Optimal. Leaf size=17 \[ \frac {1}{x-1}-\frac {1}{(1-x)^2}+\tan ^{-1}(x) \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {801, 203} \begin {gather*} \frac {1}{x-1}-\frac {1}{(1-x)^2}+\tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 801
Rubi steps
\begin {align*} \int \frac {2+2 x}{(-1+x)^3 \left (1+x^2\right )} \, dx &=\int \left (\frac {2}{(-1+x)^3}-\frac {1}{(-1+x)^2}+\frac {1}{1+x^2}\right ) \, dx\\ &=-\frac {1}{(1-x)^2}+\frac {1}{-1+x}+\int \frac {1}{1+x^2} \, dx\\ &=-\frac {1}{(1-x)^2}+\frac {1}{-1+x}+\tan ^{-1}(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {x+(x-1)^2 \tan ^{-1}(x)-2}{(x-1)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2+2 x}{(-1+x)^3 \left (1+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.24, size = 25, normalized size = 1.47 \begin {gather*} \frac {{\left (x^{2} - 2 \, x + 1\right )} \arctan \relax (x) + x - 2}{x^{2} - 2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.24, size = 12, normalized size = 0.71 \begin {gather*} \frac {x - 2}{{\left (x - 1\right )}^{2}} + \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 16, normalized size = 0.94 \begin {gather*} \arctan \relax (x )-\frac {1}{\left (x -1\right )^{2}}+\frac {1}{x -1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.74, size = 17, normalized size = 1.00 \begin {gather*} \frac {x - 2}{x^{2} - 2 \, x + 1} + \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 17, normalized size = 1.00 \begin {gather*} \mathrm {atan}\relax (x)+\frac {x-2}{x^2-2\,x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.12, size = 14, normalized size = 0.82 \begin {gather*} \frac {x - 2}{x^{2} - 2 x + 1} + \operatorname {atan}{\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________