Optimal. Leaf size=43 \[ \frac {1}{3} \tan ^{-1}\left (\frac {1-x^3}{\sqrt [4]{x^6+1}}\right )-\frac {1}{3} \tanh ^{-1}\left (\frac {x^3-1}{\sqrt [4]{x^6+1}}\right ) \]
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Rubi [F] time = 1.14, 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 {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {2+x+x^2}{x \sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {1}{2 x \sqrt [4]{1+x^2}}+\frac {7-2 x+x^2}{2 \sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )}\right ) \, dx,x,x^3\right )\\ &=-\left (\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{1+x^2}} \, dx,x,x^3\right )\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {7-2 x+x^2}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )\\ &=-\left (\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{1+x}} \, dx,x,x^6\right )\right )+\frac {1}{6} \operatorname {Subst}\left (\int \left (\frac {7}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )}-\frac {2 x}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )}+\frac {x^2}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )}\right ) \, dx,x,x^3\right )\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{1+x^6}\right )+\frac {7}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^6}\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^6}\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )+\frac {7}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )\\ &=-\frac {1}{6} \tan ^{-1}\left (\sqrt [4]{1+x^6}\right )+\frac {1}{6} \tanh ^{-1}\left (\sqrt [4]{1+x^6}\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )+\frac {7}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^2} \left (-4+5 x-4 x^2+x^3\right )} \, dx,x,x^3\right )\\ \end {align*}
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Mathematica [F] time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 15.63, size = 43, normalized size = 1.00 \begin {gather*} \frac {1}{3} \tan ^{-1}\left (\frac {1-x^3}{\sqrt [4]{x^6+1}}\right )-\frac {1}{3} \tanh ^{-1}\left (\frac {x^3-1}{\sqrt [4]{x^6+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 12.49, size = 167, normalized size = 3.88 \begin {gather*} \frac {1}{6} \, \arctan \left (\frac {2 \, {\left ({\left (x^{6} + 1\right )}^{\frac {3}{4}} {\left (x^{3} - 1\right )} + {\left (x^{9} - 3 \, x^{6} + 3 \, x^{3} - 1\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}}\right )}}{x^{12} - 4 \, x^{9} + 5 \, x^{6} - 4 \, x^{3}}\right ) + \frac {1}{6} \, \log \left (-\frac {x^{12} - 4 \, x^{9} + 7 \, x^{6} - 4 \, x^{3} - 2 \, {\left (x^{6} + 1\right )}^{\frac {3}{4}} {\left (x^{3} - 1\right )} + 2 \, {\left (x^{6} - 2 \, x^{3} + 1\right )} \sqrt {x^{6} + 1} - 2 \, {\left (x^{9} - 3 \, x^{6} + 3 \, x^{3} - 1\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} + 2}{x^{12} - 4 \, x^{9} + 5 \, x^{6} - 4 \, x^{3}}\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^{6} + x^{3} + 2}{{\left (x^{9} - 4 \, x^{6} + 5 \, x^{3} - 4\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 112.80, size = 346, normalized size = 8.05 \begin {gather*} -\frac {\ln \left (-\frac {x^{12}+2 \left (x^{6}+1\right )^{\frac {1}{4}} x^{9}-4 x^{9}+2 \sqrt {x^{6}+1}\, x^{6}-6 \left (x^{6}+1\right )^{\frac {1}{4}} x^{6}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x^{3}+7 x^{6}-4 x^{3} \sqrt {x^{6}+1}+6 \left (x^{6}+1\right )^{\frac {1}{4}} x^{3}-2 \left (x^{6}+1\right )^{\frac {3}{4}}-4 x^{3}+2 \sqrt {x^{6}+1}-2 \left (x^{6}+1\right )^{\frac {1}{4}}+2}{\left (x^{9}-4 x^{6}+5 x^{3}-4\right ) x^{3}}\right )}{6}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{12}-2 \left (x^{6}+1\right )^{\frac {1}{4}} x^{9}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{9}-2 \sqrt {x^{6}+1}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}+6 \left (x^{6}+1\right )^{\frac {1}{4}} x^{6}+7 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x^{3}+4 \sqrt {x^{6}+1}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-6 \left (x^{6}+1\right )^{\frac {1}{4}} x^{3}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{6}+1\right )^{\frac {3}{4}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+1}+2 \left (x^{6}+1\right )^{\frac {1}{4}}+2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (x^{9}-4 x^{6}+5 x^{3}-4\right ) x^{3}}\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 {x^{6} + x^{3} + 2}{{\left (x^{9} - 4 \, x^{6} + 5 \, x^{3} - 4\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x}\,{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 = -0.02 \begin {gather*} \int \frac {x^6+x^3+2}{x\,{\left (x^6+1\right )}^{1/4}\,\left (x^9-4\,x^6+5\,x^3-4\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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