Integrand size = 14, antiderivative size = 590 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=\frac {x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {i \log \left (\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (-\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (-\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \log \left (\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {a \log \left (1-a^2 x^2\right )}{4 c \left (a^2 c+d\right )}-\frac {a \log \left (c+d x^2\right )}{4 c \left (a^2 c+d\right )}+\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}} \]
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Time = 0.65 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {205, 211, 6124, 6857, 531, 455, 36, 31, 5028, 2456, 2441, 2440, 2438} \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=\frac {a \log \left (1-a^2 x^2\right )}{4 c \left (a^2 c+d\right )}-\frac {a \log \left (c+d x^2\right )}{4 c \left (a^2 c+d\right )}+\frac {\coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {c} a+i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (i \sqrt {d} x+\sqrt {c}\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (i \sqrt {d} x+\sqrt {c}\right )}{\sqrt {c} a+i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {\sqrt {d} (1-a x)}{\sqrt {d}+i a \sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {\sqrt {d} (a x+1)}{-\sqrt {d}+i a \sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {\sqrt {d} (1-a x)}{-\sqrt {d}+i a \sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {\sqrt {d} (a x+1)}{\sqrt {d}+i a \sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )} \]
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Rule 31
Rule 36
Rule 205
Rule 211
Rule 455
Rule 531
Rule 2438
Rule 2440
Rule 2441
Rule 2456
Rule 5028
Rule 6124
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}-a \int \frac {\frac {x}{2 c \left (c+d x^2\right )}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}}{1-a^2 x^2} \, dx \\ & = \frac {x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}-a \int \left (-\frac {x}{2 c (-1+a x) (1+a x) \left (c+d x^2\right )}-\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d} \left (-1+a^2 x^2\right )}\right ) \, dx \\ & = \frac {x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {a \int \frac {x}{(-1+a x) (1+a x) \left (c+d x^2\right )} \, dx}{2 c}+\frac {a \int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{-1+a^2 x^2} \, dx}{2 c^{3/2} \sqrt {d}} \\ & = \frac {x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {a \int \frac {x}{\left (-1+a^2 x^2\right ) \left (c+d x^2\right )} \, dx}{2 c}+\frac {(i a) \int \frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{-1+a^2 x^2} \, dx}{4 c^{3/2} \sqrt {d}}-\frac {(i a) \int \frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{-1+a^2 x^2} \, dx}{4 c^{3/2} \sqrt {d}} \\ & = \frac {x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {a \text {Subst}\left (\int \frac {1}{\left (-1+a^2 x\right ) (c+d x)} \, dx,x,x^2\right )}{4 c}+\frac {(i a) \int \left (-\frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{2 (1-a x)}-\frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{2 (1+a x)}\right ) \, dx}{4 c^{3/2} \sqrt {d}}-\frac {(i a) \int \left (-\frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{2 (1-a x)}-\frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{2 (1+a x)}\right ) \, dx}{4 c^{3/2} \sqrt {d}} \\ & = \frac {x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}-\frac {(i a) \int \frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1-a x} \, dx}{8 c^{3/2} \sqrt {d}}-\frac {(i a) \int \frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1+a x} \, dx}{8 c^{3/2} \sqrt {d}}+\frac {(i a) \int \frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1-a x} \, dx}{8 c^{3/2} \sqrt {d}}+\frac {(i a) \int \frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1+a x} \, dx}{8 c^{3/2} \sqrt {d}}+\frac {a^3 \text {Subst}\left (\int \frac {1}{-1+a^2 x} \, dx,x,x^2\right )}{4 c \left (a^2 c+d\right )}-\frac {(a d) \text {Subst}\left (\int \frac {1}{c+d x} \, dx,x,x^2\right )}{4 c \left (a^2 c+d\right )} \\ & = \frac {x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {i \log \left (\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (-\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (-\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \log \left (\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {a \log \left (1-a^2 x^2\right )}{4 c \left (a^2 c+d\right )}-\frac {a \log \left (c+d x^2\right )}{4 c \left (a^2 c+d\right )}-\frac {\int \frac {\log \left (-\frac {i \sqrt {d} (1-a x)}{\sqrt {c} \left (a-\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1-\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{8 c^2}-\frac {\int \frac {\log \left (\frac {i \sqrt {d} (1-a x)}{\sqrt {c} \left (a+\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1+\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{8 c^2}+\frac {\int \frac {\log \left (-\frac {i \sqrt {d} (1+a x)}{\sqrt {c} \left (-a-\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1-\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{8 c^2}+\frac {\int \frac {\log \left (\frac {i \sqrt {d} (1+a x)}{\sqrt {c} \left (-a+\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1+\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{8 c^2} \\ & = \frac {x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {i \log \left (\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (-\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (-\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \log \left (\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {a \log \left (1-a^2 x^2\right )}{4 c \left (a^2 c+d\right )}-\frac {a \log \left (c+d x^2\right )}{4 c \left (a^2 c+d\right )}+\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {a x}{-a-\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \text {Subst}\left (\int \frac {\log \left (1-\frac {a x}{a-\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {a x}{-a+\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \text {Subst}\left (\int \frac {\log \left (1-\frac {a x}{a+\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}} \\ & = \frac {x \coth ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {i \log \left (\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (-\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \log \left (-\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \log \left (\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {a \log \left (1-a^2 x^2\right )}{4 c \left (a^2 c+d\right )}-\frac {a \log \left (c+d x^2\right )}{4 c \left (a^2 c+d\right )}+\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{8 c^{3/2} \sqrt {d}} \\ \end{align*}
Time = 5.78 (sec) , antiderivative size = 755, normalized size of antiderivative = 1.28 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=-\frac {a \left (\frac {2 \log \left (1-\frac {\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )}{a^2 c-d}\right )}{a^2 c+d}+\frac {2 i \arccos \left (\frac {a^2 c-d}{a^2 c+d}\right ) \arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right )-4 \coth ^{-1}(a x) \arctan \left (\frac {a d x}{\sqrt {a^2 c d}}\right )+\left (\arccos \left (\frac {a^2 c-d}{a^2 c+d}\right )+2 \arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right )\right ) \log \left (\frac {2 d \left (a^2 c-i \sqrt {a^2 c d}\right ) (-1+a x)}{\left (a^2 c+d\right ) \left (i \sqrt {a^2 c d}+a d x\right )}\right )+\left (\arccos \left (\frac {a^2 c-d}{a^2 c+d}\right )-2 \arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right )\right ) \log \left (\frac {2 d \left (a^2 c+i \sqrt {a^2 c d}\right ) (1+a x)}{\left (a^2 c+d\right ) \left (i \sqrt {a^2 c d}+a d x\right )}\right )-\left (\arccos \left (\frac {a^2 c-d}{a^2 c+d}\right )+2 \left (\arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right )+\arctan \left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{-\coth ^{-1}(a x)}}{\sqrt {a^2 c+d} \sqrt {-a^2 c+d+\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )}}\right )-\left (\arccos \left (\frac {a^2 c-d}{a^2 c+d}\right )-2 \left (\arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right )+\arctan \left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{\coth ^{-1}(a x)}}{\sqrt {a^2 c+d} \sqrt {-a^2 c+d+\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )}}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (a^2 c-d-2 i \sqrt {a^2 c d}\right ) \left (\sqrt {a^2 c d}+i a d x\right )}{\left (a^2 c+d\right ) \left (\sqrt {a^2 c d}-i a d x\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (a^2 c-d+2 i \sqrt {a^2 c d}\right ) \left (\sqrt {a^2 c d}+i a d x\right )}{\left (a^2 c+d\right ) \left (\sqrt {a^2 c d}-i a d x\right )}\right )\right )}{\sqrt {a^2 c d}}-\frac {4 \coth ^{-1}(a x) \sinh \left (2 \coth ^{-1}(a x)\right )}{-a^2 c+d+\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )}\right )}{8 c} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1925\) vs. \(2(430)=860\).
Time = 1.21 (sec) , antiderivative size = 1926, normalized size of antiderivative = 3.26
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1926\) |
derivativedivides | \(\text {Expression too large to display}\) | \(2071\) |
default | \(\text {Expression too large to display}\) | \(2071\) |
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\[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
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\[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=\int \frac {\operatorname {acoth}{\left (a x \right )}}{\left (c + d x^{2}\right )^{2}}\, dx \]
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none
Time = 0.41 (sec) , antiderivative size = 550, normalized size of antiderivative = 0.93 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=\frac {1}{2} \, {\left (\frac {x}{c d x^{2} + c^{2}} + \frac {\arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} c}\right )} \operatorname {arcoth}\left (a x\right ) - \frac {{\left (2 \, a c d \log \left (d x^{2} + c\right ) - 2 \, a c d \log \left (a x + 1\right ) - 2 \, a c d \log \left (a x - 1\right ) + {\left ({\left (a^{2} c + d\right )} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} + 2 \, a d x + d}{a^{2} c + d}\right ) - {\left (a^{2} c + d\right )} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} - 2 \, a d x + d}{a^{2} c + d}\right ) + {\left (i \, a^{2} c + i \, d\right )} {\rm Li}_2\left (\frac {a^{2} c + a d x - {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + {\left (i \, a^{2} c + i \, d\right )} {\rm Li}_2\left (\frac {a^{2} c - a d x + {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + {\left (-i \, a^{2} c - i \, d\right )} {\rm Li}_2\left (\frac {a^{2} c + a d x + {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + {\left (-i \, a^{2} c - i \, d\right )} {\rm Li}_2\left (\frac {a^{2} c - a d x - {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) - {\left ({\left (a^{2} c + d\right )} \arctan \left (\frac {{\left (a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, \frac {a d x + d}{a^{2} c + d}\right ) - {\left (a^{2} c + d\right )} \arctan \left (\frac {{\left (a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, -\frac {a d x - d}{a^{2} c + d}\right )\right )} \log \left (d x^{2} + c\right )\right )} \sqrt {c} \sqrt {d}\right )} a}{8 \, {\left (a^{3} c^{3} d + a c^{2} d^{2}\right )}} \]
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\[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=\int \frac {\mathrm {acoth}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^2} \,d x \]
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