Integrand size = 14, antiderivative size = 403 \[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 i \sqrt {c} \sqrt {d} (i-a x)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 i \sqrt {c} \sqrt {d} (i+a x)}{\left (a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 i \sqrt {c} \sqrt {d} (i-a x)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,1+\frac {2 i \sqrt {c} \sqrt {d} (i+a x)}{\left (a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}} \]
[Out]
Time = 0.70 (sec) , antiderivative size = 403, 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 = {5029, 211, 2520, 12, 266, 6820, 4996, 4940, 2438, 4966, 2449, 2352, 2497} \[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=\frac {i \log \left (1-\frac {i}{a x}\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \log \left (1+\frac {i}{a x}\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 i \sqrt {c} \sqrt {d} (-a x+i)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 i \sqrt {c} \sqrt {d} (a x+i)}{\left (a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 i \sqrt {c} \sqrt {d} (i-a x)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {2 i \sqrt {c} \sqrt {d} (a x+i)}{\left (\sqrt {c} a+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}+1\right )}{4 \sqrt {c} \sqrt {d}} \]
[In]
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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 5029
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} i \int \frac {\log \left (1-\frac {i}{a x}\right )}{c+d x^2} \, dx-\frac {1}{2} i \int \frac {\log \left (1+\frac {i}{a x}\right )}{c+d x^2} \, dx \\ & = \frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{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 {i}{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 {i}{a x}\right ) x^2} \, dx}{2 a} \\ & = \frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\left (1-\frac {i}{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 {i}{a x}\right ) x^2} \, dx}{2 a \sqrt {c} \sqrt {d}} \\ & = \frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {a \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (-i+a x)} \, dx}{2 a \sqrt {c} \sqrt {d}}+\frac {\int \frac {a \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (i+a x)} \, dx}{2 a \sqrt {c} \sqrt {d}} \\ & = \frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (-i+a x)} \, dx}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (i+a x)} \, dx}{2 \sqrt {c} \sqrt {d}} \\ & = \frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \left (\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x}-\frac {i a \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{-i+a x}\right ) \, dx}{2 \sqrt {c} \sqrt {d}}+\frac {\int \left (-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x}+\frac {i a \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{i+a x}\right ) \, dx}{2 \sqrt {c} \sqrt {d}} \\ & = \frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {(i a) \int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{-i+a x} \, dx}{2 \sqrt {c} \sqrt {d}}+\frac {(i a) \int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{i+a x} \, dx}{2 \sqrt {c} \sqrt {d}} \\ & = \frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 i \sqrt {c} \sqrt {d} (i-a x)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 i \sqrt {c} \sqrt {d} (i+a x)}{\left (a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {i \int \frac {\log \left (\frac {2 \sqrt {d} (-i+a x)}{\sqrt {c} \left (i a-\frac {i \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 {i \int \frac {\log \left (\frac {2 \sqrt {d} (i+a x)}{\sqrt {c} \left (i a+\frac {i \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 {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 i \sqrt {c} \sqrt {d} (i-a x)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 i \sqrt {c} \sqrt {d} (i+a x)}{\left (a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 i \sqrt {c} \sqrt {d} (i-a x)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,1+\frac {2 i \sqrt {c} \sqrt {d} (i+a x)}{\left (a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}} \\ \end{align*}
Time = 1.22 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.78 \[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=\frac {a \left (-2 \arccos \left (\frac {a^2 c+d}{a^2 c-d}\right ) \text {arctanh}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )-4 \cot ^{-1}(a x) \text {arctanh}\left (\frac {a d x}{\sqrt {-a^2 c d}}\right )-\left (\arccos \left (\frac {a^2 c+d}{a^2 c-d}\right )-2 i \text {arctanh}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )\right ) \log \left (\frac {2 i d \left (i a^2 c+\sqrt {-a^2 c d}\right ) (i+a x)}{\left (a^2 c-d\right ) \left (\sqrt {-a^2 c d}-a d x\right )}\right )-\left (\arccos \left (\frac {a^2 c+d}{a^2 c-d}\right )+2 i \text {arctanh}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )\right ) \log \left (\frac {2 d \left (a^2 c+i \sqrt {-a^2 c d}\right ) (-i+a x)}{\left (a^2 c-d\right ) \left (-\sqrt {-a^2 c d}+a d x\right )}\right )+\left (\arccos \left (\frac {a^2 c+d}{a^2 c-d}\right )+2 i \text {arctanh}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )+2 i \text {arctanh}\left (\frac {a d x}{\sqrt {-a^2 c d}}\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-a^2 c d} e^{-i \cot ^{-1}(a x)}}{\sqrt {a^2 c-d} \sqrt {-a^2 c-d+\left (a^2 c-d\right ) \cos \left (2 \cot ^{-1}(a x)\right )}}\right )+\left (\arccos \left (\frac {a^2 c+d}{a^2 c-d}\right )-2 i \text {arctanh}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )-2 i \text {arctanh}\left (\frac {a d x}{\sqrt {-a^2 c d}}\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-a^2 c d} e^{i \cot ^{-1}(a x)}}{\sqrt {a^2 c-d} \sqrt {-a^2 c-d+\left (a^2 c-d\right ) \cos \left (2 \cot ^{-1}(a x)\right )}}\right )+i \left (-\operatorname {PolyLog}\left (2,\frac {\left (a^2 c+d-2 i \sqrt {-a^2 c d}\right ) \left (\sqrt {-a^2 c d}+a d x\right )}{\left (a^2 c-d\right ) \left (\sqrt {-a^2 c d}-a d x\right )}\right )+\operatorname {PolyLog}\left (2,\frac {\left (a^2 c+d+2 i \sqrt {-a^2 c d}\right ) \left (\sqrt {-a^2 c d}+a d x\right )}{\left (a^2 c-d\right ) \left (\sqrt {-a^2 c d}-a d x\right )}\right )\right )\right )}{4 \sqrt {-a^2 c d}} \]
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Time = 1.32 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {i \pi \,\operatorname {arctanh}\left (\frac {2 \left (-i a x +1\right ) d -2 d}{2 a \sqrt {c d}}\right )}{2 \sqrt {c d}}-\frac {\ln \left (-i a x +1\right ) \ln \left (\frac {a \sqrt {c d}-\left (-i a x +1\right ) d +d}{a \sqrt {c d}+d}\right )}{4 \sqrt {c d}}+\frac {\ln \left (-i a x +1\right ) \ln \left (\frac {a \sqrt {c d}+\left (-i a x +1\right ) d -d}{a \sqrt {c d}-d}\right )}{4 \sqrt {c d}}-\frac {\operatorname {dilog}\left (\frac {a \sqrt {c d}-\left (-i a x +1\right ) d +d}{a \sqrt {c d}+d}\right )}{4 \sqrt {c d}}+\frac {\operatorname {dilog}\left (\frac {a \sqrt {c d}+\left (-i a x +1\right ) d -d}{a \sqrt {c d}-d}\right )}{4 \sqrt {c d}}-\frac {\ln \left (i a x +1\right ) \ln \left (\frac {a \sqrt {c d}-\left (i a x +1\right ) d +d}{a \sqrt {c d}+d}\right )}{4 \sqrt {c d}}+\frac {\ln \left (i a x +1\right ) \ln \left (\frac {a \sqrt {c d}+\left (i a x +1\right ) d -d}{a \sqrt {c d}-d}\right )}{4 \sqrt {c d}}-\frac {\operatorname {dilog}\left (\frac {a \sqrt {c d}-\left (i a x +1\right ) d +d}{a \sqrt {c d}+d}\right )}{4 \sqrt {c d}}+\frac {\operatorname {dilog}\left (\frac {a \sqrt {c d}+\left (i a x +1\right ) d -d}{a \sqrt {c d}-d}\right )}{4 \sqrt {c d}}\) | \(392\) |
derivativedivides | \(\frac {-\frac {i \sqrt {a^{2} c d}\, \operatorname {arccot}\left (a x \right ) \ln \left (1-\frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+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 {arccot}\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 +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c +2 \sqrt {a^{2} c d}+d \right )}\right )}{4 c d}+\frac {i \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 +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) \operatorname {arccot}\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 {arccot}\left (a x \right )^{2}}{2 d \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}-\frac {i \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 +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) \operatorname {arccot}\left (a x \right ) a^{2}}{a^{4} c^{2}-2 a^{2} c d +d^{2}}+\frac {i \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 +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) \operatorname {arccot}\left (a x \right )}{2 c \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 +i\right )^{2}}{\left (a^{2} x^{2}+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 ) a^{2} \operatorname {arccot}\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 {arccot}\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 +i\right )^{2}}{\left (a^{2} x^{2}+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 +i\right )^{2}}{\left (a^{2} x^{2}+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}\) | \(977\) |
default | \(\frac {-\frac {i \sqrt {a^{2} c d}\, \operatorname {arccot}\left (a x \right ) \ln \left (1-\frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+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 {arccot}\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 +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c +2 \sqrt {a^{2} c d}+d \right )}\right )}{4 c d}+\frac {i \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 +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) \operatorname {arccot}\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 {arccot}\left (a x \right )^{2}}{2 d \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}-\frac {i \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 +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) \operatorname {arccot}\left (a x \right ) a^{2}}{a^{4} c^{2}-2 a^{2} c d +d^{2}}+\frac {i \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 +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) \operatorname {arccot}\left (a x \right )}{2 c \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 +i\right )^{2}}{\left (a^{2} x^{2}+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 ) a^{2} \operatorname {arccot}\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 {arccot}\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 +i\right )^{2}}{\left (a^{2} x^{2}+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 +i\right )^{2}}{\left (a^{2} x^{2}+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}\) | \(977\) |
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\[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )}{d x^{2} + c} \,d x } \]
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\[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=\int \frac {\operatorname {acot}{\left (a x \right )}}{c + d x^{2}}\, dx \]
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Time = 0.35 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.31 \[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=-\frac {a {\left (\frac {8 \, \arctan \left (a x\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{a} - \frac {4 \, \arctan \left (a x\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) + 4 \, \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \arctan \left (-\frac {a \sqrt {d} x}{a \sqrt {c} - \sqrt {d}}, -\frac {\sqrt {d}}{a \sqrt {c} - \sqrt {d}}\right ) + \log \left (d x^{2} + c\right ) \log \left (\frac {a^{2} c d + {\left (a^{4} c d + a^{2} d^{2}\right )} x^{2} + 2 \, {\left (a^{3} d x^{2} + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}{a^{4} c^{2} + 6 \, a^{2} c d + 4 \, {\left (a^{3} c + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}\right ) - \log \left (d x^{2} + c\right ) \log \left (\frac {a^{2} c d + {\left (a^{4} c d + a^{2} d^{2}\right )} x^{2} - 2 \, {\left (a^{3} d x^{2} + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}{a^{4} c^{2} + 6 \, a^{2} c d - 4 \, {\left (a^{3} c + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}\right ) + 2 \, {\rm Li}_2\left (\frac {a^{2} c + i \, a d x + {\left (i \, a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 \, a \sqrt {c} \sqrt {d} + d}\right ) + 2 \, {\rm Li}_2\left (\frac {a^{2} c - i \, a d x - {\left (i \, a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 \, a \sqrt {c} \sqrt {d} + d}\right ) - 2 \, {\rm Li}_2\left (\frac {a^{2} c + i \, a d x - {\left (i \, a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 \, a \sqrt {c} \sqrt {d} + d}\right ) - 2 \, {\rm Li}_2\left (\frac {a^{2} c - i \, a d x + {\left (i \, a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 \, a \sqrt {c} \sqrt {d} + d}\right )}{a}\right )}}{8 \, \sqrt {c d}} + \frac {\operatorname {arccot}\left (a x\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}} + \frac {\arctan \left (a x\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}} \]
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\[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )}{d x^{2} + c} \,d x } \]
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Timed out. \[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=\int \frac {\mathrm {acot}\left (a\,x\right )}{d\,x^2+c} \,d x \]
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