Optimal. Leaf size=92 \[ -\frac {1}{4} d \log \left (x^2-x+1\right )+\frac {1}{4} d \log \left (x^2+x+1\right )-\frac {d \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {d \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {e \tan ^{-1}\left (\frac {2 x^2+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1673, 12, 1094, 634, 618, 204, 628, 1107} \begin {gather*} -\frac {1}{4} d \log \left (x^2-x+1\right )+\frac {1}{4} d \log \left (x^2+x+1\right )-\frac {d \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {d \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {e \tan ^{-1}\left (\frac {2 x^2+1}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 204
Rule 618
Rule 628
Rule 634
Rule 1094
Rule 1107
Rule 1673
Rubi steps
\begin {align*} \int \frac {d+e x}{1+x^2+x^4} \, dx &=\int \frac {d}{1+x^2+x^4} \, dx+\int \frac {e x}{1+x^2+x^4} \, dx\\ &=d \int \frac {1}{1+x^2+x^4} \, dx+e \int \frac {x}{1+x^2+x^4} \, dx\\ &=\frac {1}{2} d \int \frac {1-x}{1-x+x^2} \, dx+\frac {1}{2} d \int \frac {1+x}{1+x+x^2} \, dx+\frac {1}{2} e \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^2\right )\\ &=\frac {1}{4} d \int \frac {1}{1-x+x^2} \, dx-\frac {1}{4} d \int \frac {-1+2 x}{1-x+x^2} \, dx+\frac {1}{4} d \int \frac {1}{1+x+x^2} \, dx+\frac {1}{4} d \int \frac {1+2 x}{1+x+x^2} \, dx-e \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^2\right )\\ &=\frac {e \tan ^{-1}\left (\frac {1+2 x^2}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{4} d \log \left (1-x+x^2\right )+\frac {1}{4} d \log \left (1+x+x^2\right )-\frac {1}{2} d \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )-\frac {1}{2} d \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=-\frac {d \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {d \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {e \tan ^{-1}\left (\frac {1+2 x^2}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{4} d \log \left (1-x+x^2\right )+\frac {1}{4} d \log \left (1+x+x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.18, size = 98, normalized size = 1.07 \begin {gather*} \frac {1}{6} i \left (\sqrt {6-6 i \sqrt {3}} d \tan ^{-1}\left (\frac {1}{2} \left (\sqrt {3}-i\right ) x\right )-\sqrt {6+6 i \sqrt {3}} d \tan ^{-1}\left (\frac {1}{2} \left (\sqrt {3}+i\right ) x\right )+2 i \sqrt {3} e \tan ^{-1}\left (\frac {\sqrt {3}}{2 x^2+1}\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x}{1+x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.12, size = 65, normalized size = 0.71 \begin {gather*} \frac {1}{6} \, \sqrt {3} {\left (d - 2 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} {\left (d + 2 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \, d \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, d \log \left (x^{2} - x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.38, size = 67, normalized size = 0.73 \begin {gather*} \frac {1}{6} \, \sqrt {3} {\left (d - 2 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} {\left (d + 2 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \, d \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, d \log \left (x^{2} - x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 92, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3}\, d \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\sqrt {3}\, d \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}-\frac {d \ln \left (x^{2}-x +1\right )}{4}+\frac {d \ln \left (x^{2}+x +1\right )}{4}-\frac {\sqrt {3}\, e \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{3}+\frac {\sqrt {3}\, e \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.21, size = 65, normalized size = 0.71 \begin {gather*} \frac {1}{6} \, \sqrt {3} {\left (d - 2 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} {\left (d + 2 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \, d \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, d \log \left (x^{2} - x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.24, size = 118, normalized size = 1.28 \begin {gather*} -\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {d}{4}+\frac {\sqrt {3}\,d\,1{}\mathrm {i}}{12}+\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{6}\right )+\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {d}{4}-\frac {\sqrt {3}\,d\,1{}\mathrm {i}}{12}+\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{6}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {d}{4}+\frac {\sqrt {3}\,d\,1{}\mathrm {i}}{12}+\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{6}\right )+\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {d}{4}+\frac {\sqrt {3}\,d\,1{}\mathrm {i}}{12}-\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{6}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 2.89, size = 923, normalized size = 10.03
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________