Optimal. Leaf size=82 \[ -\frac {1}{4} \tan ^{-1}\left (\sqrt {3}-2 x^2\right )+\frac {1}{4} \tan ^{-1}\left (2 x^2+\sqrt {3}\right )+\frac {\log \left (x^4-\sqrt {3} x^2+1\right )}{8 \sqrt {3}}-\frac {\log \left (x^4+\sqrt {3} x^2+1\right )}{8 \sqrt {3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1359, 1127, 1161, 618, 204, 1164, 628} \[ \frac {\log \left (x^4-\sqrt {3} x^2+1\right )}{8 \sqrt {3}}-\frac {\log \left (x^4+\sqrt {3} x^2+1\right )}{8 \sqrt {3}}-\frac {1}{4} \tan ^{-1}\left (\sqrt {3}-2 x^2\right )+\frac {1}{4} \tan ^{-1}\left (2 x^2+\sqrt {3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 618
Rule 628
Rule 1127
Rule 1161
Rule 1164
Rule 1359
Rubi steps
\begin {align*} \int \frac {x^5}{1-x^4+x^8} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{1-x^2+x^4} \, dx,x,x^2\right )\\ &=-\left (\frac {1}{4} \operatorname {Subst}\left (\int \frac {1-x^2}{1-x^2+x^4} \, dx,x,x^2\right )\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1+x^2}{1-x^2+x^4} \, dx,x,x^2\right )\\ &=\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {3} x+x^2} \, dx,x,x^2\right )+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {3} x+x^2} \, dx,x,x^2\right )+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {3}+2 x}{-1-\sqrt {3} x-x^2} \, dx,x,x^2\right )}{8 \sqrt {3}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {3}-2 x}{-1+\sqrt {3} x-x^2} \, dx,x,x^2\right )}{8 \sqrt {3}}\\ &=\frac {\log \left (1-\sqrt {3} x^2+x^4\right )}{8 \sqrt {3}}-\frac {\log \left (1+\sqrt {3} x^2+x^4\right )}{8 \sqrt {3}}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+2 x^2\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+2 x^2\right )\\ &=-\frac {1}{4} \tan ^{-1}\left (\sqrt {3}-2 x^2\right )+\frac {1}{4} \tan ^{-1}\left (\sqrt {3}+2 x^2\right )+\frac {\log \left (1-\sqrt {3} x^2+x^4\right )}{8 \sqrt {3}}-\frac {\log \left (1+\sqrt {3} x^2+x^4\right )}{8 \sqrt {3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.13, size = 98, normalized size = 1.20 \[ \frac {\sqrt {-1-i \sqrt {3}} \left (\sqrt {3}+i\right ) \tan ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) x^2\right )+\sqrt {-1+i \sqrt {3}} \left (\sqrt {3}-i\right ) \tan ^{-1}\left (\frac {1}{2} \left (1+i \sqrt {3}\right ) x^2\right )}{4 \sqrt {6}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.91, size = 171, normalized size = 2.09 \[ -\frac {1}{12} \, \sqrt {6} \sqrt {3} \sqrt {2} \arctan \left (-\frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2} x^{2} + \frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2 \, x^{4} + \sqrt {6} \sqrt {2} x^{2} + 2} - \sqrt {3}\right ) - \frac {1}{12} \, \sqrt {6} \sqrt {3} \sqrt {2} \arctan \left (-\frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2} x^{2} + \frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2 \, x^{4} - \sqrt {6} \sqrt {2} x^{2} + 2} + \sqrt {3}\right ) - \frac {1}{48} \, \sqrt {6} \sqrt {2} \log \left (2 \, x^{4} + \sqrt {6} \sqrt {2} x^{2} + 2\right ) + \frac {1}{48} \, \sqrt {6} \sqrt {2} \log \left (2 \, x^{4} - \sqrt {6} \sqrt {2} x^{2} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.35, size = 76, normalized size = 0.93 \[ \frac {1}{24} \, \sqrt {3} x^{4} \log \left (x^{4} + \sqrt {3} x^{2} + 1\right ) - \frac {1}{24} \, \sqrt {3} x^{4} \log \left (x^{4} - \sqrt {3} x^{2} + 1\right ) + \frac {1}{4} \, x^{4} \arctan \left (2 \, x^{2} + \sqrt {3}\right ) + \frac {1}{4} \, x^{4} \arctan \left (2 \, x^{2} - \sqrt {3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 65, normalized size = 0.79 \[ \frac {\arctan \left (2 x^{2}-\sqrt {3}\right )}{4}+\frac {\arctan \left (2 x^{2}+\sqrt {3}\right )}{4}+\frac {\sqrt {3}\, \ln \left (x^{4}-\sqrt {3}\, x^{2}+1\right )}{24}-\frac {\sqrt {3}\, \ln \left (x^{4}+\sqrt {3}\, x^{2}+1\right )}{24} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{x^{8} - x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 53, normalized size = 0.65 \[ -\mathrm {atan}\left (\frac {2\,x^2}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\mathrm {atan}\left (\frac {2\,x^2}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.21, size = 70, normalized size = 0.85 \[ \frac {\sqrt {3} \log {\left (x^{4} - \sqrt {3} x^{2} + 1 \right )}}{24} - \frac {\sqrt {3} \log {\left (x^{4} + \sqrt {3} x^{2} + 1 \right )}}{24} + \frac {\operatorname {atan}{\left (2 x^{2} - \sqrt {3} \right )}}{4} + \frac {\operatorname {atan}{\left (2 x^{2} + \sqrt {3} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________