Integrand size = 14, antiderivative size = 35 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx=-\frac {1}{a \sqrt {a+a x^2}}+\frac {x \cot ^{-1}(x)}{a \sqrt {a+a x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {5015} \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx=\frac {x \cot ^{-1}(x)}{a \sqrt {a x^2+a}}-\frac {1}{a \sqrt {a x^2+a}} \]
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Rule 5015
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{a \sqrt {a+a x^2}}+\frac {x \cot ^{-1}(x)}{a \sqrt {a+a x^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.60 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx=\frac {-1+x \cot ^{-1}(x)}{a \sqrt {a \left (1+x^2\right )}} \]
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Result contains complex when optimal does not.
Time = 0.57 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.57
method | result | size |
risch | \(\frac {i x \ln \left (i x +1\right )}{2 a \sqrt {a \left (x^{2}+1\right )}}+\frac {-i x \ln \left (-i x +1\right )+\pi x -2}{2 a \sqrt {a \left (x^{2}+1\right )}}\) | \(55\) |
default | \(\frac {\left (\operatorname {arccot}\left (x \right )+i\right ) \left (i+x \right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}}{2 \left (x^{2}+1\right ) a^{2}}+\frac {\sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (x -i\right ) \left (\operatorname {arccot}\left (x \right )-i\right )}{2 \left (x^{2}+1\right ) a^{2}}\) | \(68\) |
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none
Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx=\frac {\sqrt {a x^{2} + a} {\left (x \operatorname {arccot}\left (x\right ) - 1\right )}}{a^{2} x^{2} + a^{2}} \]
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\[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {acot}{\left (x \right )}}{\left (a \left (x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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none
Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx=\frac {x \operatorname {arccot}\left (x\right )}{\sqrt {a x^{2} + a} a} - \frac {1}{\sqrt {a x^{2} + a} a} \]
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none
Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx=\frac {x \arctan \left (\frac {1}{x}\right )}{\sqrt {a x^{2} + a} a} - \frac {1}{\sqrt {a x^{2} + a} a} \]
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Timed out. \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx=\int \frac {\mathrm {acot}\left (x\right )}{{\left (a\,x^2+a\right )}^{3/2}} \,d x \]
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