Integrand size = 16, antiderivative size = 134 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {a}{3 c \left (a^2 c-d\right ) \sqrt {c+d x^2}}+\frac {x \cot ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \cot ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}-\frac {\left (3 a^2 c-2 d\right ) \text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{3 c^2 \left (a^2 c-d\right )^{3/2}} \]
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Time = 0.24 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {198, 197, 5033, 6820, 12, 585, 79, 65, 214} \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=-\frac {\left (3 a^2 c-2 d\right ) \text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{3 c^2 \left (a^2 c-d\right )^{3/2}}+\frac {a}{3 c \left (a^2 c-d\right ) \sqrt {c+d x^2}}+\frac {2 x \cot ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \cot ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}} \]
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Rule 12
Rule 65
Rule 79
Rule 197
Rule 198
Rule 214
Rule 585
Rule 5033
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \frac {x \cot ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \cot ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+a \int \frac {\frac {x}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x}{3 c^2 \sqrt {c+d x^2}}}{1+a^2 x^2} \, dx \\ & = \frac {x \cot ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \cot ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+a \int \frac {x \left (3 c+2 d x^2\right )}{3 c^2 \left (1+a^2 x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx \\ & = \frac {x \cot ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \cot ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {a \int \frac {x \left (3 c+2 d x^2\right )}{\left (1+a^2 x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{3 c^2} \\ & = \frac {x \cot ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \cot ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {a \text {Subst}\left (\int \frac {3 c+2 d x}{\left (1+a^2 x\right ) (c+d x)^{3/2}} \, dx,x,x^2\right )}{6 c^2} \\ & = \frac {a}{3 c \left (a^2 c-d\right ) \sqrt {c+d x^2}}+\frac {x \cot ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \cot ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {\left (a \left (3 a^2 c-2 d\right )\right ) \text {Subst}\left (\int \frac {1}{\left (1+a^2 x\right ) \sqrt {c+d x}} \, dx,x,x^2\right )}{6 c^2 \left (a^2 c-d\right )} \\ & = \frac {a}{3 c \left (a^2 c-d\right ) \sqrt {c+d x^2}}+\frac {x \cot ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \cot ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {\left (a \left (3 a^2 c-2 d\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {a^2 c}{d}+\frac {a^2 x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{3 c^2 \left (a^2 c-d\right ) d} \\ & = \frac {a}{3 c \left (a^2 c-d\right ) \sqrt {c+d x^2}}+\frac {x \cot ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \cot ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}-\frac {\left (3 a^2 c-2 d\right ) \text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{3 c^2 \left (a^2 c-d\right )^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.96 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=-\frac {-\frac {2 a c}{\left (a^2 c-d\right ) \sqrt {c+d x^2}}-\frac {2 x \left (3 c+2 d x^2\right ) \cot ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}}+\frac {\left (3 a^2 c-2 d\right ) \log \left (\frac {12 a c^2 \sqrt {a^2 c-d} \left (a c-i d x+\sqrt {a^2 c-d} \sqrt {c+d x^2}\right )}{\left (3 a^2 c-2 d\right ) (i+a x)}\right )}{\left (a^2 c-d\right )^{3/2}}+\frac {\left (3 a^2 c-2 d\right ) \log \left (\frac {12 a c^2 \sqrt {a^2 c-d} \left (a c+i d x+\sqrt {a^2 c-d} \sqrt {c+d x^2}\right )}{\left (3 a^2 c-2 d\right ) (-i+a x)}\right )}{\left (a^2 c-d\right )^{3/2}}}{6 c^2} \]
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\[\int \frac {\operatorname {arccot}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {5}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (114) = 228\).
Time = 0.31 (sec) , antiderivative size = 712, normalized size of antiderivative = 5.31 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\left [\frac {{\left (3 \, a^{2} c^{3} + {\left (3 \, a^{2} c d^{2} - 2 \, d^{3}\right )} x^{4} - 2 \, c^{2} d + 2 \, {\left (3 \, a^{2} c^{2} d - 2 \, c d^{2}\right )} x^{2}\right )} \sqrt {a^{2} c - d} \log \left (\frac {a^{4} d^{2} x^{4} + 8 \, a^{4} c^{2} - 8 \, a^{2} c d + 2 \, {\left (4 \, a^{4} c d - 3 \, a^{2} d^{2}\right )} x^{2} - 4 \, {\left (a^{3} d x^{2} + 2 \, a^{3} c - a d\right )} \sqrt {a^{2} c - d} \sqrt {d x^{2} + c} + d^{2}}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) + 4 \, {\left (a^{3} c^{3} - a c^{2} d + {\left (a^{3} c^{2} d - a c d^{2}\right )} x^{2} + {\left (2 \, {\left (a^{4} c^{2} d - 2 \, a^{2} c d^{2} + d^{3}\right )} x^{3} + 3 \, {\left (a^{4} c^{3} - 2 \, a^{2} c^{2} d + c d^{2}\right )} x\right )} \operatorname {arccot}\left (a x\right )\right )} \sqrt {d x^{2} + c}}{12 \, {\left (a^{4} c^{6} - 2 \, a^{2} c^{5} d + c^{4} d^{2} + {\left (a^{4} c^{4} d^{2} - 2 \, a^{2} c^{3} d^{3} + c^{2} d^{4}\right )} x^{4} + 2 \, {\left (a^{4} c^{5} d - 2 \, a^{2} c^{4} d^{2} + c^{3} d^{3}\right )} x^{2}\right )}}, -\frac {{\left (3 \, a^{2} c^{3} + {\left (3 \, a^{2} c d^{2} - 2 \, d^{3}\right )} x^{4} - 2 \, c^{2} d + 2 \, {\left (3 \, a^{2} c^{2} d - 2 \, c d^{2}\right )} x^{2}\right )} \sqrt {-a^{2} c + d} \arctan \left (-\frac {{\left (a^{2} d x^{2} + 2 \, a^{2} c - d\right )} \sqrt {-a^{2} c + d} \sqrt {d x^{2} + c}}{2 \, {\left (a^{3} c^{2} - a c d + {\left (a^{3} c d - a d^{2}\right )} x^{2}\right )}}\right ) - 2 \, {\left (a^{3} c^{3} - a c^{2} d + {\left (a^{3} c^{2} d - a c d^{2}\right )} x^{2} + {\left (2 \, {\left (a^{4} c^{2} d - 2 \, a^{2} c d^{2} + d^{3}\right )} x^{3} + 3 \, {\left (a^{4} c^{3} - 2 \, a^{2} c^{2} d + c d^{2}\right )} x\right )} \operatorname {arccot}\left (a x\right )\right )} \sqrt {d x^{2} + c}}{6 \, {\left (a^{4} c^{6} - 2 \, a^{2} c^{5} d + c^{4} d^{2} + {\left (a^{4} c^{4} d^{2} - 2 \, a^{2} c^{3} d^{3} + c^{2} d^{4}\right )} x^{4} + 2 \, {\left (a^{4} c^{5} d - 2 \, a^{2} c^{4} d^{2} + c^{3} d^{3}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {acot}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.94 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {1}{3} \, a {\left (\frac {{\left (3 \, a^{2} c - 2 \, d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} a}{\sqrt {-a^{2} c + d}}\right )}{{\left (a^{2} c^{3} - c^{2} d\right )} \sqrt {-a^{2} c + d} a} + \frac {1}{{\left (a^{2} c^{2} - c d\right )} \sqrt {d x^{2} + c}}\right )} + \frac {x {\left (\frac {2 \, d x^{2}}{c^{2}} + \frac {3}{c}\right )} \arctan \left (\frac {1}{a x}\right )}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \]
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Timed out. \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\mathrm {acot}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \]
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