Integrand size = 16, antiderivative size = 200 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\frac {a}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}+\frac {a \left (7 a^2 c+4 d\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{15 c^3 \left (a^2 c+d\right )^{5/2}} \]
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Time = 0.75 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {198, 197, 6124, 6820, 12, 6847, 911, 1275, 214} \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\frac {a \left (7 a^2 c+4 d\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {a}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}-\frac {\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{15 c^3 \left (a^2 c+d\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}} \]
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Rule 12
Rule 197
Rule 198
Rule 214
Rule 911
Rule 1275
Rule 6124
Rule 6820
Rule 6847
Rubi steps \begin{align*} \text {integral}& = \frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-a \int \frac {\frac {x}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x}{15 c^3 \sqrt {c+d x^2}}}{1-a^2 x^2} \, dx \\ & = \frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-a \int \frac {x \left (15 c^2+20 c d x^2+8 d^2 x^4\right )}{15 c^3 \left (1-a^2 x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx \\ & = \frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {a \int \frac {x \left (15 c^2+20 c d x^2+8 d^2 x^4\right )}{\left (1-a^2 x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx}{15 c^3} \\ & = \frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {a \text {Subst}\left (\int \frac {15 c^2+20 c d x+8 d^2 x^2}{\left (1-a^2 x\right ) (c+d x)^{5/2}} \, dx,x,x^2\right )}{30 c^3} \\ & = \frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {a \text {Subst}\left (\int \frac {3 c^2+4 c x^2+8 x^4}{x^4 \left (\frac {a^2 c+d}{d}-\frac {a^2 x^2}{d}\right )} \, dx,x,\sqrt {c+d x^2}\right )}{15 c^3 d} \\ & = \frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {a \text {Subst}\left (\int \left (\frac {3 c^2 d}{\left (a^2 c+d\right ) x^4}+\frac {c d \left (7 a^2 c+4 d\right )}{\left (a^2 c+d\right )^2 x^2}+\frac {d \left (15 a^4 c^2+20 a^2 c d+8 d^2\right )}{\left (a^2 c+d\right )^2 \left (a^2 c+d-a^2 x^2\right )}\right ) \, dx,x,\sqrt {c+d x^2}\right )}{15 c^3 d} \\ & = \frac {a}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}+\frac {a \left (7 a^2 c+4 d\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {\left (a \left (15 a^4 c^2+20 a^2 c d+8 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{a^2 c+d-a^2 x^2} \, dx,x,\sqrt {c+d x^2}\right )}{15 c^3 \left (a^2 c+d\right )^2} \\ & = \frac {a}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}+\frac {a \left (7 a^2 c+4 d\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{15 c^3 \left (a^2 c+d\right )^{5/2}} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.64 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\frac {2 a c \sqrt {a^2 c+d} \left (c+d x^2\right ) \left (d \left (5 c+4 d x^2\right )+a^2 c \left (8 c+7 d x^2\right )\right )+2 \left (a^2 c+d\right )^{5/2} x \left (15 c^2+20 c d x^2+8 d^2 x^4\right ) \coth ^{-1}(a x)+\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^{5/2} \log (1-a x)+\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^{5/2} \log (1+a x)-\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^{5/2} \log \left (a c-d x+\sqrt {a^2 c+d} \sqrt {c+d x^2}\right )-\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^{5/2} \log \left (a c+d x+\sqrt {a^2 c+d} \sqrt {c+d x^2}\right )}{30 c^3 \left (a^2 c+d\right )^{5/2} \left (c+d x^2\right )^{5/2}} \]
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\[\int \frac {\operatorname {arccoth}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {7}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (172) = 344\).
Time = 0.34 (sec) , antiderivative size = 1278, normalized size of antiderivative = 6.39 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\operatorname {acoth}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {7}{2}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (172) = 344\).
Time = 0.30 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.00 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\frac {1}{30} \, a {\left (\frac {\frac {3 \, a^{3} d \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{{\left (a^{4} c^{3} + 2 \, a^{2} c^{2} d + c d^{2}\right )} \sqrt {a^{2} c + d}} + \frac {2 \, {\left (3 \, {\left (d x^{2} + c\right )} a^{2} d + a^{2} c d + d^{2}\right )}}{{\left (a^{4} c^{3} + 2 \, a^{2} c^{2} d + c d^{2}\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}}{d} + \frac {4 \, {\left (\frac {a d \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{{\left (a^{2} c^{3} + c^{2} d\right )} \sqrt {a^{2} c + d}} + \frac {2 \, d}{{\left (a^{2} c^{3} + c^{2} d\right )} \sqrt {d x^{2} + c}}\right )}}{d} + \frac {8 \, \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{\sqrt {a^{2} c + d} a c^{3}}\right )} + \frac {1}{15} \, {\left (\frac {8 \, x}{\sqrt {d x^{2} + c} c^{3}} + \frac {4 \, x}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{2}} + \frac {3 \, x}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} c}\right )} \operatorname {arcoth}\left (a x\right ) \]
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Time = 0.31 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.13 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\frac {1}{15} \, a {\left (\frac {{\left (15 \, a^{4} c^{2} + 20 \, a^{2} c d + 8 \, d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} a}{\sqrt {-a^{2} c - d}}\right )}{{\left (a^{4} c^{5} + 2 \, a^{2} c^{4} d + c^{3} d^{2}\right )} \sqrt {-a^{2} c - d} a} + \frac {7 \, {\left (d x^{2} + c\right )} a^{2} c + a^{2} c^{2} + 4 \, {\left (d x^{2} + c\right )} d + c d}{{\left (a^{4} c^{4} + 2 \, a^{2} c^{3} d + c^{2} d^{2}\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}\right )} + \frac {{\left (4 \, x^{2} {\left (\frac {2 \, d^{2} x^{2}}{c^{3}} + \frac {5 \, d}{c^{2}}\right )} + \frac {15}{c}\right )} x \log \left (-\frac {\frac {1}{a x} + 1}{\frac {1}{a x} - 1}\right )}{30 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \]
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Timed out. \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\mathrm {acoth}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{7/2}} \,d x \]
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