Optimal. Leaf size=25 \[ \frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} \log (1+x)-\frac {1}{4} \log \left (1+x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {720, 31, 649,
209, 266} \begin {gather*} \frac {\text {ArcTan}(x)}{2}-\frac {1}{4} \log \left (x^2+1\right )+\frac {1}{2} \log (x+1) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 209
Rule 266
Rule 649
Rule 720
Rubi steps
\begin {align*} \int \frac {1}{(1+x) \left (1+x^2\right )} \, dx &=\frac {1}{2} \int \frac {1}{1+x} \, dx+\frac {1}{2} \int \frac {1-x}{1+x^2} \, dx\\ &=\frac {1}{2} \log (1+x)+\frac {1}{2} \int \frac {1}{1+x^2} \, dx-\frac {1}{2} \int \frac {x}{1+x^2} \, dx\\ &=\frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} \log (1+x)-\frac {1}{4} \log \left (1+x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} \frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} \log (1+x)-\frac {1}{4} \log \left (1+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.21, size = 20, normalized size = 0.80
method | result | size |
default | \(\frac {\arctan \left (x \right )}{2}+\frac {\ln \left (1+x \right )}{2}-\frac {\ln \left (x^{2}+1\right )}{4}\) | \(20\) |
risch | \(\frac {\arctan \left (x \right )}{2}+\frac {\ln \left (1+x \right )}{2}-\frac {\ln \left (x^{2}+1\right )}{4}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.48, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, \arctan \left (x\right ) - \frac {1}{4} \, \log \left (x^{2} + 1\right ) + \frac {1}{2} \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, \arctan \left (x\right ) - \frac {1}{4} \, \log \left (x^{2} + 1\right ) + \frac {1}{2} \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.05, size = 19, normalized size = 0.76 \begin {gather*} \frac {\log {\left (x + 1 \right )}}{2} - \frac {\log {\left (x^{2} + 1 \right )}}{4} + \frac {\operatorname {atan}{\left (x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 4.69, size = 20, normalized size = 0.80 \begin {gather*} \frac {1}{2} \, \arctan \left (x\right ) - \frac {1}{4} \, \log \left (x^{2} + 1\right ) + \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.23, size = 25, normalized size = 1.00 \begin {gather*} \frac {\ln \left (x+1\right )}{2}+\ln \left (x-\mathrm {i}\right )\,\left (-\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (-\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________