Optimal. Leaf size=25 \[ -\frac {1}{(1-x)^2}+\frac {2}{1-x}+\log (1-x) \]
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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {711}
\begin {gather*} \frac {2}{1-x}-\frac {1}{(1-x)^2}+\log (1-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 711
Rubi steps
\begin {align*} \int \frac {1+x^2}{(-1+x)^3} \, dx &=\int \left (\frac {2}{(-1+x)^3}+\frac {2}{(-1+x)^2}+\frac {1}{-1+x}\right ) \, dx\\ &=-\frac {1}{(1-x)^2}+\frac {2}{1-x}+\log (1-x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 0.64 \begin {gather*} \frac {1-2 x}{(-1+x)^2}+\log (-1+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 20, normalized size = 0.80
method | result | size |
norman | \(\frac {1-2 x}{\left (-1+x \right )^{2}}+\ln \left (-1+x \right )\) | \(17\) |
risch | \(\frac {1-2 x}{\left (-1+x \right )^{2}}+\ln \left (-1+x \right )\) | \(17\) |
default | \(-\frac {1}{\left (-1+x \right )^{2}}+\ln \left (-1+x \right )-\frac {2}{-1+x}\) | \(20\) |
meijerg | \(-\frac {x \left (2-x \right )}{2 \left (1-x \right )^{2}}+\frac {x \left (-9 x +6\right )}{6 \left (1-x \right )^{2}}+\ln \left (1-x \right )\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.00, size = 22, normalized size = 0.88 \begin {gather*} -\frac {2 \, x - 1}{x^{2} - 2 \, x + 1} + \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.93, size = 29, normalized size = 1.16 \begin {gather*} \frac {{\left (x^{2} - 2 \, x + 1\right )} \log \left (x - 1\right ) - 2 \, x + 1}{x^{2} - 2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 17, normalized size = 0.68 \begin {gather*} \frac {1 - 2 x}{x^{2} - 2 x + 1} + \log {\left (x - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.58, size = 18, normalized size = 0.72 \begin {gather*} -\frac {2 \, x - 1}{{\left (x - 1\right )}^{2}} + \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 22, normalized size = 0.88 \begin {gather*} \ln \left (x-1\right )-\frac {2\,x-1}{x^2-2\,x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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