Optimal. Leaf size=88 \[ \frac {1}{2} \log \left (\sqrt [3]{x^5-x}-x\right )-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^5-x}+x}\right )-\frac {1}{4} \log \left (\sqrt [3]{x^5-x} x+\left (x^5-x\right )^{2/3}+x^2\right ) \]
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Rubi [C] time = 1.16, antiderivative size = 305, normalized size of antiderivative = 3.47, number of steps used = 22, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2056, 6715, 6728, 246, 245, 1438, 430, 429, 465, 511, 510} \begin {gather*} -\frac {3 \sqrt [3]{1-x^4} x F_1\left (\frac {1}{6};\frac {1}{3},1;\frac {7}{6};x^4,\frac {2 x^4}{3-\sqrt {5}}\right )}{2 \sqrt [3]{x^5-x}}-\frac {3 \sqrt [3]{1-x^4} x F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};\frac {2 x^4}{3+\sqrt {5}},x^4\right )}{2 \sqrt [3]{x^5-x}}-\frac {3 \left (1-\sqrt {5}\right ) \sqrt [3]{1-x^4} x^3 F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x^4,\frac {2 x^4}{3-\sqrt {5}}\right )}{8 \left (3-\sqrt {5}\right ) \sqrt [3]{x^5-x}}-\frac {3 \left (1+\sqrt {5}\right ) \sqrt [3]{1-x^4} x^3 F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x^4,\frac {2 x^4}{3+\sqrt {5}}\right )}{8 \left (3+\sqrt {5}\right ) \sqrt [3]{x^5-x}}+\frac {3 \sqrt [3]{1-x^4} x \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^4\right )}{2 \sqrt [3]{x^5-x}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 245
Rule 246
Rule 429
Rule 430
Rule 465
Rule 510
Rule 511
Rule 1438
Rule 2056
Rule 6715
Rule 6728
Rubi steps
\begin {align*} \int \frac {1+x^4}{\left (-1-x^2+x^4\right ) \sqrt [3]{-x+x^5}} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \int \frac {1+x^4}{\sqrt [3]{x} \sqrt [3]{-1+x^4} \left (-1-x^2+x^4\right )} \, dx}{\sqrt [3]{-x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1+x^6}{\sqrt [3]{-1+x^6} \left (-1-x^3+x^6\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [3]{-1+x^6}}+\frac {2+x^3}{\sqrt [3]{-1+x^6} \left (-1-x^3+x^6\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^6}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {2+x^3}{\sqrt [3]{-1+x^6} \left (-1-x^3+x^6\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1-x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x^6}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1+\sqrt {5}}{\left (-1-\sqrt {5}+2 x^3\right ) \sqrt [3]{-1+x^6}}+\frac {1-\sqrt {5}}{\left (-1+\sqrt {5}+2 x^3\right ) \sqrt [3]{-1+x^6}}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}\\ &=\frac {3 x \sqrt [3]{1-x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^4\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+\sqrt {5}+2 x^3\right ) \sqrt [3]{-1+x^6}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1-\sqrt {5}+2 x^3\right ) \sqrt [3]{-1+x^6}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}\\ &=\frac {3 x \sqrt [3]{1-x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^4\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1-\sqrt {5}}{2 \sqrt [3]{-1+x^6} \left (-3+\sqrt {5}+2 x^6\right )}+\frac {x^3}{\sqrt [3]{-1+x^6} \left (-3+\sqrt {5}+2 x^6\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {-1-\sqrt {5}}{2 \left (3+\sqrt {5}-2 x^6\right ) \sqrt [3]{-1+x^6}}+\frac {x^3}{\sqrt [3]{-1+x^6} \left (-3-\sqrt {5}+2 x^6\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}\\ &=\frac {3 x \sqrt [3]{1-x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^4\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{-1+x^6} \left (-3+\sqrt {5}+2 x^6\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1-\sqrt {5}\right )^2 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^6} \left (-3+\sqrt {5}+2 x^6\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{-1+x^6} \left (-3-\sqrt {5}+2 x^6\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (-1-\sqrt {5}\right ) \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (3+\sqrt {5}-2 x^6\right ) \sqrt [3]{-1+x^6}} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-x+x^5}}\\ &=\frac {3 x \sqrt [3]{1-x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^4\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1-\sqrt {5}\right )^2 \sqrt [3]{x} \sqrt [3]{1-x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x^6} \left (-3+\sqrt {5}+2 x^6\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (-1-\sqrt {5}\right ) \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{1-x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (3+\sqrt {5}-2 x^6\right ) \sqrt [3]{1-x^6}} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^3} \left (-3+\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^3} \left (-3-\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}}\\ &=-\frac {3 x \sqrt [3]{1-x^4} F_1\left (\frac {1}{6};\frac {1}{3},1;\frac {7}{6};x^4,\frac {2 x^4}{3-\sqrt {5}}\right )}{2 \sqrt [3]{-x+x^5}}-\frac {3 x \sqrt [3]{1-x^4} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};\frac {2 x^4}{3+\sqrt {5}},x^4\right )}{2 \sqrt [3]{-x+x^5}}+\frac {3 x \sqrt [3]{1-x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^4\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{1-x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1-x^3} \left (-3+\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{1-x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1-x^3} \left (-3-\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}}\\ &=-\frac {3 x \sqrt [3]{1-x^4} F_1\left (\frac {1}{6};\frac {1}{3},1;\frac {7}{6};x^4,\frac {2 x^4}{3-\sqrt {5}}\right )}{2 \sqrt [3]{-x+x^5}}-\frac {3 x \sqrt [3]{1-x^4} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};\frac {2 x^4}{3+\sqrt {5}},x^4\right )}{2 \sqrt [3]{-x+x^5}}-\frac {3 \left (1-\sqrt {5}\right ) x^3 \sqrt [3]{1-x^4} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x^4,\frac {2 x^4}{3-\sqrt {5}}\right )}{8 \left (3-\sqrt {5}\right ) \sqrt [3]{-x+x^5}}-\frac {3 \left (1+\sqrt {5}\right ) x^3 \sqrt [3]{1-x^4} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x^4,\frac {2 x^4}{3+\sqrt {5}}\right )}{8 \left (3+\sqrt {5}\right ) \sqrt [3]{-x+x^5}}+\frac {3 x \sqrt [3]{1-x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^4\right )}{2 \sqrt [3]{-x+x^5}}\\ \end {align*}
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Mathematica [F] time = 1.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^4}{\left (-1-x^2+x^4\right ) \sqrt [3]{-x+x^5}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.45, size = 88, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log \left (\sqrt [3]{x^5-x}-x\right )-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^5-x}+x}\right )-\frac {1}{4} \log \left (\sqrt [3]{x^5-x} x+\left (x^5-x\right )^{2/3}+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.20, size = 109, normalized size = 1.24 \begin {gather*} -\frac {1}{2} \, \sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{4} - 1\right )} - 2 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {2}{3}}}{x^{4} + 8 \, x^{2} - 1}\right ) + \frac {1}{4} \, \log \left (\frac {x^{4} - x^{2} + 3 \, {\left (x^{5} - x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{5} - x\right )}^{\frac {2}{3}} - 1}{x^{4} - x^{2} - 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^{4} + 1}{{\left (x^{5} - x\right )}^{\frac {1}{3}} {\left (x^{4} - x^{2} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 5.48, size = 591, normalized size = 6.72 \begin {gather*} \frac {\ln \left (\frac {11545253597330080 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{4}+31816769631648996 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}-43294700989987800 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+10442381802726383 x^{4}-41156279639771682 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {2}{3}}+4180130780591496 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {1}{3}} x -10715344468797670 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}-11545253597330080 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}-2090065390295748 \left (x^{5}-x \right )^{\frac {2}{3}}-18488074429590093 x \left (x^{5}-x \right )^{\frac {1}{3}}+2198396168995028 x^{2}-31816769631648996 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-10442381802726383}{x^{4}-x^{2}-1}\right )}{2}-\frac {\ln \left (-\frac {2751668921349968 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{4}-9556171565521246 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}-10318758455062380 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+8243985633731355 x^{4}-41156279639771682 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {2}{3}}+36976148859180186 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {1}{3}} x +9952757579256536 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}-2751668921349968 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}-18488074429590093 \left (x^{5}-x \right )^{\frac {2}{3}}-2090065390295748 x \left (x^{5}-x \right )^{\frac {1}{3}}+10442381802726383 x^{2}+9556171565521246 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-8243985633731355}{x^{4}-x^{2}-1}\right )}{2}-\ln \left (-\frac {2751668921349968 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{4}-9556171565521246 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}-10318758455062380 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+8243985633731355 x^{4}-41156279639771682 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {2}{3}}+36976148859180186 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {1}{3}} x +9952757579256536 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}-2751668921349968 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}-18488074429590093 \left (x^{5}-x \right )^{\frac {2}{3}}-2090065390295748 x \left (x^{5}-x \right )^{\frac {1}{3}}+10442381802726383 x^{2}+9556171565521246 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-8243985633731355}{x^{4}-x^{2}-1}\right ) \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \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 {x^{4} + 1}{{\left (x^{5} - x\right )}^{\frac {1}{3}} {\left (x^{4} - x^{2} - 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^4+1}{{\left (x^5-x\right )}^{1/3}\,\left (-x^4+x^2+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^{4} + 1}{\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{4} - x^{2} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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