Optimal. Leaf size=23 \[ \frac{1}{2} \log \left (x^2+1\right )+\frac{1}{2} \cot ^{-1}(x)^2+x \cot ^{-1}(x) \]
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Rubi [A] time = 0.0481948, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {4917, 4847, 260, 4885} \[ \frac{1}{2} \log \left (x^2+1\right )+\frac{1}{2} \cot ^{-1}(x)^2+x \cot ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 4917
Rule 4847
Rule 260
Rule 4885
Rubi steps
\begin{align*} \int \frac{x^2 \cot ^{-1}(x)}{1+x^2} \, dx &=\int \cot ^{-1}(x) \, dx-\int \frac{\cot ^{-1}(x)}{1+x^2} \, dx\\ &=x \cot ^{-1}(x)+\frac{1}{2} \cot ^{-1}(x)^2+\int \frac{x}{1+x^2} \, dx\\ &=x \cot ^{-1}(x)+\frac{1}{2} \cot ^{-1}(x)^2+\frac{1}{2} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0133661, size = 23, normalized size = 1. \[ \frac{1}{2} \log \left (x^2+1\right )+\frac{1}{2} \cot ^{-1}(x)^2+x \cot ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 26, normalized size = 1.1 \begin{align*} -{\rm arccot} \left (x\right )\arctan \left ( x \right ) +x{\rm arccot} \left (x\right )+{\frac{\ln \left ({x}^{2}+1 \right ) }{2}}-{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54104, size = 32, normalized size = 1.39 \begin{align*}{\left (x - \arctan \left (x\right )\right )} \operatorname{arccot}\left (x\right ) - \frac{1}{2} \, \arctan \left (x\right )^{2} + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86692, size = 68, normalized size = 2.96 \begin{align*} x \operatorname{arccot}\left (x\right ) + \frac{1}{2} \, \operatorname{arccot}\left (x\right )^{2} + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.38859, size = 19, normalized size = 0.83 \begin{align*} x \operatorname{acot}{\left (x \right )} + \frac{\log{\left (x^{2} + 1 \right )}}{2} + \frac{\operatorname{acot}^{2}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1239, size = 58, normalized size = 2.52 \begin{align*} \frac{1}{2} \, i x \log \left (-\frac{i - x}{i + x}\right ) - \frac{1}{8} \, \log \left (-\frac{i - x}{i + x}\right )^{2} + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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