Optimal. Leaf size=40 \[ \frac {1}{3} x^3 \cot ^{-1}(x)+\frac {x^2}{6}-\frac {2}{3} \log \left (x^2+1\right )-x \cot ^{-1}(x)-\frac {1}{2} \cot ^{-1}(x)^2 \]
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Rubi [A] time = 0.10, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {4917, 4853, 266, 43, 4847, 260, 4885} \[ \frac {x^2}{6}-\frac {2}{3} \log \left (x^2+1\right )+\frac {1}{3} x^3 \cot ^{-1}(x)-x \cot ^{-1}(x)-\frac {1}{2} \cot ^{-1}(x)^2 \]
Antiderivative was successfully verified.
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Rule 43
Rule 260
Rule 266
Rule 4847
Rule 4853
Rule 4885
Rule 4917
Rubi steps
\begin {align*} \int \frac {x^4 \cot ^{-1}(x)}{1+x^2} \, dx &=\int x^2 \cot ^{-1}(x) \, dx-\int \frac {x^2 \cot ^{-1}(x)}{1+x^2} \, dx\\ &=\frac {1}{3} x^3 \cot ^{-1}(x)+\frac {1}{3} \int \frac {x^3}{1+x^2} \, dx-\int \cot ^{-1}(x) \, dx+\int \frac {\cot ^{-1}(x)}{1+x^2} \, dx\\ &=-x \cot ^{-1}(x)+\frac {1}{3} x^3 \cot ^{-1}(x)-\frac {1}{2} \cot ^{-1}(x)^2+\frac {1}{6} \operatorname {Subst}\left (\int \frac {x}{1+x} \, dx,x,x^2\right )-\int \frac {x}{1+x^2} \, dx\\ &=-x \cot ^{-1}(x)+\frac {1}{3} x^3 \cot ^{-1}(x)-\frac {1}{2} \cot ^{-1}(x)^2-\frac {1}{2} \log \left (1+x^2\right )+\frac {1}{6} \operatorname {Subst}\left (\int \left (1+\frac {1}{-1-x}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{6}-x \cot ^{-1}(x)+\frac {1}{3} x^3 \cot ^{-1}(x)-\frac {1}{2} \cot ^{-1}(x)^2-\frac {2}{3} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 32, normalized size = 0.80 \[ \frac {1}{6} \left (x^2-4 \log \left (x^2+1\right )+2 \left (x^2-3\right ) x \cot ^{-1}(x)-3 \cot ^{-1}(x)^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 31, normalized size = 0.78 \[ \frac {1}{6} \, x^{2} + \frac {1}{3} \, {\left (x^{3} - 3 \, x\right )} \operatorname {arccot}\relax (x) - \frac {1}{2} \, \operatorname {arccot}\relax (x)^{2} - \frac {2}{3} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \operatorname {arccot}\relax (x)}{x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 38, normalized size = 0.95 \[ \frac {x^{3} \mathrm {arccot}\relax (x )}{3}-x \,\mathrm {arccot}\relax (x )+\mathrm {arccot}\relax (x ) \arctan \relax (x )+\frac {x^{2}}{6}-\frac {2 \ln \left (x^{2}+1\right )}{3}+\frac {\arctan \relax (x )^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 35, normalized size = 0.88 \[ \frac {1}{6} \, x^{2} + \frac {1}{3} \, {\left (x^{3} - 3 \, x + 3 \, \arctan \relax (x)\right )} \operatorname {arccot}\relax (x) + \frac {1}{2} \, \arctan \relax (x)^{2} - \frac {2}{3} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.73, size = 32, normalized size = 0.80 \[ \frac {x^3\,\mathrm {acot}\relax (x)}{3}-\frac {2\,\ln \left (x^2+1\right )}{3}-\frac {{\mathrm {acot}\relax (x)}^2}{2}-x\,\mathrm {acot}\relax (x)+\frac {x^2}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.51, size = 34, normalized size = 0.85 \[ \frac {x^{3} \operatorname {acot}{\relax (x )}}{3} + \frac {x^{2}}{6} - x \operatorname {acot}{\relax (x )} - \frac {2 \log {\left (x^{2} + 1 \right )}}{3} - \frac {\operatorname {acot}^{2}{\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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