Optimal. Leaf size=164 \[ -\frac {1}{4} \sqrt {2+\sqrt {3}} \log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )+\frac {1}{4} \sqrt {2+\sqrt {3}} \log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )-\frac {1}{2} \sqrt {2+\sqrt {3}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{2} \sqrt {2+\sqrt {3}} \tan ^{-1}\left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1423, 1161, 618, 204, 1164, 628} \begin {gather*} -\frac {1}{4} \sqrt {2+\sqrt {3}} \log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )+\frac {1}{4} \sqrt {2+\sqrt {3}} \log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )-\frac {1}{2} \sqrt {2+\sqrt {3}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{2} \sqrt {2+\sqrt {3}} \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 {1+\left (1+\sqrt {3}\right ) x^4}{1-x^4+x^8} \, dx &=\frac {\int \frac {\sqrt {3}+\sqrt {3} x^2}{1-\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}+\frac {\int \frac {\sqrt {3}-\sqrt {3} x^2}{1+\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}\\ &=\frac {1}{4} \int \frac {1}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx+\frac {1}{4} \int \frac {1}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx-\frac {1}{4} \sqrt {2+\sqrt {3}} \int \frac {\sqrt {2-\sqrt {3}}+2 x}{-1-\sqrt {2-\sqrt {3}} x-x^2} \, dx-\frac {1}{4} \sqrt {2+\sqrt {3}} \int \frac {\sqrt {2-\sqrt {3}}-2 x}{-1+\sqrt {2-\sqrt {3}} x-x^2} \, dx\\ &=-\frac {1}{4} \sqrt {2+\sqrt {3}} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )+\frac {1}{4} \sqrt {2+\sqrt {3}} \log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,-\sqrt {2+\sqrt {3}}+2 x\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,\sqrt {2+\sqrt {3}}+2 x\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2-\sqrt {3}}}-\frac {1}{4} \sqrt {2+\sqrt {3}} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )+\frac {1}{4} \sqrt {2+\sqrt {3}} \log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.04, size = 72, normalized size = 0.44 \begin {gather*} \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\&,\frac {\sqrt {3} \text {$\#$1}^4 \log (x-\text {$\#$1})+\text {$\#$1}^4 \log (x-\text {$\#$1})+\log (x-\text {$\#$1})}{2 \text {$\#$1}^7-\text {$\#$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 {1+\left (1+\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 = 1.23, size = 111, normalized size = 0.68 \begin {gather*} \frac {1}{2} \, \sqrt {\sqrt {3} + 2} \arctan \left (x^{3} \sqrt {\sqrt {3} + 2} - x \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 1\right )}\right ) + \frac {1}{2} \, \sqrt {\sqrt {3} + 2} \arctan \left (x \sqrt {\sqrt {3} + 2}\right ) + \frac {1}{4} \, \sqrt {\sqrt {3} + 2} \log \left (-\frac {x \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} - x^{2} - 1}{x \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} + x^{2} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.43, size = 123, normalized size = 0.75 \begin {gather*} \frac {1}{4} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{4} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{8} \, {\left (\sqrt {6} + \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} + \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 = 0.04, size = 62, normalized size = 0.38 \begin {gather*} \frac {\left (2 \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 (1+\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} + 1\right )} + 1}{x^{8} - x^{4} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.19, 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]
________________________________________________________________________________________