Optimal. Leaf size=69 \[ \frac {1}{3} 2^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \left (x^4-x\right )^{3/4}}{x^3-1}\right )+\frac {1}{3} 2^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \left (x^4-x\right )^{3/4}}{x^3-1}\right ) \]
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Rubi [A] time = 0.14, antiderivative size = 111, normalized size of antiderivative = 1.61, number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2056, 466, 465, 377, 212, 206, 203} \begin {gather*} \frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{x^3-1} \tan ^{-1}\left (\frac {\sqrt [4]{2} x^{3/4}}{\sqrt [4]{x^3-1}}\right )}{3 \sqrt [4]{x^4-x}}+\frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{x^3-1} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x^{3/4}}{\sqrt [4]{x^3-1}}\right )}{3 \sqrt [4]{x^4-x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 377
Rule 465
Rule 466
Rule 2056
Rubi steps
\begin {align*} \int \frac {1}{\left (1+x^3\right ) \sqrt [4]{-x+x^4}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{-1+x^3} \left (1+x^3\right )} \, dx}{\sqrt [4]{-x+x^4}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{-1+x^{12}} \left (1+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^4}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^4} \left (1+x^4\right )} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{-x+x^4}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-2 x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}\\ &=\frac {\left (2 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}+\frac {\left (2 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}\\ &=\frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{-1+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{2} x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}+\frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{-1+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{3 \sqrt [4]{-x+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 58, normalized size = 0.84 \begin {gather*} \frac {4 x \sqrt [4]{1-x^3} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {2 x^3}{x^3+1}\right )}{3 \sqrt [4]{x \left (x^3-1\right )} \sqrt [4]{x^3+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.32, size = 69, normalized size = 1.00 \begin {gather*} \frac {1}{3} 2^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \left (x^4-x\right )^{3/4}}{x^3-1}\right )+\frac {1}{3} 2^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \left (x^4-x\right )^{3/4}}{x^3-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.94, size = 218, normalized size = 3.16 \begin {gather*} -\frac {1}{3} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {4 \cdot 2^{\frac {3}{4}} {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (2 \cdot 2^{\frac {3}{4}} \sqrt {x^{4} - x} x + 2^{\frac {1}{4}} {\left (3 \, x^{3} - 1\right )}\right )} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{4} - x\right )}^{\frac {3}{4}}}{2 \, {\left (x^{3} + 1\right )}}\right ) + \frac {1}{12} \cdot 2^{\frac {3}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (3 \, x^{3} - 1\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - x} x + 4 \, {\left (x^{4} - x\right )}^{\frac {3}{4}}}{x^{3} + 1}\right ) - \frac {1}{12} \cdot 2^{\frac {3}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (3 \, x^{3} - 1\right )} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - x} x + 4 \, {\left (x^{4} - x\right )}^{\frac {3}{4}}}{x^{3} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 62, normalized size = 0.90 \begin {gather*} \frac {1}{3} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{6} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{6} \cdot 2^{\frac {3}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 6.10, size = 229, normalized size = 3.32 \begin {gather*} -\frac {\RootOf \left (\textit {\_Z}^{4}-8\right ) \ln \left (-\frac {-\sqrt {x^{4}-x}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x +2 \left (x^{4}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-3 \RootOf \left (\textit {\_Z}^{4}-8\right ) x^{3}+4 \left (x^{4}-x \right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}-8\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{6}-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (-\frac {\sqrt {x^{4}-x}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x -2 \left (x^{4}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}+4 \left (x^{4}-x \right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{6} \end {gather*}
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 {1}{{\left (x^{4} - x\right )}^{\frac {1}{4}} {\left (x^{3} + 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 {1}{{\left (x^4-x\right )}^{1/4}\,\left (x^3+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 {1}{\sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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