Optimal. Leaf size=18 \[ \frac {1}{x+1}+\log (x+1)+\frac {1}{2} \tan ^{-1}(x)^2 \]
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Rubi [A] time = 0.15, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6725, 43, 4884} \[ \frac {1}{x+1}+\log (x+1)+\frac {1}{2} \tan ^{-1}(x)^2 \]
Antiderivative was successfully verified.
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Rule 43
Rule 4884
Rule 6725
Rubi steps
\begin {align*} \int \frac {x+x^3+(1+x)^2 \tan ^{-1}(x)}{(1+x)^2 \left (1+x^2\right )} \, dx &=\int \left (\frac {x}{(1+x)^2}+\frac {\tan ^{-1}(x)}{1+x^2}\right ) \, dx\\ &=\int \frac {x}{(1+x)^2} \, dx+\int \frac {\tan ^{-1}(x)}{1+x^2} \, dx\\ &=\frac {1}{2} \tan ^{-1}(x)^2+\int \left (-\frac {1}{(1+x)^2}+\frac {1}{1+x}\right ) \, dx\\ &=\frac {1}{1+x}+\frac {1}{2} \tan ^{-1}(x)^2+\log (1+x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 18, normalized size = 1.00 \[ \frac {1}{x+1}+\log (x+1)+\frac {1}{2} \tan ^{-1}(x)^2 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 26, normalized size = 1.44 \[ \frac {{\left (x + 1\right )} \arctan \relax (x)^{2} + 2 \, {\left (x + 1\right )} \log \left (x + 1\right ) + 2}{2 \, {\left (x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 104, normalized size = 5.78 \[ \frac {{\left (x + 1\right )} {\left (\frac {1}{x + 1} - 1\right )} \arctan \left ({\left (x + 1\right )} {\left (\frac {1}{x + 1} - 1\right )}\right )^{2} + 2 \, {\left (x + 1\right )} {\left (\frac {1}{x + 1} - 1\right )} \log \left (-{\left (x + 1\right )} {\left (\frac {1}{x + 1} - 1\right )} + 1\right ) - \arctan \left ({\left (x + 1\right )} {\left (\frac {1}{x + 1} - 1\right )}\right )^{2} - 2 \, \log \left (-{\left (x + 1\right )} {\left (\frac {1}{x + 1} - 1\right )} + 1\right ) - 2}{2 \, {\left ({\left (x + 1\right )} {\left (\frac {1}{x + 1} - 1\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 17, normalized size = 0.94 \[ \frac {1}{x +1}+\frac {\arctan \relax (x )^{2}}{2}+\ln \left (x +1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 16, normalized size = 0.89 \[ \frac {1}{2} \, \arctan \relax (x)^{2} + \frac {1}{x + 1} + \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 16, normalized size = 0.89 \[ \ln \left (x+1\right )+\frac {1}{x+1}+\frac {{\mathrm {atan}\relax (x)}^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.62, size = 53, normalized size = 2.94 \[ \frac {2 x \log {\left (x + 1 \right )}}{2 x + 2} + \frac {x \operatorname {atan}^{2}{\relax (x )}}{2 x + 2} + \frac {2 \log {\left (x + 1 \right )}}{2 x + 2} + \frac {\operatorname {atan}^{2}{\relax (x )}}{2 x + 2} + \frac {2}{2 x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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