Optimal. Leaf size=72 \[ -\frac {x \left (50+51 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (254+125 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac {369}{8} \tan ^{-1}(x)+\frac {267 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{4 \sqrt {2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1682, 1692,
1180, 209} \begin {gather*} -\frac {369 \text {ArcTan}(x)}{8}+\frac {267 \text {ArcTan}\left (\frac {x}{\sqrt {2}}\right )}{4 \sqrt {2}}-\frac {x \left (51 x^2+50\right )}{4 \left (x^4+3 x^2+2\right )^2}+\frac {x \left (125 x^2+254\right )}{8 \left (x^4+3 x^2+2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 209
Rule 1180
Rule 1682
Rule 1692
Rubi steps
\begin {align*} \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx &=-\frac {x \left (50+51 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac {1}{8} \int \frac {-100+294 x^2+96 x^4-40 x^6}{\left (2+3 x^2+x^4\right )^2} \, dx\\ &=-\frac {x \left (50+51 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (254+125 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac {1}{32} \int \frac {-816+660 x^2}{2+3 x^2+x^4} \, dx\\ &=-\frac {x \left (50+51 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (254+125 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac {369}{8} \int \frac {1}{1+x^2} \, dx+\frac {267}{4} \int \frac {1}{2+x^2} \, dx\\ &=-\frac {x \left (50+51 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (254+125 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac {369}{8} \tan ^{-1}(x)+\frac {267 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{4 \sqrt {2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 55, normalized size = 0.76 \begin {gather*} \frac {1}{8} \left (\frac {x \left (408+910 x^2+629 x^4+125 x^6\right )}{\left (2+3 x^2+x^4\right )^2}-369 \tan ^{-1}(x)+267 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.03, size = 54, normalized size = 0.75
method | result | size |
risch | \(\frac {\frac {125}{8} x^{7}+\frac {629}{8} x^{5}+\frac {455}{4} x^{3}+51 x}{\left (x^{4}+3 x^{2}+2\right )^{2}}-\frac {369 \arctan \left (x \right )}{8}+\frac {267 \arctan \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}}{8}\) | \(50\) |
default | \(-\frac {-\frac {23}{8} x^{3}-\frac {25}{8} x}{\left (x^{2}+1\right )^{2}}-\frac {369 \arctan \left (x \right )}{8}+\frac {\frac {51}{4} x^{3}+\frac {77}{2} x}{\left (x^{2}+2\right )^{2}}+\frac {267 \arctan \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}}{8}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 60, normalized size = 0.83 \begin {gather*} \frac {267}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + \frac {125 \, x^{7} + 629 \, x^{5} + 910 \, x^{3} + 408 \, x}{8 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} - \frac {369}{8} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 99, normalized size = 1.38 \begin {gather*} \frac {125 \, x^{7} + 629 \, x^{5} + 910 \, x^{3} + 267 \, \sqrt {2} {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 369 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (x\right ) + 408 \, x}{8 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.10, size = 65, normalized size = 0.90 \begin {gather*} \frac {125 x^{7} + 629 x^{5} + 910 x^{3} + 408 x}{8 x^{8} + 48 x^{6} + 104 x^{4} + 96 x^{2} + 32} - \frac {369 \operatorname {atan}{\left (x \right )}}{8} + \frac {267 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.10, size = 50, normalized size = 0.69 \begin {gather*} \frac {267}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + \frac {125 \, x^{7} + 629 \, x^{5} + 910 \, x^{3} + 408 \, x}{8 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} - \frac {369}{8} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.93, size = 59, normalized size = 0.82 \begin {gather*} \frac {267\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{8}-\frac {369\,\mathrm {atan}\left (x\right )}{8}+\frac {\frac {125\,x^7}{8}+\frac {629\,x^5}{8}+\frac {455\,x^3}{4}+51\,x}{x^8+6\,x^6+13\,x^4+12\,x^2+4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________