Optimal. Leaf size=288 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4+x^2}-x}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4+x^2}+x}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4+x^2}-x}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4+x^2}+x}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {2}{3} \tanh ^{-1}\left (\frac {x}{\sqrt [3]{x^4+x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^4+x^2}}\right )}{3 \sqrt [3]{2}}-\frac {1}{3} \tanh ^{-1}\left (\frac {x^2+\left (x^4+x^2\right )^{2/3}}{x \sqrt [3]{x^4+x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{2} x^2+\frac {\left (x^4+x^2\right )^{2/3}}{\sqrt [3]{2}}}{x \sqrt [3]{x^4+x^2}}\right )}{6 \sqrt [3]{2}} \]
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Rubi [F] time = 2.46, 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 {1+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {1+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {1+x^6}{x^{2/3} \sqrt [3]{1+x^2} \left (-1+x^6\right )} \, dx}{\sqrt [3]{x^2+x^4}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {\left (1+x^2\right )^{2/3} \left (1-x^2+x^4\right )}{x^{2/3} \left (-1+x^6\right )} \, dx}{\sqrt [3]{x^2+x^4}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3} \left (1-x^6+x^{12}\right )}{-1+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {\left (1+x^6\right )^{2/3}}{18 (1-x)}-\frac {\left (1+x^6\right )^{2/3}}{18 (1+x)}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1-\sqrt [9]{-1} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1+\sqrt [9]{-1} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1-(-1)^{2/9} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1+(-1)^{2/9} x\right )}-\frac {\left (1+x^6\right )^{2/3}}{18 \left (1-\sqrt [3]{-1} x\right )}-\frac {\left (1+x^6\right )^{2/3}}{18 \left (1+\sqrt [3]{-1} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1-(-1)^{4/9} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1+(-1)^{4/9} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1-(-1)^{5/9} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1+(-1)^{5/9} x\right )}-\frac {\left (1+x^6\right )^{2/3}}{18 \left (1-(-1)^{2/3} x\right )}-\frac {\left (1+x^6\right )^{2/3}}{18 \left (1+(-1)^{2/3} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1-(-1)^{7/9} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1+(-1)^{7/9} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1-(-1)^{8/9} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1+(-1)^{8/9} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1-x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1-\sqrt [3]{-1} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1+\sqrt [3]{-1} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1-(-1)^{2/3} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1+(-1)^{2/3} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1-\sqrt [9]{-1} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1+\sqrt [9]{-1} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1-(-1)^{4/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1+(-1)^{4/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1-(-1)^{7/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1+(-1)^{7/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1-(-1)^{2/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1+(-1)^{2/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1-(-1)^{5/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1+(-1)^{5/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1-(-1)^{8/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1+(-1)^{8/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}\\ \end {align*}
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Mathematica [F] time = 0.72, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.90, size = 288, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4+x^2}-x}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4+x^2}+x}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4+x^2}-x}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4+x^2}+x}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {2}{3} \tanh ^{-1}\left (\frac {x}{\sqrt [3]{x^4+x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^4+x^2}}\right )}{3 \sqrt [3]{2}}-\frac {1}{3} \tanh ^{-1}\left (\frac {x^2+\left (x^4+x^2\right )^{2/3}}{x \sqrt [3]{x^4+x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{2} x^2+\frac {\left (x^4+x^2\right )^{2/3}}{\sqrt [3]{2}}}{x \sqrt [3]{x^4+x^2}}\right )}{6 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 7.01, size = 456, normalized size = 1.58 \begin {gather*} \frac {1}{12} \, \sqrt {6} 2^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {2^{\frac {1}{6}} {\left (4 \, \sqrt {6} 2^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} {\left (x^{2} + 2 \, x + 1\right )} + 8 \, \sqrt {6} \left (-1\right )^{\frac {1}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - 2 \, x^{2} + x\right )} - \sqrt {6} 2^{\frac {1}{3}} {\left (x^{5} - 8 \, x^{4} - 2 \, x^{3} - 8 \, x^{2} + x\right )}\right )}}{6 \, {\left (x^{5} + 8 \, x^{4} - 2 \, x^{3} + 8 \, x^{2} + x\right )}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {4 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x - 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + 2 \, x^{2} + x\right )} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} - 2 \, x^{2} + x}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - 2 \, x^{2} + x\right )} + 2 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x}{x^{3} - 2 \, x^{2} + x}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} {\left (x^{2} + x + 1\right )} - 4 \, \sqrt {3} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - x^{2} + x\right )} - \sqrt {3} {\left (x^{5} - 4 \, x^{4} + x^{3} - 4 \, x^{2} + x\right )}}{3 \, {\left (x^{5} + 4 \, x^{4} + x^{3} + 4 \, x^{2} + x\right )}}\right ) + \frac {1}{3} \, \log \left (\frac {x^{3} - x^{2} + 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x + x - 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} + x^{2} + x}\right ) + \frac {1}{6} \, \log \left (\frac {x^{3} - x^{2} + 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x + x - 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} - x^{2} + x}\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} + 1}{{\left (x^{6} - 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 55.67, size = 7357, normalized size = 25.55 \begin {gather*} \text {output too large to display} \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} + 1}{{\left (x^{6} - 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}}\,{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 = -0.00 \begin {gather*} \int \frac {x^6+1}{{\left (x^4+x^2\right )}^{1/3}\,\left (x^6-1\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 {\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}{\sqrt [3]{x^{2} \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - 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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