Integrand size = 16, antiderivative size = 128 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {a}{3 c \left (a^2 c+d\right ) \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \coth ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}-\frac {\left (3 a^2 c+2 d\right ) \text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{3 c^2 \left (a^2 c+d\right )^{3/2}} \]
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Time = 0.24 (sec) , antiderivative size = 128, 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, 6124, 6820, 12, 585, 79, 65, 214} \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=-\frac {\left (3 a^2 c+2 d\right ) \text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{3 c^2 \left (a^2 c+d\right )^{3/2}}+\frac {a}{3 c \left (a^2 c+d\right ) \sqrt {c+d x^2}}+\frac {2 x \coth ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}} \]
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Rule 12
Rule 65
Rule 79
Rule 197
Rule 198
Rule 214
Rule 585
Rule 6124
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \frac {x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \coth ^{-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 \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \coth ^{-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 \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \coth ^{-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 \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \coth ^{-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 \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \coth ^{-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 \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \coth ^{-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 d \left (a^2 c+d\right )} \\ & = \frac {a}{3 c \left (a^2 c+d\right ) \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \coth ^{-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*}
Time = 0.20 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.77 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {\frac {2 a c}{\left (a^2 c+d\right ) \sqrt {c+d x^2}}+\frac {2 x \left (3 c+2 d x^2\right ) \coth ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}}+\frac {\left (3 a^2 c+2 d\right ) \log (1-a x)}{\left (a^2 c+d\right )^{3/2}}+\frac {\left (3 a^2 c+2 d\right ) \log (1+a x)}{\left (a^2 c+d\right )^{3/2}}-\frac {\left (3 a^2 c+2 d\right ) \log \left (a c-d x+\sqrt {a^2 c+d} \sqrt {c+d x^2}\right )}{\left (a^2 c+d\right )^{3/2}}-\frac {\left (3 a^2 c+2 d\right ) \log \left (a c+d x+\sqrt {a^2 c+d} \sqrt {c+d x^2}\right )}{\left (a^2 c+d\right )^{3/2}}}{6 c^2} \]
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\[\int \frac {\operatorname {arccoth}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {5}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (108) = 216\).
Time = 0.30 (sec) , antiderivative size = 728, normalized size of antiderivative = 5.69 \[ \int \frac {\coth ^{-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 ) + 2 \, {\left (2 \, a^{3} c^{3} + 2 \, a c^{2} d + 2 \, {\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 )} \log \left (\frac {a x + 1}{a x - 1}\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 ) + {\left (2 \, a^{3} c^{3} + 2 \, a c^{2} d + 2 \, {\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 )} \log \left (\frac {a x + 1}{a x - 1}\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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\[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {acoth}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (108) = 216\).
Time = 0.29 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.74 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {1}{6} \, a {\left (\frac {\frac {a d \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{{\left (a^{2} c^{2} + c d\right )} \sqrt {a^{2} c + d}} + \frac {2 \, d}{{\left (a^{2} c^{2} + c d\right )} \sqrt {d x^{2} + c}}}{d} + \frac {2 \, \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{\sqrt {a^{2} c + d} a c^{2}}\right )} + \frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {d x^{2} + c} c^{2}} + \frac {x}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} c}\right )} \operatorname {arcoth}\left (a x\right ) \]
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Time = 0.30 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.12 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {1}{3} \, a {\left (\frac {{\left (3 \, a^{2} c + 2 \, d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} a}{\sqrt {-a^{2} c - d}}\right )}{{\left (a^{2} c^{3} + c^{2} d\right )} \sqrt {-a^{2} c - d} a} + \frac {1}{{\left (a^{2} c^{2} + c d\right )} \sqrt {d x^{2} + c}}\right )} + \frac {x {\left (\frac {2 \, d x^{2}}{c^{2}} + \frac {3}{c}\right )} \log \left (-\frac {\frac {1}{a x} + 1}{\frac {1}{a x} - 1}\right )}{6 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \]
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Timed out. \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\mathrm {acoth}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \]
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