Optimal. Leaf size=32 \[ \frac {x}{4 \left (1+x^2\right )}+\frac {1}{4} \tan ^{-1}(x)-\frac {\tan ^{-1}(x)}{2 \left (1+x^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5050, 205, 209}
\begin {gather*} -\frac {\text {ArcTan}(x)}{2 \left (x^2+1\right )}+\frac {\text {ArcTan}(x)}{4}+\frac {x}{4 \left (x^2+1\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 209
Rule 5050
Rubi steps
\begin {align*} \int \frac {x \tan ^{-1}(x)}{\left (1+x^2\right )^2} \, dx &=-\frac {\tan ^{-1}(x)}{2 \left (1+x^2\right )}+\frac {1}{2} \int \frac {1}{\left (1+x^2\right )^2} \, dx\\ &=\frac {x}{4 \left (1+x^2\right )}-\frac {\tan ^{-1}(x)}{2 \left (1+x^2\right )}+\frac {1}{4} \int \frac {1}{1+x^2} \, dx\\ &=\frac {x}{4 \left (1+x^2\right )}+\frac {1}{4} \tan ^{-1}(x)-\frac {\tan ^{-1}(x)}{2 \left (1+x^2\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 21, normalized size = 0.66 \begin {gather*} \frac {x+\left (-1+x^2\right ) \tan ^{-1}(x)}{4 \left (1+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 27, normalized size = 0.84
method | result | size |
default | \(\frac {x}{4 x^{2}+4}+\frac {\arctan \left (x \right )}{4}-\frac {\arctan \left (x \right )}{2 \left (x^{2}+1\right )}\) | \(27\) |
risch | \(\frac {i \ln \left (i x +1\right )}{4 x^{2}+4}-\frac {i \left (2 \ln \left (-i x +1\right )+x^{2} \ln \left (x -i\right )+\ln \left (x -i\right )-\ln \left (x +i\right ) x^{2}-\ln \left (x +i\right )+2 i x \right )}{8 \left (x -i\right ) \left (x +i\right )}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 1.58, size = 26, normalized size = 0.81 \begin {gather*} \frac {x}{4 \, {\left (x^{2} + 1\right )}} - \frac {\arctan \left (x\right )}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{4} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.52, size = 19, normalized size = 0.59 \begin {gather*} \frac {{\left (x^{2} - 1\right )} \arctan \left (x\right ) + x}{4 \, {\left (x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.22, size = 31, normalized size = 0.97 \begin {gather*} \frac {x^{2} \operatorname {atan}{\left (x \right )}}{4 x^{2} + 4} + \frac {x}{4 x^{2} + 4} - \frac {\operatorname {atan}{\left (x \right )}}{4 x^{2} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.65, size = 26, normalized size = 0.81 \begin {gather*} \frac {x}{4 \, {\left (x^{2} + 1\right )}} - \frac {\arctan \left (x\right )}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{4} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.08, size = 21, normalized size = 0.66 \begin {gather*} \frac {\mathrm {atan}\left (x\right )}{4}+\frac {\frac {x}{4}-\frac {\mathrm {atan}\left (x\right )}{2}}{x^2+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________