Integrand size = 14, antiderivative size = 657 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^3} \, dx=\frac {a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac {x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}+\frac {3 i \log \left (\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (-\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (-\frac {\sqrt {d} (1-a x)}{i a \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \log \left (\frac {\sqrt {d} (1+a x)}{i a \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac {a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}+\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}} \]
[Out]
Time = 0.75 (sec) , antiderivative size = 657, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {205, 211, 6124, 6857, 585, 78, 5028, 2456, 2441, 2440, 2438} \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^3} \, dx=\frac {a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac {a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}+\frac {a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac {3 \coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}+\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {c} a+i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (i \sqrt {d} x+\sqrt {c}\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (i \sqrt {d} x+\sqrt {c}\right )}{\sqrt {c} a+i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {\sqrt {d} (1-a x)}{\sqrt {d}+i a \sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {\sqrt {d} (a x+1)}{-\sqrt {d}+i a \sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {\sqrt {d} (1-a x)}{-\sqrt {d}+i a \sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {\sqrt {d} (a x+1)}{\sqrt {d}+i a \sqrt {c}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2} \]
[In]
[Out]
Rule 78
Rule 205
Rule 211
Rule 585
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)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}-a \int \frac {\frac {x}{4 c \left (c+d x^2\right )^2}+\frac {3 x}{8 c^2 \left (c+d x^2\right )}+\frac {3 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}}{1-a^2 x^2} \, dx \\ & = \frac {x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}-a \int \left (-\frac {x \left (5 c+3 d x^2\right )}{8 c^2 \left (-1+a^2 x^2\right ) \left (c+d x^2\right )^2}-\frac {3 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d} \left (-1+a^2 x^2\right )}\right ) \, dx \\ & = \frac {x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}+\frac {a \int \frac {x \left (5 c+3 d x^2\right )}{\left (-1+a^2 x^2\right ) \left (c+d x^2\right )^2} \, dx}{8 c^2}+\frac {(3 a) \int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{-1+a^2 x^2} \, dx}{8 c^{5/2} \sqrt {d}} \\ & = \frac {x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}+\frac {a \text {Subst}\left (\int \frac {5 c+3 d x}{\left (-1+a^2 x\right ) (c+d x)^2} \, dx,x,x^2\right )}{16 c^2}+\frac {(3 i a) \int \frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{-1+a^2 x^2} \, dx}{16 c^{5/2} \sqrt {d}}-\frac {(3 i a) \int \frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{-1+a^2 x^2} \, dx}{16 c^{5/2} \sqrt {d}} \\ & = \frac {x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}+\frac {a \text {Subst}\left (\int \left (\frac {a^2 \left (5 a^2 c+3 d\right )}{\left (a^2 c+d\right )^2 \left (-1+a^2 x\right )}-\frac {2 c d}{\left (a^2 c+d\right ) (c+d x)^2}-\frac {d \left (5 a^2 c+3 d\right )}{\left (a^2 c+d\right )^2 (c+d x)}\right ) \, dx,x,x^2\right )}{16 c^2}+\frac {(3 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}{16 c^{5/2} \sqrt {d}}-\frac {(3 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}{16 c^{5/2} \sqrt {d}} \\ & = \frac {a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac {x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}+\frac {a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac {a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac {(3 i a) \int \frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1-a x} \, dx}{32 c^{5/2} \sqrt {d}}-\frac {(3 i a) \int \frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1+a x} \, dx}{32 c^{5/2} \sqrt {d}}+\frac {(3 i a) \int \frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1-a x} \, dx}{32 c^{5/2} \sqrt {d}}+\frac {(3 i a) \int \frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1+a x} \, dx}{32 c^{5/2} \sqrt {d}} \\ & = \frac {a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac {x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}+\frac {3 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 )}{32 c^{5/2} \sqrt {d}}-\frac {3 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 )}{32 c^{5/2} \sqrt {d}}-\frac {3 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 )}{32 c^{5/2} \sqrt {d}}+\frac {3 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 )}{32 c^{5/2} \sqrt {d}}+\frac {a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac {a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac {3 \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}{32 c^3}-\frac {3 \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}{32 c^3}+\frac {3 \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}{32 c^3}+\frac {3 \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}{32 c^3} \\ & = \frac {a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac {x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}+\frac {3 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 )}{32 c^{5/2} \sqrt {d}}-\frac {3 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 )}{32 c^{5/2} \sqrt {d}}-\frac {3 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 )}{32 c^{5/2} \sqrt {d}}+\frac {3 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 )}{32 c^{5/2} \sqrt {d}}+\frac {a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac {a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}+\frac {(3 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 )}{32 c^{5/2} \sqrt {d}}-\frac {(3 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 )}{32 c^{5/2} \sqrt {d}}-\frac {(3 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 )}{32 c^{5/2} \sqrt {d}}+\frac {(3 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 )}{32 c^{5/2} \sqrt {d}} \\ & = \frac {a}{8 c \left (a^2 c+d\right ) \left (c+d x^2\right )}+\frac {x \coth ^{-1}(a x)}{4 c \left (c+d x^2\right )^2}+\frac {3 x \coth ^{-1}(a x)}{8 c^2 \left (c+d x^2\right )}+\frac {3 \coth ^{-1}(a x) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} \sqrt {d}}+\frac {3 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 )}{32 c^{5/2} \sqrt {d}}-\frac {3 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 )}{32 c^{5/2} \sqrt {d}}-\frac {3 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 )}{32 c^{5/2} \sqrt {d}}+\frac {3 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 )}{32 c^{5/2} \sqrt {d}}+\frac {a \left (5 a^2 c+3 d\right ) \log \left (1-a^2 x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}-\frac {a \left (5 a^2 c+3 d\right ) \log \left (c+d x^2\right )}{16 c^2 \left (a^2 c+d\right )^2}+\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}-i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}+\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}-i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}}-\frac {3 i \operatorname {PolyLog}\left (2,\frac {a \left (\sqrt {c}+i \sqrt {d} x\right )}{a \sqrt {c}+i \sqrt {d}}\right )}{32 c^{5/2} \sqrt {d}} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1559\) vs. \(2(657)=1314\).
Time = 9.81 (sec) , antiderivative size = 1559, normalized size of antiderivative = 2.37 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^3} \, dx=-\frac {a \left (10 a^2 c \log \left (1-\frac {\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )}{a^2 c-d}\right )+6 d \log \left (1-\frac {\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )}{a^2 c-d}\right )-\frac {3 d \left (a^2 c+d\right ) \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 )}{\sqrt {a^2 c d}}-\frac {3 \sqrt {a^2 c d} \left (a^2 c+d\right ) \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 )}{d}+\frac {16 a^2 c d \left (a^2 c+d\right ) \coth ^{-1}(a x) \sinh \left (2 \coth ^{-1}(a x)\right )}{\left (-a^2 c+d+\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )\right )^2}+\frac {8 a^2 c d-4 \left (5 a^4 c^2+8 a^2 c d+3 d^2\right ) \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 )}{32 c^2 \left (a^2 c+d\right )^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(3790\) vs. \(2(493)=986\).
Time = 1.48 (sec) , antiderivative size = 3791, normalized size of antiderivative = 5.77
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(3791\) |
default | \(\text {Expression too large to display}\) | \(3791\) |
risch | \(\text {Expression too large to display}\) | \(4508\) |
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\[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^3} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )}{{\left (d x^{2} + c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1087 vs. \(2 (463) = 926\).
Time = 0.42 (sec) , antiderivative size = 1087, normalized size of antiderivative = 1.65 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
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\[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^3} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )}{{\left (d x^{2} + c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^3} \, dx=\int \frac {\mathrm {acoth}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^3} \,d x \]
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