Optimal. Leaf size=52 \[ \frac {7}{12} \tan ^{-1}\left (\sqrt {x^6-1}+x^3\right )+\frac {\sqrt {x^6-1} \left (32 x^9+21 x^6+16 x^3+6\right )}{72 x^{12}} \]
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Rubi [F] time = 0.62, 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 (1+x^3\right ) \sqrt {-1+x^6}}{x^{13} \left (-1+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (1+x^3\right ) \sqrt {-1+x^6}}{x^{13} \left (-1+x^3\right )} \, dx &=\int \left (\frac {2 \sqrt {-1+x^6}}{3 (-1+x)}-\frac {\sqrt {-1+x^6}}{x^{13}}-\frac {2 \sqrt {-1+x^6}}{x^{10}}-\frac {2 \sqrt {-1+x^6}}{x^7}-\frac {2 \sqrt {-1+x^6}}{x^4}-\frac {2 \sqrt {-1+x^6}}{x}+\frac {2 (1+2 x) \sqrt {-1+x^6}}{3 \left (1+x+x^2\right )}\right ) \, dx\\ &=\frac {2}{3} \int \frac {\sqrt {-1+x^6}}{-1+x} \, dx+\frac {2}{3} \int \frac {(1+2 x) \sqrt {-1+x^6}}{1+x+x^2} \, dx-2 \int \frac {\sqrt {-1+x^6}}{x^{10}} \, dx-2 \int \frac {\sqrt {-1+x^6}}{x^7} \, dx-2 \int \frac {\sqrt {-1+x^6}}{x^4} \, dx-2 \int \frac {\sqrt {-1+x^6}}{x} \, dx-\int \frac {\sqrt {-1+x^6}}{x^{13}} \, dx\\ &=-\frac {2 \left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x^3} \, dx,x,x^6\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x^2} \, dx,x,x^6\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,x^6\right )+\frac {2}{3} \int \frac {\sqrt {-1+x^6}}{-1+x} \, dx+\frac {2}{3} \int \left (\frac {2 \sqrt {-1+x^6}}{1-i \sqrt {3}+2 x}+\frac {2 \sqrt {-1+x^6}}{1+i \sqrt {3}+2 x}\right ) \, dx-\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^2}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {2}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{12 x^{12}}+\frac {\sqrt {-1+x^6}}{3 x^6}+\frac {2 \sqrt {-1+x^6}}{3 x^3}-\frac {2 \left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{24} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^2} \, dx,x,x^6\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )+\frac {2}{3} \int \frac {\sqrt {-1+x^6}}{-1+x} \, dx-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{1-i \sqrt {3}+2 x} \, dx+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{1+i \sqrt {3}+2 x} \, dx\\ &=-\frac {2}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{12 x^{12}}+\frac {7 \sqrt {-1+x^6}}{24 x^6}+\frac {2 \sqrt {-1+x^6}}{3 x^3}-\frac {2 \left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{48} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )+\frac {2}{3} \int \frac {\sqrt {-1+x^6}}{-1+x} \, dx-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{1-i \sqrt {3}+2 x} \, dx+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{1+i \sqrt {3}+2 x} \, dx\\ &=-\frac {2}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{12 x^{12}}+\frac {7 \sqrt {-1+x^6}}{24 x^6}+\frac {2 \sqrt {-1+x^6}}{3 x^3}-\frac {2 \left (-1+x^6\right )^{3/2}}{9 x^9}+\frac {1}{3} \tan ^{-1}\left (\sqrt {-1+x^6}\right )-\frac {2}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )-\frac {1}{24} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )+\frac {2}{3} \int \frac {\sqrt {-1+x^6}}{-1+x} \, dx+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{1-i \sqrt {3}+2 x} \, dx+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{1+i \sqrt {3}+2 x} \, dx\\ &=-\frac {2}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{12 x^{12}}+\frac {7 \sqrt {-1+x^6}}{24 x^6}+\frac {2 \sqrt {-1+x^6}}{3 x^3}-\frac {2 \left (-1+x^6\right )^{3/2}}{9 x^9}+\frac {7}{24} \tan ^{-1}\left (\sqrt {-1+x^6}\right )-\frac {2}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )+\frac {2}{3} \int \frac {\sqrt {-1+x^6}}{-1+x} \, dx+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{1-i \sqrt {3}+2 x} \, dx+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{1+i \sqrt {3}+2 x} \, dx\\ \end {align*}
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Mathematica [A] time = 0.14, size = 47, normalized size = 0.90 \begin {gather*} \frac {1}{72} \left (21 \tan ^{-1}\left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1} \left (32 x^9+21 x^6+16 x^3+6\right )}{x^{12}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.21, size = 54, normalized size = 1.04 \begin {gather*} \frac {\sqrt {-1+x^6} \left (6+16 x^3+21 x^6+32 x^9\right )}{72 x^{12}}-\frac {7}{12} \tan ^{-1}\left (x^3-\sqrt {-1+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 55, normalized size = 1.06 \begin {gather*} \frac {42 \, x^{12} \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + 32 \, x^{12} + {\left (32 \, x^{9} + 21 \, x^{6} + 16 \, x^{3} + 6\right )} \sqrt {x^{6} - 1}}{72 \, x^{12}} \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 {\sqrt {x^{6} - 1} {\left (x^{3} + 1\right )}}{{\left (x^{3} - 1\right )} x^{13}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 47, normalized size = 0.90
method | result | size |
risch | \(\frac {32 x^{15}+21 x^{12}-16 x^{9}-15 x^{6}-16 x^{3}-6}{72 x^{12} \sqrt {x^{6}-1}}-\frac {7 \arcsin \left (\frac {1}{x^{3}}\right )}{24}\) | \(47\) |
trager | \(\frac {\sqrt {x^{6}-1}\, \left (32 x^{9}+21 x^{6}+16 x^{3}+6\right )}{72 x^{12}}+\frac {7 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{24}\) | \(58\) |
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 {\sqrt {x^{6} - 1} {\left (x^{3} + 1\right )}}{{\left (x^{3} - 1\right )} x^{13}}\,{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 {\left (x^3+1\right )\,\sqrt {x^6-1}}{x^{13}\,\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 {\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}{x^{13} \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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