Integrand size = 14, antiderivative size = 79 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=-\frac {1}{9 a \left (a+a x^2\right )^{3/2}}-\frac {2}{3 a^2 \sqrt {a+a x^2}}+\frac {x \cot ^{-1}(x)}{3 a \left (a+a x^2\right )^{3/2}}+\frac {2 x \cot ^{-1}(x)}{3 a^2 \sqrt {a+a x^2}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5017, 5015} \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=-\frac {2}{3 a^2 \sqrt {a x^2+a}}+\frac {2 x \cot ^{-1}(x)}{3 a^2 \sqrt {a x^2+a}}-\frac {1}{9 a \left (a x^2+a\right )^{3/2}}+\frac {x \cot ^{-1}(x)}{3 a \left (a x^2+a\right )^{3/2}} \]
[In]
[Out]
Rule 5015
Rule 5017
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{9 a \left (a+a x^2\right )^{3/2}}+\frac {x \cot ^{-1}(x)}{3 a \left (a+a x^2\right )^{3/2}}+\frac {2 \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx}{3 a} \\ & = -\frac {1}{9 a \left (a+a x^2\right )^{3/2}}-\frac {2}{3 a^2 \sqrt {a+a x^2}}+\frac {x \cot ^{-1}(x)}{3 a \left (a+a x^2\right )^{3/2}}+\frac {2 x \cot ^{-1}(x)}{3 a^2 \sqrt {a+a x^2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.47 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=\frac {-7-6 x^2+\left (9 x+6 x^3\right ) \cot ^{-1}(x)}{9 a \left (a \left (1+x^2\right )\right )^{3/2}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.30 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.28
method | result | size |
risch | \(\frac {i x \left (2 x^{2}+3\right ) \ln \left (i x +1\right )}{6 a^{2} \left (x^{2}+1\right ) \sqrt {a \left (x^{2}+1\right )}}+\frac {-6 i x^{3} \ln \left (-i x +1\right )+6 \pi \,x^{3}-9 i x \ln \left (-i x +1\right )+9 \pi x -12 x^{2}-14}{18 a^{2} \left (x^{2}+1\right ) \sqrt {a \left (x^{2}+1\right )}}\) | \(101\) |
default | \(-\frac {\left (i+3 \,\operatorname {arccot}\left (x \right )\right ) \left (x^{3}+3 i x^{2}-3 x -i\right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}}{72 \left (x^{2}+1\right )^{2} a^{3}}+\frac {3 \left (\operatorname {arccot}\left (x \right )+i\right ) \left (i+x \right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}}{8 a^{3} \left (x^{2}+1\right )}+\frac {3 \sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (x -i\right ) \left (\operatorname {arccot}\left (x \right )-i\right )}{8 a^{3} \left (x^{2}+1\right )}-\frac {\left (-i+3 \,\operatorname {arccot}\left (x \right )\right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (x^{3}-3 i x^{2}-3 x +i\right )}{72 \left (x^{4}+2 x^{2}+1\right ) a^{3}}\) | \(165\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.66 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=-\frac {\sqrt {a x^{2} + a} {\left (6 \, x^{2} - 3 \, {\left (2 \, x^{3} + 3 \, x\right )} \operatorname {arccot}\left (x\right ) + 7\right )}}{9 \, {\left (a^{3} x^{4} + 2 \, a^{3} x^{2} + a^{3}\right )}} \]
[In]
[Out]
\[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {acot}{\left (x \right )}}{\left (a \left (x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.80 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=\frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {a x^{2} + a} a^{2}} + \frac {x}{{\left (a x^{2} + a\right )}^{\frac {3}{2}} a}\right )} \operatorname {arccot}\left (x\right ) - \frac {2}{3 \, \sqrt {a x^{2} + a} a^{2}} - \frac {1}{9 \, {\left (a x^{2} + a\right )}^{\frac {3}{2}} a} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=\frac {x {\left (\frac {2 \, x^{2}}{a} + \frac {3}{a}\right )} \arctan \left (\frac {1}{x}\right )}{3 \, {\left (a x^{2} + a\right )}^{\frac {3}{2}}} - \frac {6 \, a x^{2} + 7 \, a}{9 \, {\left (a x^{2} + a\right )}^{\frac {3}{2}} a^{2}} \]
[In]
[Out]
Timed out. \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=\int \frac {\mathrm {acot}\left (x\right )}{{\left (a\,x^2+a\right )}^{5/2}} \,d x \]
[In]
[Out]