Optimal. Leaf size=37 \[ \frac {1}{6} \log \left (x^2+1\right )-\frac {1}{6} \log \left (x^2+4\right )-\frac {1}{3} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {2}{3} \tan ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1010, 391, 203, 444, 36, 31} \[ \frac {1}{6} \log \left (x^2+1\right )-\frac {1}{6} \log \left (x^2+4\right )-\frac {1}{3} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {2}{3} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 36
Rule 203
Rule 391
Rule 444
Rule 1010
Rubi steps
\begin {align*} \int \frac {2+x}{\left (1+x^2\right ) \left (4+x^2\right )} \, dx &=2 \int \frac {1}{\left (1+x^2\right ) \left (4+x^2\right )} \, dx+\int \frac {x}{\left (1+x^2\right ) \left (4+x^2\right )} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(1+x) (4+x)} \, dx,x,x^2\right )+\frac {2}{3} \int \frac {1}{1+x^2} \, dx-\frac {2}{3} \int \frac {1}{4+x^2} \, dx\\ &=-\frac {1}{3} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {2}{3} \tan ^{-1}(x)+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^2\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{4+x} \, dx,x,x^2\right )\\ &=-\frac {1}{3} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {2}{3} \tan ^{-1}(x)+\frac {1}{6} \log \left (1+x^2\right )-\frac {1}{6} \log \left (4+x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 37, normalized size = 1.00 \[ \frac {1}{6} \log \left (x^2+1\right )-\frac {1}{6} \log \left (x^2+4\right )-\frac {1}{3} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {2}{3} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.75, size = 27, normalized size = 0.73 \[ -\frac {1}{3} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {2}{3} \, \arctan \relax (x) - \frac {1}{6} \, \log \left (x^{2} + 4\right ) + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.29, size = 27, normalized size = 0.73 \[ -\frac {1}{3} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {2}{3} \, \arctan \relax (x) - \frac {1}{6} \, \log \left (x^{2} + 4\right ) + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 28, normalized size = 0.76 \[ \frac {2 \arctan \relax (x )}{3}-\frac {\arctan \left (\frac {x}{2}\right )}{3}+\frac {\ln \left (x^{2}+1\right )}{6}-\frac {\ln \left (x^{2}+4\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.00, size = 27, normalized size = 0.73 \[ -\frac {1}{3} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {2}{3} \, \arctan \relax (x) - \frac {1}{6} \, \log \left (x^{2} + 4\right ) + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.14, size = 37, normalized size = 1.00 \[ \ln \left (x-\mathrm {i}\right )\,\left (\frac {1}{6}-\frac {1}{3}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (\frac {1}{6}+\frac {1}{3}{}\mathrm {i}\right )+\ln \left (x-2{}\mathrm {i}\right )\,\left (-\frac {1}{6}+\frac {1}{6}{}\mathrm {i}\right )+\ln \left (x+2{}\mathrm {i}\right )\,\left (-\frac {1}{6}-\frac {1}{6}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.17, size = 29, normalized size = 0.78 \[ \frac {\log {\left (x^{2} + 1 \right )}}{6} - \frac {\log {\left (x^{2} + 4 \right )}}{6} - \frac {\operatorname {atan}{\left (\frac {x}{2} \right )}}{3} + \frac {2 \operatorname {atan}{\relax (x )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________