Integrand size = 14, antiderivative size = 390 \[ \int \frac {\coth ^{-1}(a x)}{c+d x^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (i a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \sqrt {d} (1+a x)}{\left (i a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,1+\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (i a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {c} \sqrt {d} (1+a x)}{\left (i a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}} \]
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Time = 0.73 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {6120, 211, 2520, 12, 266, 6820, 4996, 4940, 2438, 4966, 2449, 2352, 2497} \[ \int \frac {\coth ^{-1}(a x)}{c+d x^2} \, dx=-\frac {\log \left (1-\frac {1}{a x}\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\log \left (\frac {1}{a x}+1\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (-\sqrt {d}+i a \sqrt {c}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \sqrt {d} (a x+1)}{\left (\sqrt {d}+i a \sqrt {c}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (i a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}+1\right )}{4 \sqrt {c} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {c} \sqrt {d} (a x+1)}{\left (i \sqrt {c} a+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}} \]
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Rule 12
Rule 211
Rule 266
Rule 2352
Rule 2438
Rule 2449
Rule 2497
Rule 2520
Rule 4940
Rule 4966
Rule 4996
Rule 6120
Rule 6820
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \int \frac {\log \left (1-\frac {1}{a x}\right )}{c+d x^2} \, dx\right )+\frac {1}{2} \int \frac {\log \left (1+\frac {1}{a x}\right )}{c+d x^2} \, dx \\ & = -\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d} \left (1-\frac {1}{a x}\right ) x^2} \, dx}{2 a}+\frac {\int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d} \left (1+\frac {1}{a x}\right ) x^2} \, dx}{2 a} \\ & = -\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\left (1-\frac {1}{a x}\right ) x^2} \, dx}{2 a \sqrt {c} \sqrt {d}}+\frac {\int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\left (1+\frac {1}{a x}\right ) x^2} \, dx}{2 a \sqrt {c} \sqrt {d}} \\ & = -\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {a \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (-1+a x)} \, dx}{2 a \sqrt {c} \sqrt {d}}+\frac {\int \frac {a \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (1+a x)} \, dx}{2 a \sqrt {c} \sqrt {d}} \\ & = -\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (-1+a x)} \, dx}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (1+a x)} \, dx}{2 \sqrt {c} \sqrt {d}} \\ & = -\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \left (-\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x}+\frac {a \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{-1+a x}\right ) \, dx}{2 \sqrt {c} \sqrt {d}}+\frac {\int \left (\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x}-\frac {a \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{1+a x}\right ) \, dx}{2 \sqrt {c} \sqrt {d}} \\ & = -\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {a \int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{-1+a x} \, dx}{2 \sqrt {c} \sqrt {d}}-\frac {a \int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{1+a x} \, dx}{2 \sqrt {c} \sqrt {d}} \\ & = -\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (i a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \sqrt {d} (1+a x)}{\left (i a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\int \frac {\log \left (\frac {2 \sqrt {d} (-1+a x)}{\sqrt {c} \left (i a-\frac {\sqrt {d}}{\sqrt {c}}\right ) \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}\right )}{1+\frac {d x^2}{c}} \, dx}{2 c}+\frac {\int \frac {\log \left (\frac {2 \sqrt {d} (1+a x)}{\sqrt {c} \left (i a+\frac {\sqrt {d}}{\sqrt {c}}\right ) \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}\right )}{1+\frac {d x^2}{c}} \, dx}{2 c} \\ & = -\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (i a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \sqrt {d} (1+a x)}{\left (i a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,1+\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (i a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {c} \sqrt {d} (1+a x)}{\left (i a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}} \\ \end{align*}
Time = 1.29 (sec) , antiderivative size = 671, normalized size of antiderivative = 1.72 \[ \int \frac {\coth ^{-1}(a x)}{c+d x^2} \, dx=\frac {a \left (-2 i \arccos \left (\frac {a^2 c-d}{a^2 c+d}\right ) \arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right )+4 \coth ^{-1}(a x) \arctan \left (\frac {a d x}{\sqrt {a^2 c d}}\right )-\left (\arccos \left (\frac {a^2 c-d}{a^2 c+d}\right )+2 \arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right )\right ) \log \left (\frac {2 d \left (a^2 c-i \sqrt {a^2 c d}\right ) (-1+a x)}{\left (a^2 c+d\right ) \left (i \sqrt {a^2 c d}+a d x\right )}\right )-\left (\arccos \left (\frac {a^2 c-d}{a^2 c+d}\right )-2 \arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right )\right ) \log \left (\frac {2 d \left (a^2 c+i \sqrt {a^2 c d}\right ) (1+a x)}{\left (a^2 c+d\right ) \left (i \sqrt {a^2 c d}+a d x\right )}\right )+\left (\arccos \left (\frac {a^2 c-d}{a^2 c+d}\right )+2 \left (\arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right )+\arctan \left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{-\coth ^{-1}(a x)}}{\sqrt {a^2 c+d} \sqrt {-a^2 c+d+\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )}}\right )+\left (\arccos \left (\frac {a^2 c-d}{a^2 c+d}\right )-2 \left (\arctan \left (\frac {a c}{\sqrt {a^2 c d} x}\right )+\arctan \left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{\coth ^{-1}(a x)}}{\sqrt {a^2 c+d} \sqrt {-a^2 c+d+\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )}}\right )+i \left (-\operatorname {PolyLog}\left (2,\frac {\left (a^2 c-d-2 i \sqrt {a^2 c d}\right ) \left (\sqrt {a^2 c d}+i a d x\right )}{\left (a^2 c+d\right ) \left (\sqrt {a^2 c d}-i a d x\right )}\right )+\operatorname {PolyLog}\left (2,\frac {\left (a^2 c-d+2 i \sqrt {a^2 c d}\right ) \left (\sqrt {a^2 c d}+i a d x\right )}{\left (a^2 c+d\right ) \left (\sqrt {a^2 c d}-i a d x\right )}\right )\right )\right )}{4 \sqrt {a^2 c d}} \]
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Time = 1.05 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.92
method | result | size |
risch | \(-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a \sqrt {-c d}-\left (a x -1\right ) d -d}{a \sqrt {-c d}-d}\right )}{4 \sqrt {-c d}}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a \sqrt {-c d}+\left (a x -1\right ) d +d}{a \sqrt {-c d}+d}\right )}{4 \sqrt {-c d}}-\frac {\operatorname {dilog}\left (\frac {a \sqrt {-c d}-\left (a x -1\right ) d -d}{a \sqrt {-c d}-d}\right )}{4 \sqrt {-c d}}+\frac {\operatorname {dilog}\left (\frac {a \sqrt {-c d}+\left (a x -1\right ) d +d}{a \sqrt {-c d}+d}\right )}{4 \sqrt {-c d}}+\frac {\ln \left (a x +1\right ) \ln \left (\frac {a \sqrt {-c d}-\left (a x +1\right ) d +d}{a \sqrt {-c d}+d}\right )}{4 \sqrt {-c d}}-\frac {\ln \left (a x +1\right ) \ln \left (\frac {a \sqrt {-c d}+\left (a x +1\right ) d -d}{a \sqrt {-c d}-d}\right )}{4 \sqrt {-c d}}+\frac {\operatorname {dilog}\left (\frac {a \sqrt {-c d}-\left (a x +1\right ) d +d}{a \sqrt {-c d}+d}\right )}{4 \sqrt {-c d}}-\frac {\operatorname {dilog}\left (\frac {a \sqrt {-c d}+\left (a x +1\right ) d -d}{a \sqrt {-c d}-d}\right )}{4 \sqrt {-c d}}\) | \(358\) |
derivativedivides | \(\frac {\frac {\sqrt {-a^{2} c d}\, \operatorname {arccoth}\left (a x \right ) \ln \left (1-\frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c +2 \sqrt {-a^{2} c d}-d \right )}\right )}{2 c d}-\frac {\sqrt {-a^{2} c d}\, \operatorname {arccoth}\left (a x \right )^{2}}{2 c d}+\frac {\sqrt {-a^{2} c d}\, \operatorname {polylog}\left (2, \frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c +2 \sqrt {-a^{2} c d}-d \right )}\right )}{4 c d}+\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) a^{2} \ln \left (1-\frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c -2 \sqrt {-a^{2} c d}-d \right )}\right ) \operatorname {arccoth}\left (a x \right )}{2 d \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) a^{2} \operatorname {arccoth}\left (a x \right )^{2}}{2 d \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}+\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) a^{2} \operatorname {polylog}\left (2, \frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c -2 \sqrt {-a^{2} c d}-d \right )}\right )}{4 d \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\left (a^{2} c +2 \sqrt {-a^{2} c d}-d \right ) \ln \left (1-\frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c -2 \sqrt {-a^{2} c d}-d \right )}\right ) \operatorname {arccoth}\left (a x \right ) a^{2}}{a^{4} c^{2}+2 a^{2} c d +d^{2}}-\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) \ln \left (1-\frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c -2 \sqrt {-a^{2} c d}-d \right )}\right ) \operatorname {arccoth}\left (a x \right )}{2 c \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}+\frac {\left (a^{2} c +2 \sqrt {-a^{2} c d}-d \right ) a^{2} \operatorname {arccoth}\left (a x \right )^{2}}{a^{4} c^{2}+2 a^{2} c d +d^{2}}+\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) \operatorname {arccoth}\left (a x \right )^{2}}{2 c \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\left (a^{2} c +2 \sqrt {-a^{2} c d}-d \right ) \operatorname {polylog}\left (2, \frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c -2 \sqrt {-a^{2} c d}-d \right )}\right ) a^{2}}{2 \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) \operatorname {polylog}\left (2, \frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c -2 \sqrt {-a^{2} c d}-d \right )}\right )}{4 c \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}}{a}\) | \(954\) |
default | \(\frac {\frac {\sqrt {-a^{2} c d}\, \operatorname {arccoth}\left (a x \right ) \ln \left (1-\frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c +2 \sqrt {-a^{2} c d}-d \right )}\right )}{2 c d}-\frac {\sqrt {-a^{2} c d}\, \operatorname {arccoth}\left (a x \right )^{2}}{2 c d}+\frac {\sqrt {-a^{2} c d}\, \operatorname {polylog}\left (2, \frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c +2 \sqrt {-a^{2} c d}-d \right )}\right )}{4 c d}+\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) a^{2} \ln \left (1-\frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c -2 \sqrt {-a^{2} c d}-d \right )}\right ) \operatorname {arccoth}\left (a x \right )}{2 d \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) a^{2} \operatorname {arccoth}\left (a x \right )^{2}}{2 d \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}+\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) a^{2} \operatorname {polylog}\left (2, \frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c -2 \sqrt {-a^{2} c d}-d \right )}\right )}{4 d \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\left (a^{2} c +2 \sqrt {-a^{2} c d}-d \right ) \ln \left (1-\frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c -2 \sqrt {-a^{2} c d}-d \right )}\right ) \operatorname {arccoth}\left (a x \right ) a^{2}}{a^{4} c^{2}+2 a^{2} c d +d^{2}}-\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) \ln \left (1-\frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c -2 \sqrt {-a^{2} c d}-d \right )}\right ) \operatorname {arccoth}\left (a x \right )}{2 c \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}+\frac {\left (a^{2} c +2 \sqrt {-a^{2} c d}-d \right ) a^{2} \operatorname {arccoth}\left (a x \right )^{2}}{a^{4} c^{2}+2 a^{2} c d +d^{2}}+\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) \operatorname {arccoth}\left (a x \right )^{2}}{2 c \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\left (a^{2} c +2 \sqrt {-a^{2} c d}-d \right ) \operatorname {polylog}\left (2, \frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c -2 \sqrt {-a^{2} c d}-d \right )}\right ) a^{2}}{2 \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}-\frac {\left (-\sqrt {-a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {-a^{2} c d}\, d \right ) \operatorname {polylog}\left (2, \frac {\left (a^{2} c +d \right ) \left (a x +1\right )}{\left (a x -1\right ) \left (a^{2} c -2 \sqrt {-a^{2} c d}-d \right )}\right )}{4 c \left (a^{4} c^{2}+2 a^{2} c d +d^{2}\right )}}{a}\) | \(954\) |
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\[ \int \frac {\coth ^{-1}(a x)}{c+d x^2} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )}{d x^{2} + c} \,d x } \]
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\[ \int \frac {\coth ^{-1}(a x)}{c+d x^2} \, dx=\int \frac {\operatorname {acoth}{\left (a x \right )}}{c + d x^{2}}\, dx \]
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Time = 0.38 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.04 \[ \int \frac {\coth ^{-1}(a x)}{c+d x^2} \, dx=\frac {\operatorname {arcoth}\left (a x\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}} + \frac {{\left (\arctan \left (\frac {{\left (a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, \frac {a d x + d}{a^{2} c + d}\right ) - \arctan \left (\frac {{\left (a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, -\frac {a d x - d}{a^{2} c + d}\right )\right )} \log \left (d x^{2} + c\right ) - \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} + 2 \, a d x + d}{a^{2} c + d}\right ) + \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} - 2 \, a d x + d}{a^{2} c + d}\right ) - i \, {\rm Li}_2\left (\frac {a^{2} c + a d x - {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) - i \, {\rm Li}_2\left (\frac {a^{2} c - a d x + {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + i \, {\rm Li}_2\left (\frac {a^{2} c + a d x + {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + i \, {\rm Li}_2\left (\frac {a^{2} c - a d x - {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right )}{4 \, \sqrt {c d}} \]
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\[ \int \frac {\coth ^{-1}(a x)}{c+d x^2} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )}{d x^{2} + c} \,d x } \]
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Timed out. \[ \int \frac {\coth ^{-1}(a x)}{c+d x^2} \, dx=\int \frac {\mathrm {acoth}\left (a\,x\right )}{d\,x^2+c} \,d x \]
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