Optimal. Leaf size=17 \[ -\frac {1}{(1-x)^2}+\frac {1}{-1+x}+\tan ^{-1}(x) \]
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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 = {815, 209}
\begin {gather*} \text {ArcTan}(x)+\frac {1}{x-1}-\frac {1}{(1-x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 815
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*}
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Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {-2+x+(-1+x)^2 \tan ^{-1}(x)}{(-1+x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 16, normalized size = 0.94
method | result | size |
risch | \(\frac {x -2}{\left (-1+x \right )^{2}}+\arctan \left (x \right )\) | \(13\) |
default | \(-\frac {1}{\left (-1+x \right )^{2}}+\frac {1}{-1+x}+\arctan \left (x \right )\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 17, normalized size = 1.00 \begin {gather*} \frac {x - 2}{x^{2} - 2 \, x + 1} + \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 25, normalized size = 1.47 \begin {gather*} \frac {{\left (x^{2} - 2 \, x + 1\right )} \arctan \left (x\right ) + x - 2}{x^{2} - 2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 14, normalized size = 0.82 \begin {gather*} \frac {x - 2}{x^{2} - 2 x + 1} + \operatorname {atan}{\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.31, size = 12, normalized size = 0.71 \begin {gather*} \frac {x - 2}{{\left (x - 1\right )}^{2}} + \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 17, normalized size = 1.00 \begin {gather*} \mathrm {atan}\left (x\right )+\frac {x-2}{x^2-2\,x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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