Integrand size = 14, antiderivative size = 801 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=\frac {x \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}-\frac {i a \log \left (\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (1+\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i a \log \left (\frac {\sqrt {d} \left (1+\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {a \log \left (1+a^2 x^2\right )}{4 c \left (a^2 c-d\right )}-\frac {a \log \left (c+d x^2\right )}{4 c \left (a^2 c-d\right )}-\frac {i a \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}-i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}+i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i a \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (\sqrt {c}+i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}-i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (\sqrt {c}+i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}+i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}} \]
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Time = 0.91 (sec) , antiderivative size = 801, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {205, 211, 5033, 6857, 455, 36, 31, 5028, 2456, 2441, 2440, 2438} \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \cot ^{-1}(a x)}{2 c^{3/2} \sqrt {d}}+\frac {x \cot ^{-1}(a x)}{2 c \left (d x^2+c\right )}-\frac {i a \log \left (\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (\sqrt {-a^2} x+1\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (\frac {i \sqrt {d} x}{\sqrt {c}}+1\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i a \log \left (\frac {\sqrt {d} \left (\sqrt {-a^2} x+1\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (\frac {i \sqrt {d} x}{\sqrt {c}}+1\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {a \log \left (a^2 x^2+1\right )}{4 c \left (a^2 c-d\right )}-\frac {a \log \left (d x^2+c\right )}{4 c \left (a^2 c-d\right )}-\frac {i a \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}-i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}+i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i a \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (i \sqrt {d} x+\sqrt {c}\right )}{\sqrt {-a^2} \sqrt {c}-i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (i \sqrt {d} x+\sqrt {c}\right )}{\sqrt {-a^2} \sqrt {c}+i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}} \]
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Rule 31
Rule 36
Rule 205
Rule 211
Rule 455
Rule 2438
Rule 2440
Rule 2441
Rule 2456
Rule 5028
Rule 5033
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {x \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-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 \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-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 \left (1+a^2 x^2\right ) \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 \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-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 {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 \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-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 \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 \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 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 {(i a) \int \left (\frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{2 \left (1-\sqrt {-a^2} x\right )}+\frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{2 \left (1+\sqrt {-a^2} x\right )}\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 \left (1-\sqrt {-a^2} x\right )}+\frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{2 \left (1+\sqrt {-a^2} x\right )}\right ) \, dx}{4 c^{3/2} \sqrt {d}}-\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 \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 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 a) \int \frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1-\sqrt {-a^2} 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+\sqrt {-a^2} 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-\sqrt {-a^2} 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+\sqrt {-a^2} x} \, dx}{8 c^{3/2} \sqrt {d}} \\ & = \frac {x \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}-\frac {i a \log \left (\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (1+\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i a \log \left (\frac {\sqrt {d} \left (1+\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} 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 {a \int \frac {\log \left (-\frac {i \sqrt {d} \left (1-\sqrt {-a^2} x\right )}{\sqrt {c} \left (\sqrt {-a^2}-\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1-\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{8 \sqrt {-a^2} c^2}+\frac {a \int \frac {\log \left (\frac {i \sqrt {d} \left (1-\sqrt {-a^2} x\right )}{\sqrt {c} \left (\sqrt {-a^2}+\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1+\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{8 \sqrt {-a^2} c^2}-\frac {a \int \frac {\log \left (-\frac {i \sqrt {d} \left (1+\sqrt {-a^2} x\right )}{\sqrt {c} \left (-\sqrt {-a^2}-\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1-\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{8 \sqrt {-a^2} c^2}-\frac {a \int \frac {\log \left (\frac {i \sqrt {d} \left (1+\sqrt {-a^2} x\right )}{\sqrt {c} \left (-\sqrt {-a^2}+\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1+\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{8 \sqrt {-a^2} c^2} \\ & = \frac {x \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}-\frac {i a \log \left (\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (1+\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i a \log \left (\frac {\sqrt {d} \left (1+\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} 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 a) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-a^2} x}{-\sqrt {-a^2}-\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {(i a) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-a^2} x}{\sqrt {-a^2}-\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {(i a) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-a^2} x}{-\sqrt {-a^2}+\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {(i a) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-a^2} x}{\sqrt {-a^2}+\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}} \\ & = \frac {x \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}-\frac {i a \log \left (\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (1+\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i a \log \left (\frac {\sqrt {d} \left (1+\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} 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 a \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}-i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}+i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i a \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (\sqrt {c}+i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}-i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (\sqrt {c}+i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}+i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}} \\ \end{align*}
Time = 5.30 (sec) , antiderivative size = 806, normalized size of antiderivative = 1.01 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=-\frac {a \left (\frac {2 \log \left (\frac {a^2 c+d+\left (-a^2 c+d\right ) \cos \left (2 \cot ^{-1}(a x)\right )}{a^2 c+d}\right )}{a^2 c-d}+\frac {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 )}{\sqrt {-a^2 c d}}-\frac {4 \cot ^{-1}(a x) \sin \left (2 \cot ^{-1}(a x)\right )}{a^2 c+d+\left (-a^2 c+d\right ) \cos \left (2 \cot ^{-1}(a x)\right )}\right )}{8 c} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2140 vs. \(2 (593 ) = 1186\).
Time = 1.59 (sec) , antiderivative size = 2141, normalized size of antiderivative = 2.67
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2141\) |
derivativedivides | \(\text {Expression too large to display}\) | \(2275\) |
default | \(\text {Expression too large to display}\) | \(2275\) |
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\[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
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Time = 0.38 (sec) , antiderivative size = 628, normalized size of antiderivative = 0.78 \[ \int \frac {\cot ^{-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 {arccot}\left (a x\right ) + \frac {{\left (4 \, a c d \log \left (a^{2} x^{2} + 1\right ) - 4 \, a c d \log \left (d x^{2} + c\right ) + {\left (4 \, {\left (a^{2} c - d\right )} \arctan \left (a x\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) + 4 \, {\left (a^{2} c - d\right )} \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 ) + {\left (a^{2} c - 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 ) - {\left (a^{2} c - 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 ) + 2 \, {\left (a^{2} c - d\right )} {\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 \, {\left (a^{2} c - d\right )} {\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 \, {\left (a^{2} c - d\right )} {\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 \, {\left (a^{2} c - d\right )} {\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 )\right )} \sqrt {c} \sqrt {d}\right )} a}{16 \, {\left (a^{3} c^{3} d - a c^{2} d^{2}\right )}} \]
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\[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx=\int \frac {\mathrm {acot}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^2} \,d x \]
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