Integrand size = 16, antiderivative size = 62 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {x \coth ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{c \sqrt {a^2 c+d}} \]
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Time = 0.07 (sec) , antiderivative size = 62, 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, 6124, 12, 455, 65, 214} \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {x \coth ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{c \sqrt {a^2 c+d}} \]
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Rule 12
Rule 65
Rule 197
Rule 214
Rule 455
Rule 6124
Rubi steps \begin{align*} \text {integral}& = \frac {x \coth ^{-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 \coth ^{-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 \coth ^{-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 \coth ^{-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 \coth ^{-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*}
Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.92 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\frac {2 x \coth ^{-1}(a x)}{\sqrt {c+d x^2}}+\frac {\log (1-a x)+\log (1+a x)-\log \left (a c-d x+\sqrt {a^2 c+d} \sqrt {c+d x^2}\right )-\log \left (a c+d x+\sqrt {a^2 c+d} \sqrt {c+d x^2}\right )}{\sqrt {a^2 c+d}}}{2 c} \]
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\[\int \frac {\operatorname {arccoth}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {3}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (54) = 108\).
Time = 0.28 (sec) , antiderivative size = 354, normalized size of antiderivative = 5.71 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx=\left [\frac {2 \, {\left (a^{2} c + d\right )} \sqrt {d x^{2} + c} x \log \left (\frac {a x + 1}{a x - 1}\right ) + \sqrt {a^{2} c + d} {\left (d x^{2} + c\right )} \log \left (\frac {a^{4} d^{2} x^{4} + 8 \, a^{4} c^{2} + 8 \, a^{2} c d + 2 \, {\left (4 \, a^{4} c d + 3 \, a^{2} d^{2}\right )} x^{2} - 4 \, {\left (a^{3} d x^{2} + 2 \, a^{3} c + a d\right )} \sqrt {a^{2} c + d} \sqrt {d x^{2} + c} + d^{2}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1}\right )}{4 \, {\left (a^{2} c^{3} + c^{2} d + {\left (a^{2} c^{2} d + c d^{2}\right )} x^{2}\right )}}, \frac {{\left (a^{2} c + d\right )} \sqrt {d x^{2} + c} x \log \left (\frac {a x + 1}{a x - 1}\right ) + \sqrt {-a^{2} c - d} {\left (d x^{2} + c\right )} \arctan \left (\frac {{\left (a^{2} d x^{2} + 2 \, a^{2} c + d\right )} \sqrt {-a^{2} c - d} \sqrt {d x^{2} + c}}{2 \, {\left (a^{3} c^{2} + a c d + {\left (a^{3} c d + a d^{2}\right )} x^{2}\right )}}\right )}{2 \, {\left (a^{2} c^{3} + c^{2} d + {\left (a^{2} c^{2} d + c d^{2}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {acoth}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (54) = 108\).
Time = 0.22 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.47 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {a^{2} {\left (\frac {\operatorname {arsinh}\left (-\frac {2 \, a^{2} c}{\sqrt {c d} {\left | 2 \, a^{2} x + 2 \, a \right |}} + \frac {2 \, a d x}{\sqrt {c d} {\left | 2 \, a^{2} x + 2 \, a \right |}}\right )}{a^{3} \sqrt {c + \frac {d}{a^{2}}}} - \frac {\operatorname {arsinh}\left (\frac {2 \, a^{2} c}{\sqrt {c d} {\left | 2 \, a^{2} x - 2 \, a \right |}} + \frac {2 \, a d x}{\sqrt {c d} {\left | 2 \, a^{2} x - 2 \, a \right |}}\right )}{a^{3} \sqrt {c + \frac {d}{a^{2}}}}\right )}}{2 \, c} + \frac {x \operatorname {arcoth}\left (a x\right )}{\sqrt {d x^{2} + c} c} \]
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none
Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.27 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {x \log \left (-\frac {\frac {1}{a x} + 1}{\frac {1}{a x} - 1}\right )}{2 \, \sqrt {d x^{2} + c} c} + \frac {\arctan \left (\frac {\sqrt {d x^{2} + c} a}{\sqrt {-a^{2} c - d}}\right )}{\sqrt {-a^{2} c - d} c} \]
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Timed out. \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\mathrm {acoth}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \]
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