Optimal. Leaf size=49 \[ \frac {\left (25 x^2+24\right ) x}{2 \left (x^4+3 x^2+2\right )}+5 x-\frac {15}{2} \tan ^{-1}(x)-\frac {7 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1668, 1676, 1166, 203} \[ \frac {\left (25 x^2+24\right ) x}{2 \left (x^4+3 x^2+2\right )}+5 x-\frac {15}{2} \tan ^{-1}(x)-\frac {7 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 1166
Rule 1668
Rule 1676
Rubi steps
\begin {align*} \int \frac {x^2 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx &=\frac {x \left (24+25 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{4} \int \frac {48-2 x^2-20 x^4}{2+3 x^2+x^4} \, dx\\ &=\frac {x \left (24+25 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{4} \int \left (-20+\frac {2 \left (44+29 x^2\right )}{2+3 x^2+x^4}\right ) \, dx\\ &=5 x+\frac {x \left (24+25 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \int \frac {44+29 x^2}{2+3 x^2+x^4} \, dx\\ &=5 x+\frac {x \left (24+25 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-7 \int \frac {1}{2+x^2} \, dx-\frac {15}{2} \int \frac {1}{1+x^2} \, dx\\ &=5 x+\frac {x \left (24+25 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac {15}{2} \tan ^{-1}(x)-\frac {7 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 50, normalized size = 1.02 \[ \frac {25 x^3+24 x}{2 \left (x^4+3 x^2+2\right )}+5 x-\frac {15}{2} \tan ^{-1}(x)-\frac {7 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.91, size = 64, normalized size = 1.31 \[ \frac {10 \, x^{5} + 55 \, x^{3} - 7 \, \sqrt {2} {\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 15 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \relax (x) + 44 \, x}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.34, size = 43, normalized size = 0.88 \[ -\frac {7}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 5 \, x + \frac {25 \, x^{3} + 24 \, x}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac {15}{2} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 41, normalized size = 0.84 \[ 5 x -\frac {x}{2 \left (x^{2}+1\right )}+\frac {13 x}{x^{2}+2}-\frac {15 \arctan \relax (x )}{2}-\frac {7 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.63, size = 43, normalized size = 0.88 \[ -\frac {7}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 5 \, x + \frac {25 \, x^{3} + 24 \, x}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac {15}{2} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.07, size = 42, normalized size = 0.86 \[ 5\,x-\frac {15\,\mathrm {atan}\relax (x)}{2}-\frac {7\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{2}+\frac {\frac {25\,x^3}{2}+12\,x}{x^4+3\,x^2+2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.21, size = 48, normalized size = 0.98 \[ 5 x + \frac {25 x^{3} + 24 x}{2 x^{4} + 6 x^{2} + 4} - \frac {15 \operatorname {atan}{\relax (x )}}{2} - \frac {7 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________