 93, normalized size = 2.38 \begin {gather*} \arctan \left (\frac {2 \, {\left ({\left (x^{2} - x\right )}^{\frac {1}{4}} x^{2} + {\left (x^{2} - x\right )}^{\frac {3}{4}}\right )}}{x^{3} - x + 1}\right ) + \log \left (-\frac {x^{3} - 2 \, {\left (x^{2} - x\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{2} - x} x + x - 2 \, {\left (x^{2} - x\right )}^{\frac {3}{4}} - 1}{x^{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 {2 \, x - 3}{{\left (x^{3} - x + 1\right )} {\left (x^{2} - x\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.80, size = 150, normalized size = 3.85
method | result | size |
trager | \(-\ln \left (-\frac {2 \left (x^{2}-x \right )^{\frac {3}{4}}+2 x \sqrt {x^{2}-x}+2 \left (x^{2}-x \right )^{\frac {1}{4}} x^{2}+x^{3}+x -1}{x^{3}-x +1}\right )+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{2}-x}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{2}-x \right )^{\frac {3}{4}}-2 \left (x^{2}-x \right )^{\frac {1}{4}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x +\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{3}-x +1}\right )\) | \(150\) |
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 {2 \, x - 3}{{\left (x^{3} - x + 1\right )} {\left (x^{2} - x\right )}^{\frac {1}{4}}}\,{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 = -0.03 \begin {gather*} \int \frac {2\,x-3}{{\left (x^2-x\right )}^{1/4}\,\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(-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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