Optimal. Leaf size=43 \[ \frac {7 x}{16 \left (x^2+1\right )}-\frac {x}{24 \left (x^2+1\right )^2}+\frac {x}{6 \left (x^2+1\right )^3}+\frac {7}{16} \tan ^{-1}(x) \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1157, 385, 199, 203} \[ \frac {7 x}{16 \left (x^2+1\right )}-\frac {x}{24 \left (x^2+1\right )^2}+\frac {x}{6 \left (x^2+1\right )^3}+\frac {7}{16} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 199
Rule 203
Rule 385
Rule 1157
Rubi steps
\begin {align*} \int \frac {1+x^2+x^4}{\left (1+x^2\right )^4} \, dx &=\frac {x}{6 \left (1+x^2\right )^3}-\frac {1}{6} \int \frac {-5-6 x^2}{\left (1+x^2\right )^3} \, dx\\ &=\frac {x}{6 \left (1+x^2\right )^3}-\frac {x}{24 \left (1+x^2\right )^2}+\frac {7}{8} \int \frac {1}{\left (1+x^2\right )^2} \, dx\\ &=\frac {x}{6 \left (1+x^2\right )^3}-\frac {x}{24 \left (1+x^2\right )^2}+\frac {7 x}{16 \left (1+x^2\right )}+\frac {7}{16} \int \frac {1}{1+x^2} \, dx\\ &=\frac {x}{6 \left (1+x^2\right )^3}-\frac {x}{24 \left (1+x^2\right )^2}+\frac {7 x}{16 \left (1+x^2\right )}+\frac {7}{16} \tan ^{-1}(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 30, normalized size = 0.70 \[ \frac {1}{48} \left (\frac {x \left (21 x^4+40 x^2+27\right )}{\left (x^2+1\right )^3}+21 \tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.02, size = 31, normalized size = 0.72 \[ \frac {x \left (21 x^4+40 x^2+27\right )}{48 \left (x^2+1\right )^3}+\frac {7}{16} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.75, size = 52, normalized size = 1.21 \[ \frac {21 \, x^{5} + 40 \, x^{3} + 21 \, {\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )} \arctan \relax (x) + 27 \, x}{48 \, {\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.91, size = 28, normalized size = 0.65 \[ \frac {21 \, x^{5} + 40 \, x^{3} + 27 \, x}{48 \, {\left (x^{2} + 1\right )}^{3}} + \frac {7}{16} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.28, size = 28, normalized size = 0.65
method | result | size |
default | \(\frac {\frac {7}{16} x^{5}+\frac {5}{6} x^{3}+\frac {9}{16} x}{\left (x^{2}+1\right )^{3}}+\frac {7 \arctan \relax (x )}{16}\) | \(28\) |
risch | \(\frac {\frac {7}{16} x^{5}+\frac {5}{6} x^{3}+\frac {9}{16} x}{\left (x^{2}+1\right )^{3}}+\frac {7 \arctan \relax (x )}{16}\) | \(28\) |
meijerg | \(\frac {x \left (15 x^{4}+40 x^{2}+33\right )}{48 \left (x^{2}+1\right )^{3}}+\frac {7 \arctan \relax (x )}{16}-\frac {x \left (-15 x^{4}+40 x^{2}+15\right )}{240 \left (x^{2}+1\right )^{3}}-\frac {x \left (-3 x^{4}-8 x^{2}+3\right )}{48 \left (x^{2}+1\right )^{3}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.97, size = 38, normalized size = 0.88 \[ \frac {21 \, x^{5} + 40 \, x^{3} + 27 \, x}{48 \, {\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )}} + \frac {7}{16} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.18, size = 27, normalized size = 0.63 \[ \frac {7\,\mathrm {atan}\relax (x)}{16}+\frac {\frac {7\,x^5}{16}+\frac {5\,x^3}{6}+\frac {9\,x}{16}}{{\left (x^2+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.15, size = 36, normalized size = 0.84 \[ \frac {21 x^{5} + 40 x^{3} + 27 x}{48 x^{6} + 144 x^{4} + 144 x^{2} + 48} + \frac {7 \operatorname {atan}{\relax (x )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________