Optimal. Leaf size=125 \[ \frac {4 \left (x^3+1\right )^{3/4}}{3 x^3}-2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^3+1}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [4]{x^3+1}}{x}\right )+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3+1}}{\sqrt {x^3+1}-x^2}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3+1}}{\sqrt {x^3+1}+x^2}\right ) \]
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Rubi [F] time = 1.59, 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 {\left (4+x^3\right ) \left (1+2 x^3+x^6+x^8\right )}{x^4 \sqrt [4]{1+x^3} \left (-1-2 x^3-x^6+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (4+x^3\right ) \left (1+2 x^3+x^6+x^8\right )}{x^4 \sqrt [4]{1+x^3} \left (-1-2 x^3-x^6+x^8\right )} \, dx &=\int \left (-\frac {4}{x^4 \sqrt [4]{1+x^3}}-\frac {1}{x \sqrt [4]{1+x^3}}+\frac {4+x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )}+\frac {4+x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )}\right ) \, dx\\ &=-\left (4 \int \frac {1}{x^4 \sqrt [4]{1+x^3}} \, dx\right )-\int \frac {1}{x \sqrt [4]{1+x^3}} \, dx+\int \frac {4+x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+\int \frac {4+x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{1+x}} \, dx,x,x^3\right )\right )-\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{1+x}} \, dx,x,x^3\right )+\int \left (\frac {4}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )}+\frac {x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )}\right ) \, dx+\int \left (\frac {4}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )}+\frac {x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )}\right ) \, dx\\ &=\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{1+x}} \, dx,x,x^3\right )-\frac {4}{3} \operatorname {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{1+x^3}\right )+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx\\ &=\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^3}\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^3}\right )+\frac {4}{3} \operatorname {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{1+x^3}\right )+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx\\ &=\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3}-\frac {2}{3} \tan ^{-1}\left (\sqrt [4]{1+x^3}\right )+\frac {2}{3} \tanh ^{-1}\left (\sqrt [4]{1+x^3}\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^3}\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^3}\right )+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx\\ &=\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3}+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.45, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (4+x^3\right ) \left (1+2 x^3+x^6+x^8\right )}{x^4 \sqrt [4]{1+x^3} \left (-1-2 x^3-x^6+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 14.71, size = 125, normalized size = 1.00 \begin {gather*} \frac {4 \left (x^3+1\right )^{3/4}}{3 x^3}-2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^3+1}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [4]{x^3+1}}{x}\right )+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3+1}}{\sqrt {x^3+1}-x^2}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3+1}}{\sqrt {x^3+1}+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 47.17, size = 816, normalized size = 6.53 \begin {gather*} -\frac {12 \, \sqrt {2} x^{3} \arctan \left (\frac {x^{8} + 2 \, x^{7} + x^{6} + 2 \, x^{4} + 2 \, x^{3} + 2 \, \sqrt {2} {\left (3 \, x^{5} - x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{6} - 3 \, x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{6} + x^{5} + x^{2}\right )} \sqrt {x^{3} + 1} + {\left (16 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x^{5} + 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{5} - x^{2}\right )} \sqrt {x^{3} + 1} + \sqrt {2} {\left (x^{8} + 8 \, x^{7} - x^{6} + 8 \, x^{4} - 2 \, x^{3} - 1\right )} + 4 \, {\left (x^{7} + x^{6} + x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{4} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1}{x^{4} + x^{3} + 1}} + 1}{x^{8} - 14 \, x^{7} + x^{6} - 14 \, x^{4} + 2 \, x^{3} + 1}\right ) - 12 \, \sqrt {2} x^{3} \arctan \left (\frac {x^{8} + 2 \, x^{7} + x^{6} + 2 \, x^{4} + 2 \, x^{3} - 2 \, \sqrt {2} {\left (3 \, x^{5} - x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{6} - 3 \, x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{6} + x^{5} + x^{2}\right )} \sqrt {x^{3} + 1} + {\left (16 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x^{5} - 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{5} - x^{2}\right )} \sqrt {x^{3} + 1} - \sqrt {2} {\left (x^{8} + 8 \, x^{7} - x^{6} + 8 \, x^{4} - 2 \, x^{3} - 1\right )} + 4 \, {\left (x^{7} + x^{6} + x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{4} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1}{x^{4} + x^{3} + 1}} + 1}{x^{8} - 14 \, x^{7} + x^{6} - 14 \, x^{4} + 2 \, x^{3} + 1}\right ) - 3 \, \sqrt {2} x^{3} \log \left (\frac {4 \, {\left (x^{4} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1\right )}}{x^{4} + x^{3} + 1}\right ) + 3 \, \sqrt {2} x^{3} \log \left (\frac {4 \, {\left (x^{4} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1\right )}}{x^{4} + x^{3} + 1}\right ) - 12 \, x^{3} \arctan \left (\frac {2 \, {\left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + {\left (x^{3} + 1\right )}^{\frac {3}{4}} x\right )}}{x^{4} - x^{3} - 1}\right ) - 12 \, x^{3} \log \left (\frac {x^{4} - 2 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 2 \, \sqrt {x^{3} + 1} x^{2} - 2 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1}{x^{4} - x^{3} - 1}\right ) - 16 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}}}{12 \, x^{3}} \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 {{\left (x^{8} + x^{6} + 2 \, x^{3} + 1\right )} {\left (x^{3} + 4\right )}}{{\left (x^{8} - x^{6} - 2 \, x^{3} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 4.03, size = 436, normalized size = 3.49 \begin {gather*} \frac {4 \left (x^{3}+1\right )^{\frac {3}{4}}}{3 x^{3}}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{3}+1}\, x^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x -2 \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{4}-x^{3}-1}\right )-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \sqrt {x^{3}+1}\, x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{4}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{3}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}+x^{3}+1}\right )+\RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \sqrt {x^{3}+1}\, x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}+x^{3}+1}\right )-\ln \left (-\frac {2 \left (x^{3}+1\right )^{\frac {3}{4}} x +2 x^{2} \sqrt {x^{3}+1}+2 \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+x^{4}+x^{3}+1}{x^{4}-x^{3}-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 {{\left (x^{8} + x^{6} + 2 \, x^{3} + 1\right )} {\left (x^{3} + 4\right )}}{{\left (x^{8} - x^{6} - 2 \, x^{3} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{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 = -0.01 \begin {gather*} \int -\frac {\left (x^3+4\right )\,\left (x^8+x^6+2\,x^3+1\right )}{x^4\,{\left (x^3+1\right )}^{1/4}\,\left (-x^8+x^6+2\,x^3+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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