Optimal. Leaf size=34 \[ -\frac {1}{4 \left (x^2+1\right )}+\frac {x \cot ^{-1}(x)}{2 \left (x^2+1\right )}-\frac {1}{4} \cot ^{-1}(x)^2 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4893, 261} \[ -\frac {1}{4 \left (x^2+1\right )}+\frac {x \cot ^{-1}(x)}{2 \left (x^2+1\right )}-\frac {1}{4} \cot ^{-1}(x)^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 261
Rule 4893
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(x)}{\left (1+x^2\right )^2} \, dx &=\frac {x \cot ^{-1}(x)}{2 \left (1+x^2\right )}-\frac {1}{4} \cot ^{-1}(x)^2+\frac {1}{2} \int \frac {x}{\left (1+x^2\right )^2} \, dx\\ &=-\frac {1}{4 \left (1+x^2\right )}+\frac {x \cot ^{-1}(x)}{2 \left (1+x^2\right )}-\frac {1}{4} \cot ^{-1}(x)^2\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 28, normalized size = 0.82 \[ -\frac {\left (x^2+1\right ) \cot ^{-1}(x)^2-2 x \cot ^{-1}(x)+1}{4 \left (x^2+1\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.34, size = 26, normalized size = 0.76 \[ -\frac {{\left (x^{2} + 1\right )} \operatorname {arccot}\relax (x)^{2} - 2 \, x \operatorname {arccot}\relax (x) + 1}{4 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccot}\relax (x)}{{\left (x^{2} + 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 35, normalized size = 1.03 \[ \frac {x \,\mathrm {arccot}\relax (x )}{2 x^{2}+2}+\frac {\mathrm {arccot}\relax (x ) \arctan \relax (x )}{2}-\frac {1}{4 \left (x^{2}+1\right )}+\frac {\arctan \relax (x )^{2}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 38, normalized size = 1.12 \[ \frac {1}{2} \, {\left (\frac {x}{x^{2} + 1} + \arctan \relax (x)\right )} \operatorname {arccot}\relax (x) + \frac {{\left (x^{2} + 1\right )} \arctan \relax (x)^{2} - 1}{4 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.61, size = 22, normalized size = 0.65 \[ \frac {\frac {x\,\mathrm {acot}\relax (x)}{2}-\frac {1}{4}}{x^2+1}-\frac {{\mathrm {acot}\relax (x)}^2}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RecursionError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________