Optimal. Leaf size=85 \[ \frac {\sqrt {x^6-1} \left (x^6+2\right )}{6 x^3}-\frac {5}{12} \log \left (\sqrt {x^6-1}+x^3\right )+\frac {\tan ^{-1}\left (-\frac {4 x^6}{\sqrt {3}}-\frac {4 \sqrt {x^6-1} x^3}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt {3}} \]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 79, normalized size of antiderivative = 0.93, number of steps used = 14, number of rules used = 9, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {575, 586, 580, 523, 217, 206, 377, 204, 528} \begin {gather*} \frac {1}{6} \sqrt {x^6-1} x^3+\frac {\sqrt {x^6-1}}{3 x^3}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^3}{\sqrt {x^6-1}}\right )}{4 \sqrt {3}}-\frac {5}{12} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 206
Rule 217
Rule 377
Rule 523
Rule 528
Rule 575
Rule 580
Rule 586
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x^6} \left (-1+2 x^6\right )^2}{x^4 \left (-1+4 x^6\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^2} \left (-1+2 x^2\right )^2}{x^2 \left (-1+4 x^2\right )} \, dx,x,x^3\right )\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^2} \left (-1+2 x^2\right )}{x^2 \left (-1+4 x^2\right )} \, dx,x,x^3\right )\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^2} \left (-1+2 x^2\right )}{-1+4 x^2} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1+x^6}}{3 x^3}+\frac {1}{6} x^3 \sqrt {-1+x^6}+\frac {1}{12} \operatorname {Subst}\left (\int \frac {6-12 x^2}{\sqrt {-1+x^2} \left (-1+4 x^2\right )} \, dx,x,x^3\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {-1-2 x^2}{\sqrt {-1+x^2} \left (-1+4 x^2\right )} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1+x^6}}{3 x^3}+\frac {1}{6} x^3 \sqrt {-1+x^6}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (-1+4 x^2\right )} \, dx,x,x^3\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (-1+4 x^2\right )} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1+x^6}}{3 x^3}+\frac {1}{6} x^3 \sqrt {-1+x^6}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-1-3 x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-1-3 x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )\\ &=\frac {\sqrt {-1+x^6}}{3 x^3}+\frac {1}{6} x^3 \sqrt {-1+x^6}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^3}{\sqrt {-1+x^6}}\right )}{4 \sqrt {3}}-\frac {5}{12} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.18, size = 201, normalized size = 2.36 \begin {gather*} \frac {4 \sqrt {1-x^6} x^6-8 \sqrt {1-x^6}+4 \sqrt {1-x^6} x^{12}+12 \left (x^6-1\right ) x^3 \sin ^{-1}\left (x^3\right )+\sqrt {3} \sqrt {-\left (x^6-1\right )^2} x^3 \tan ^{-1}\left (\frac {2-x^3}{\sqrt {3} \sqrt {x^6-1}}\right )-\sqrt {3} \sqrt {-\left (x^6-1\right )^2} x^3 \tan ^{-1}\left (\frac {x^3+2}{\sqrt {3} \sqrt {x^6-1}}\right )+2 \sqrt {-\left (x^6-1\right )^2} x^3 \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right )}{24 x^3 \sqrt {-\left (x^6-1\right )^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.19, size = 87, normalized size = 1.02 \begin {gather*} \frac {\sqrt {-1+x^6} \left (2+x^6\right )}{6 x^3}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {4 x^6}{\sqrt {3}}+\frac {4 x^3 \sqrt {-1+x^6}}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {5}{12} \log \left (-x^3+\sqrt {-1+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 80, normalized size = 0.94 \begin {gather*} -\frac {\sqrt {3} x^{3} \arctan \left (\frac {4}{3} \, \sqrt {3} \sqrt {x^{6} - 1} x^{3} - \frac {1}{3} \, \sqrt {3} {\left (4 \, x^{6} - 1\right )}\right ) - 5 \, x^{3} \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) - 4 \, x^{3} - 2 \, {\left (x^{6} + 2\right )} \sqrt {x^{6} - 1}}{12 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 1.16, size = 96, normalized size = 1.13
method | result | size |
trager | \(\frac {\sqrt {x^{6}-1}\, \left (x^{6}+2\right )}{6 x^{3}}+\frac {5 \ln \left (-x^{3}+\sqrt {x^{6}-1}\right )}{12}+\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {-2 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}+6 x^{3} \sqrt {x^{6}-1}-\RootOf \left (\textit {\_Z}^{2}+3\right )}{\left (2 x^{3}-1\right ) \left (2 x^{3}+1\right )}\right )}{24}\) | \(96\) |
risch | \(\frac {x^{12}+x^{6}-2}{6 x^{3} \sqrt {x^{6}-1}}+\frac {5 \ln \left (x^{3}-\sqrt {x^{6}-1}\right )}{12}-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}+6 x^{3} \sqrt {x^{6}-1}+\RootOf \left (\textit {\_Z}^{2}+3\right )}{\left (2 x^{3}-1\right ) \left (2 x^{3}+1\right )}\right )}{24}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{6} - 1\right )}^{2} \sqrt {x^{6} - 1}}{{\left (4 \, x^{6} - 1\right )} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {x^6-1}\,{\left (2\,x^6-1\right )}^2}{x^4\,\left (4\,x^6-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 (2 x^{6} - 1\right )^{2}}{x^{4} \left (2 x^{3} - 1\right ) \left (2 x^{3} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________