Integrand size = 16, antiderivative size = 66 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {x \cot ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{c \sqrt {a^2 c-d}} \]
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Time = 0.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {197, 5033, 12, 455, 65, 214} \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {x \cot ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{c \sqrt {a^2 c-d}} \]
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Rule 12
Rule 65
Rule 197
Rule 214
Rule 455
Rule 5033
Rubi steps \begin{align*} \text {integral}& = \frac {x \cot ^{-1}(a x)}{c \sqrt {c+d x^2}}+a \int \frac {x}{c \left (1+a^2 x^2\right ) \sqrt {c+d x^2}} \, dx \\ & = \frac {x \cot ^{-1}(a x)}{c \sqrt {c+d x^2}}+\frac {a \int \frac {x}{\left (1+a^2 x^2\right ) \sqrt {c+d x^2}} \, dx}{c} \\ & = \frac {x \cot ^{-1}(a x)}{c \sqrt {c+d x^2}}+\frac {a \text {Subst}\left (\int \frac {1}{\left (1+a^2 x\right ) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 c} \\ & = \frac {x \cot ^{-1}(a x)}{c \sqrt {c+d x^2}}+\frac {a \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 )}{c d} \\ & = \frac {x \cot ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{c \sqrt {a^2 c-d}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.56 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\frac {2 x \cot ^{-1}(a x)}{\sqrt {c+d x^2}}+\frac {-\log \left (\frac {4 a c \left (a c-i d x+\sqrt {a^2 c-d} \sqrt {c+d x^2}\right )}{\sqrt {a^2 c-d} (i+a x)}\right )-\log \left (\frac {4 a c \left (a c+i d x+\sqrt {a^2 c-d} \sqrt {c+d x^2}\right )}{\sqrt {a^2 c-d} (-i+a x)}\right )}{\sqrt {a^2 c-d}}}{2 c} \]
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\[\int \frac {\operatorname {arccot}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {3}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (58) = 116\).
Time = 0.30 (sec) , antiderivative size = 349, normalized size of antiderivative = 5.29 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx=\left [\frac {4 \, {\left (a^{2} c - d\right )} \sqrt {d x^{2} + c} x \operatorname {arccot}\left (a x\right ) + \sqrt {a^{2} c - d} {\left (d x^{2} + c\right )} \log \left (\frac {a^{4} d^{2} x^{4} + 8 \, a^{4} c^{2} - 8 \, a^{2} c d + 2 \, {\left (4 \, a^{4} c d - 3 \, a^{2} d^{2}\right )} x^{2} - 4 \, {\left (a^{3} d x^{2} + 2 \, a^{3} c - a d\right )} \sqrt {a^{2} c - d} \sqrt {d x^{2} + c} + d^{2}}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right )}{4 \, {\left (a^{2} c^{3} - c^{2} d + {\left (a^{2} c^{2} d - c d^{2}\right )} x^{2}\right )}}, \frac {2 \, {\left (a^{2} c - d\right )} \sqrt {d x^{2} + c} x \operatorname {arccot}\left (a x\right ) - \sqrt {-a^{2} c + d} {\left (d x^{2} + c\right )} \arctan \left (-\frac {{\left (a^{2} d x^{2} + 2 \, a^{2} c - d\right )} \sqrt {-a^{2} c + d} \sqrt {d x^{2} + c}}{2 \, {\left (a^{3} c^{2} - a c d + {\left (a^{3} c d - a d^{2}\right )} x^{2}\right )}}\right )}{2 \, {\left (a^{2} c^{3} - c^{2} d + {\left (a^{2} c^{2} d - c d^{2}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {acot}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.30 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.89 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {x \arctan \left (\frac {1}{a x}\right )}{\sqrt {d x^{2} + c} c} + \frac {\arctan \left (\frac {\sqrt {d x^{2} + c} a}{\sqrt {-a^{2} c + d}}\right )}{\sqrt {-a^{2} c + d} c} \]
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Timed out. \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\mathrm {acot}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \]
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