Integrand size = 14, antiderivative size = 118 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{7/2}} \, dx=-\frac {1}{25 a \left (a+a x^2\right )^{5/2}}-\frac {4}{45 a^2 \left (a+a x^2\right )^{3/2}}-\frac {8}{15 a^3 \sqrt {a+a x^2}}+\frac {x \cot ^{-1}(x)}{5 a \left (a+a x^2\right )^{5/2}}+\frac {4 x \cot ^{-1}(x)}{15 a^2 \left (a+a x^2\right )^{3/2}}+\frac {8 x \cot ^{-1}(x)}{15 a^3 \sqrt {a+a x^2}} \]
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Time = 0.05 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5017, 5015} \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{7/2}} \, dx=-\frac {8}{15 a^3 \sqrt {a x^2+a}}+\frac {8 x \cot ^{-1}(x)}{15 a^3 \sqrt {a x^2+a}}-\frac {4}{45 a^2 \left (a x^2+a\right )^{3/2}}+\frac {4 x \cot ^{-1}(x)}{15 a^2 \left (a x^2+a\right )^{3/2}}-\frac {1}{25 a \left (a x^2+a\right )^{5/2}}+\frac {x \cot ^{-1}(x)}{5 a \left (a x^2+a\right )^{5/2}} \]
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Rule 5015
Rule 5017
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{25 a \left (a+a x^2\right )^{5/2}}+\frac {x \cot ^{-1}(x)}{5 a \left (a+a x^2\right )^{5/2}}+\frac {4 \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx}{5 a} \\ & = -\frac {1}{25 a \left (a+a x^2\right )^{5/2}}-\frac {4}{45 a^2 \left (a+a x^2\right )^{3/2}}+\frac {x \cot ^{-1}(x)}{5 a \left (a+a x^2\right )^{5/2}}+\frac {4 x \cot ^{-1}(x)}{15 a^2 \left (a+a x^2\right )^{3/2}}+\frac {8 \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx}{15 a^2} \\ & = -\frac {1}{25 a \left (a+a x^2\right )^{5/2}}-\frac {4}{45 a^2 \left (a+a x^2\right )^{3/2}}-\frac {8}{15 a^3 \sqrt {a+a x^2}}+\frac {x \cot ^{-1}(x)}{5 a \left (a+a x^2\right )^{5/2}}+\frac {4 x \cot ^{-1}(x)}{15 a^2 \left (a+a x^2\right )^{3/2}}+\frac {8 x \cot ^{-1}(x)}{15 a^3 \sqrt {a+a x^2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.40 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{7/2}} \, dx=\frac {-149-260 x^2-120 x^4+15 x \left (15+20 x^2+8 x^4\right ) \cot ^{-1}(x)}{225 a \left (a \left (1+x^2\right )\right )^{5/2}} \]
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Result contains complex when optimal does not.
Time = 1.41 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.10
method | result | size |
risch | \(\frac {i x \left (8 x^{4}+20 x^{2}+15\right ) \ln \left (i x +1\right )}{30 a^{3} \left (x^{2}+1\right )^{2} \sqrt {a \left (x^{2}+1\right )}}+\frac {-120 i x^{5} \ln \left (-i x +1\right )+120 \pi \,x^{5}-300 i x^{3} \ln \left (-i x +1\right )+300 \pi \,x^{3}-240 x^{4}-225 i x \ln \left (-i x +1\right )+225 \pi x -520 x^{2}-298}{450 a^{3} \left (x^{2}+1\right )^{2} \sqrt {a \left (x^{2}+1\right )}}\) | \(130\) |
default | \(\frac {\left (i+5 \,\operatorname {arccot}\left (x \right )\right ) \left (x^{5}+5 i x^{4}-10 x^{3}-10 i x^{2}+5 x +i\right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}}{800 \left (x^{2}+1\right )^{3} a^{4}}+\frac {5 \left (\operatorname {arccot}\left (x \right )+i\right ) \left (i+x \right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}}{16 \left (x^{2}+1\right ) a^{4}}+\frac {5 \sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (x -i\right ) \left (\operatorname {arccot}\left (x \right )-i\right )}{16 \left (x^{2}+1\right ) a^{4}}-\frac {5 \left (-i+3 \,\operatorname {arccot}\left (x \right )\right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (x^{3}-3 i x^{2}-3 x +i\right )}{288 \left (x^{4}+2 x^{2}+1\right ) a^{4}}-\frac {\sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (x -i\right ) \left (67 i+165 \,\operatorname {arccot}\left (x \right )\right ) \cos \left (4 \,\operatorname {arccot}\left (x \right )\right )}{3600 \left (x^{2}+1\right ) a^{4}}-\frac {\sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (i x +1\right ) \left (29 i+105 \,\operatorname {arccot}\left (x \right )\right ) \sin \left (4 \,\operatorname {arccot}\left (x \right )\right )}{1800 \left (x^{2}+1\right ) a^{4}}\) | \(258\) |
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Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.59 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{7/2}} \, dx=-\frac {{\left (120 \, x^{4} + 260 \, x^{2} - 15 \, {\left (8 \, x^{5} + 20 \, x^{3} + 15 \, x\right )} \operatorname {arccot}\left (x\right ) + 149\right )} \sqrt {a x^{2} + a}}{225 \, {\left (a^{4} x^{6} + 3 \, a^{4} x^{4} + 3 \, a^{4} x^{2} + a^{4}\right )}} \]
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\[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{7/2}} \, dx=\int \frac {\operatorname {acot}{\left (x \right )}}{\left (a \left (x^{2} + 1\right )\right )^{\frac {7}{2}}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.79 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{7/2}} \, dx=\frac {1}{15} \, {\left (\frac {8 \, x}{\sqrt {a x^{2} + a} a^{3}} + \frac {4 \, x}{{\left (a x^{2} + a\right )}^{\frac {3}{2}} a^{2}} + \frac {3 \, x}{{\left (a x^{2} + a\right )}^{\frac {5}{2}} a}\right )} \operatorname {arccot}\left (x\right ) - \frac {8}{15 \, \sqrt {a x^{2} + a} a^{3}} - \frac {4}{45 \, {\left (a x^{2} + a\right )}^{\frac {3}{2}} a^{2}} - \frac {1}{25 \, {\left (a x^{2} + a\right )}^{\frac {5}{2}} a} \]
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Time = 0.30 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.70 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{7/2}} \, dx=\frac {{\left (4 \, x^{2} {\left (\frac {2 \, x^{2}}{a} + \frac {5}{a}\right )} + \frac {15}{a}\right )} x \arctan \left (\frac {1}{x}\right )}{15 \, {\left (a x^{2} + a\right )}^{\frac {5}{2}}} - \frac {120 \, {\left (a x^{2} + a\right )}^{2} + 20 \, {\left (a x^{2} + a\right )} a + 9 \, a^{2}}{225 \, {\left (a x^{2} + a\right )}^{\frac {5}{2}} a^{3}} \]
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Timed out. \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{7/2}} \, dx=\int \frac {\mathrm {acot}\left (x\right )}{{\left (a\,x^2+a\right )}^{7/2}} \,d x \]
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