Optimal. Leaf size=56 \[ -\frac {x}{4 \left (x^2+1\right )}+\frac {x \cot ^{-1}(x)^2}{2 \left (x^2+1\right )}-\frac {\cot ^{-1}(x)}{2 \left (x^2+1\right )}-\frac {1}{4} \tan ^{-1}(x)-\frac {1}{6} \cot ^{-1}(x)^3 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4893, 4931, 199, 203} \[ -\frac {x}{4 \left (x^2+1\right )}+\frac {x \cot ^{-1}(x)^2}{2 \left (x^2+1\right )}-\frac {\cot ^{-1}(x)}{2 \left (x^2+1\right )}-\frac {1}{4} \tan ^{-1}(x)-\frac {1}{6} \cot ^{-1}(x)^3 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 199
Rule 203
Rule 4893
Rule 4931
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(x)^2}{\left (1+x^2\right )^2} \, dx &=\frac {x \cot ^{-1}(x)^2}{2 \left (1+x^2\right )}-\frac {1}{6} \cot ^{-1}(x)^3+\int \frac {x \cot ^{-1}(x)}{\left (1+x^2\right )^2} \, dx\\ &=-\frac {\cot ^{-1}(x)}{2 \left (1+x^2\right )}+\frac {x \cot ^{-1}(x)^2}{2 \left (1+x^2\right )}-\frac {1}{6} \cot ^{-1}(x)^3-\frac {1}{2} \int \frac {1}{\left (1+x^2\right )^2} \, dx\\ &=-\frac {x}{4 \left (1+x^2\right )}-\frac {\cot ^{-1}(x)}{2 \left (1+x^2\right )}+\frac {x \cot ^{-1}(x)^2}{2 \left (1+x^2\right )}-\frac {1}{6} \cot ^{-1}(x)^3-\frac {1}{4} \int \frac {1}{1+x^2} \, dx\\ &=-\frac {x}{4 \left (1+x^2\right )}-\frac {\cot ^{-1}(x)}{2 \left (1+x^2\right )}+\frac {x \cot ^{-1}(x)^2}{2 \left (1+x^2\right )}-\frac {1}{6} \cot ^{-1}(x)^3-\frac {1}{4} \tan ^{-1}(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 46, normalized size = 0.82 \[ -\frac {3 \left (\left (x^2+1\right ) \tan ^{-1}(x)+x\right )+2 \left (x^2+1\right ) \cot ^{-1}(x)^3-6 x \cot ^{-1}(x)^2+6 \cot ^{-1}(x)}{12 \left (x^2+1\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.65, size = 40, normalized size = 0.71 \[ -\frac {2 \, {\left (x^{2} + 1\right )} \operatorname {arccot}\relax (x)^{3} - 6 \, x \operatorname {arccot}\relax (x)^{2} - 3 \, {\left (x^{2} - 1\right )} \operatorname {arccot}\relax (x) + 3 \, x}{12 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccot}\relax (x)^{2}}{{\left (x^{2} + 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.35, size = 61, normalized size = 1.09 \[ -\frac {\mathrm {arccot}\relax (x )^{2} \left (x^{2} \mathrm {arccot}\relax (x )+\mathrm {arccot}\relax (x )-x \right )}{2 \left (x^{2}+1\right )}+\frac {x^{2} \mathrm {arccot}\relax (x )}{2 x^{2}+2}-\frac {x}{4 \left (x^{2}+1\right )}-\frac {\mathrm {arccot}\relax (x )}{4}+\frac {\mathrm {arccot}\relax (x )^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.43, size = 75, normalized size = 1.34 \[ \frac {1}{2} \, {\left (\frac {x}{x^{2} + 1} + \arctan \relax (x)\right )} \operatorname {arccot}\relax (x)^{2} + \frac {{\left ({\left (x^{2} + 1\right )} \arctan \relax (x)^{2} - 1\right )} \operatorname {arccot}\relax (x)}{2 \, {\left (x^{2} + 1\right )}} + \frac {2 \, {\left (x^{2} + 1\right )} \arctan \relax (x)^{3} - 3 \, {\left (x^{2} + 1\right )} \arctan \relax (x) - 3 \, x}{12 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 51, normalized size = 0.91 \[ \frac {x\,{\mathrm {acot}\relax (x)}^2}{2\,\left (x^2+1\right )}-\frac {{\mathrm {acot}\relax (x)}^3}{6}-\frac {x}{4\,\left (x^2+1\right )}-\frac {\mathrm {acot}\relax (x)}{2\,\left (x^2+1\right )}-\frac {\mathrm {atan}\relax (x)}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acot}^{2}{\relax (x )}}{\left (x^{2} + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________