Optimal. Leaf size=180 \[ \frac {1}{4} \sqrt {3 \left (2-\sqrt {3}\right )} \log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )-\frac {1}{4} \sqrt {3 \left (2-\sqrt {3}\right )} \log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )+\frac {1}{2} \sqrt {3 \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{2} \sqrt {3 \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1423, 1161, 618, 204, 1164, 628} \begin {gather*} \frac {1}{4} \sqrt {3 \left (2-\sqrt {3}\right )} \log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )-\frac {1}{4} \sqrt {3 \left (2-\sqrt {3}\right )} \log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )+\frac {1}{2} \sqrt {3 \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{2} \sqrt {3 \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 618
Rule 628
Rule 1161
Rule 1164
Rule 1423
Rubi steps
\begin {align*} \int \frac {3-2 \sqrt {3}+\left (-3+\sqrt {3}\right ) x^4}{1-x^4+x^8} \, dx &=\frac {\int \frac {\sqrt {3} \left (3-2 \sqrt {3}\right )+\left (-6+3 \sqrt {3}\right ) x^2}{1-\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}+\frac {\int \frac {\sqrt {3} \left (3-2 \sqrt {3}\right )+\left (6-3 \sqrt {3}\right ) x^2}{1+\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}\\ &=\frac {1}{4} \sqrt {3 \left (2-\sqrt {3}\right )} \int \frac {\sqrt {2-\sqrt {3}}+2 x}{-1-\sqrt {2-\sqrt {3}} x-x^2} \, dx+\frac {1}{4} \sqrt {3 \left (2-\sqrt {3}\right )} \int \frac {\sqrt {2-\sqrt {3}}-2 x}{-1+\sqrt {2-\sqrt {3}} x-x^2} \, dx-\frac {1}{4} \left (-3+2 \sqrt {3}\right ) \int \frac {1}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx-\frac {1}{4} \left (-3+2 \sqrt {3}\right ) \int \frac {1}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx\\ &=\frac {1}{4} \sqrt {3 \left (2-\sqrt {3}\right )} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{4} \sqrt {3 \left (2-\sqrt {3}\right )} \log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{2} \left (3-2 \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,-\sqrt {2+\sqrt {3}}+2 x\right )-\frac {1}{2} \left (3-2 \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,\sqrt {2+\sqrt {3}}+2 x\right )\\ &=\frac {1}{2} \sqrt {6-3 \sqrt {3}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{2} \sqrt {6-3 \sqrt {3}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{4} \sqrt {3 \left (2-\sqrt {3}\right )} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{4} \sqrt {3 \left (2-\sqrt {3}\right )} \log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.05, size = 89, normalized size = 0.49 \begin {gather*} \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\&,\frac {\sqrt {3} \text {$\#$1}^4 \log (x-\text {$\#$1})-3 \text {$\#$1}^4 \log (x-\text {$\#$1})-2 \sqrt {3} \log (x-\text {$\#$1})+3 \log (x-\text {$\#$1})}{2 \text {$\#$1}^7-\text {$\#$1}^3}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3-2 \sqrt {3}+\left (-3+\sqrt {3}\right ) x^4}{1-x^4+x^8} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.55, size = 141, normalized size = 0.78 \begin {gather*} -\frac {1}{2} \, \sqrt {-3 \, \sqrt {3} + 6} \arctan \left (\frac {1}{3} \, x^{3} {\left (2 \, \sqrt {3} + 3\right )} \sqrt {-3 \, \sqrt {3} + 6} - \frac {1}{3} \, x {\left (\sqrt {3} + 3\right )} \sqrt {-3 \, \sqrt {3} + 6}\right ) - \frac {1}{2} \, \sqrt {-3 \, \sqrt {3} + 6} \arctan \left (\frac {1}{3} \, x {\left (2 \, \sqrt {3} + 3\right )} \sqrt {-3 \, \sqrt {3} + 6}\right ) + \frac {1}{4} \, \sqrt {-3 \, \sqrt {3} + 6} \log \left (\frac {3 \, x^{2} - \sqrt {3} x \sqrt {-3 \, \sqrt {3} + 6} + 3}{3 \, x^{2} + \sqrt {3} x \sqrt {-3 \, \sqrt {3} + 6} + 3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.45, size = 131, normalized size = 0.73 \begin {gather*} \frac {1}{4} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{4} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{8} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {1}{8} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.01, size = 62, normalized size = 0.34 \begin {gather*} \frac {\left (-6 \RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )^{4}+2 \sqrt {3}\, \RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )^{4}+\left (-3+\sqrt {3}\right ) \left (\sqrt {3}-1\right )\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )+x \right )}{16 \RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )^{7}-8 \RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )^{3}} \end {gather*}
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 {x^{4} {\left (\sqrt {3} - 3\right )} - 2 \, \sqrt {3} + 3}{x^{8} - x^{4} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.23, size = 1, normalized size = 0.01 \begin {gather*} 0 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________