Optimal. Leaf size=39 \[ -\frac {(x+1)^2}{4 \left (x^2+1\right )^2}-\frac {1-x}{4 \left (x^2+1\right )}+\frac {1}{4} \tan ^{-1}(x) \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {819, 639, 203} \begin {gather*} -\frac {(x+1)^2}{4 \left (x^2+1\right )^2}-\frac {1-x}{4 \left (x^2+1\right )}+\frac {1}{4} \tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 639
Rule 819
Rubi steps
\begin {align*} \int \frac {x (1+x)^2}{\left (1+x^2\right )^3} \, dx &=-\frac {(1+x)^2}{4 \left (1+x^2\right )^2}+\frac {1}{4} \int \frac {2+2 x}{\left (1+x^2\right )^2} \, dx\\ &=-\frac {(1+x)^2}{4 \left (1+x^2\right )^2}-\frac {1-x}{4 \left (1+x^2\right )}+\frac {1}{4} \int \frac {1}{1+x^2} \, dx\\ &=-\frac {(1+x)^2}{4 \left (1+x^2\right )^2}-\frac {1-x}{4 \left (1+x^2\right )}+\frac {1}{4} \tan ^{-1}(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 28, normalized size = 0.72 \begin {gather*} \frac {1}{4} \left (\frac {x^3-2 x^2-x-2}{\left (x^2+1\right )^2}+\tan ^{-1}(x)\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x (1+x)^2}{\left (1+x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 40, normalized size = 1.03 \begin {gather*} \frac {x^{3} - 2 \, x^{2} + {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \relax (x) - x - 2}{4 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 27, normalized size = 0.69 \begin {gather*} \frac {x^{3} - 2 \, x^{2} - x - 2}{4 \, {\left (x^{2} + 1\right )}^{2}} + \frac {1}{4} \, \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 29, normalized size = 0.74 \begin {gather*} \frac {\arctan \relax (x )}{4}+\frac {\frac {1}{4} x^{3}-\frac {1}{2} x^{2}-\frac {1}{4} x -\frac {1}{2}}{\left (x^{2}+1\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.18, size = 32, normalized size = 0.82 \begin {gather*} \frac {x^{3} - 2 \, x^{2} - x - 2}{4 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} + \frac {1}{4} \, \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 29, normalized size = 0.74 \begin {gather*} \frac {\mathrm {atan}\relax (x)}{4}-\frac {-\frac {x^3}{4}+\frac {x^2}{2}+\frac {x}{4}+\frac {1}{2}}{{\left (x^2+1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.13, size = 27, normalized size = 0.69 \begin {gather*} \frac {\operatorname {atan}{\relax (x )}}{4} + \frac {x^{3} - 2 x^{2} - x - 2}{4 x^{4} + 8 x^{2} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________