Optimal. Leaf size=34 \[ \frac {\sqrt {1-x^6}}{x^2}-\tanh ^{-1}\left (\frac {x^2}{\sqrt {1-x^6}}\right ) \]
________________________________________________________________________________________
Rubi [F] time = 1.08, 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 {\sqrt {1-x^6} \left (2+x^6\right )}{x^3 \left (-1+x^4+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^6} \left (2+x^6\right )}{x^3 \left (-1+x^4+x^6\right )} \, dx &=\int \left (-\frac {2 \sqrt {1-x^6}}{x^3}+\frac {x \left (2+3 x^2\right ) \sqrt {1-x^6}}{-1+x^4+x^6}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {1-x^6}}{x^3} \, dx\right )+\int \frac {x \left (2+3 x^2\right ) \sqrt {1-x^6}}{-1+x^4+x^6} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(2+3 x) \sqrt {1-x^3}}{-1+x^2+x^3} \, dx,x,x^2\right )-\operatorname {Subst}\left (\int \frac {\sqrt {1-x^3}}{x^2} \, dx,x,x^2\right )\\ &=\frac {\sqrt {1-x^6}}{x^2}+\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {2 \sqrt {1-x^3}}{-1+x^2+x^3}+\frac {3 x \sqrt {1-x^3}}{-1+x^2+x^3}\right ) \, dx,x,x^2\right )+\frac {3}{2} \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^3}} \, dx,x,x^2\right )\\ &=\frac {\sqrt {1-x^6}}{x^2}-\frac {3}{2} \operatorname {Subst}\left (\int \frac {1-\sqrt {3}-x}{\sqrt {1-x^3}} \, dx,x,x^2\right )+\frac {3}{2} \operatorname {Subst}\left (\int \frac {x \sqrt {1-x^3}}{-1+x^2+x^3} \, dx,x,x^2\right )-\left (3 \sqrt {\frac {1}{2} \left (2-\sqrt {3}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^3}} \, dx,x,x^2\right )+\operatorname {Subst}\left (\int \frac {\sqrt {1-x^3}}{-1+x^2+x^3} \, dx,x,x^2\right )\\ &=\frac {\sqrt {1-x^6}}{x^2}+\frac {3 \sqrt {1-x^6}}{1+\sqrt {3}-x^2}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+\sqrt {3}-x^2\right )^2}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}-x^2}{1+\sqrt {3}-x^2}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt {\frac {1-x^2}{\left (1+\sqrt {3}-x^2\right )^2}} \sqrt {1-x^6}}+\frac {\sqrt {2} 3^{3/4} \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+\sqrt {3}-x^2\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}-x^2}{1+\sqrt {3}-x^2}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {1-x^2}{\left (1+\sqrt {3}-x^2\right )^2}} \sqrt {1-x^6}}+\frac {3}{2} \operatorname {Subst}\left (\int \frac {x \sqrt {1-x^3}}{-1+x^2+x^3} \, dx,x,x^2\right )+\operatorname {Subst}\left (\int \frac {\sqrt {1-x^3}}{-1+x^2+x^3} \, dx,x,x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 5.54, size = 892, normalized size = 26.24
result too large to display
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 6.99, size = 34, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1-x^6}}{x^2}-\tanh ^{-1}\left (\frac {x^2}{\sqrt {1-x^6}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 58, normalized size = 1.71 \begin {gather*} \frac {x^{2} \log \left (-\frac {x^{6} - x^{4} + 2 \, \sqrt {-x^{6} + 1} x^{2} - 1}{x^{6} + x^{4} - 1}\right ) + 2 \, \sqrt {-x^{6} + 1}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 2\right )} \sqrt {-x^{6} + 1}}{{\left (x^{6} + x^{4} - 1\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.57, size = 53, normalized size = 1.56
method | result | size |
trager | \(\frac {\sqrt {-x^{6}+1}}{x^{2}}-\frac {\ln \left (\frac {-x^{6}+x^{4}+2 \sqrt {-x^{6}+1}\, x^{2}+1}{x^{6}+x^{4}-1}\right )}{2}\) | \(53\) |
risch | \(-\frac {x^{6}-1}{x^{2} \sqrt {-x^{6}+1}}-\frac {\ln \left (-\frac {-x^{6}+x^{4}+2 \sqrt {-x^{6}+1}\, x^{2}+1}{x^{6}+x^{4}-1}\right )}{2}\) | \(60\) |
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 (x^{6} + 2\right )} \sqrt {-x^{6} + 1}}{{\left (x^{6} + x^{4} - 1\right )} x^{3}}\,{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 = -0.03 \begin {gather*} \int \frac {\sqrt {1-x^6}\,\left (x^6+2\right )}{x^3\,\left (x^6+x^4-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________