Integrand size = 24, antiderivative size = 560 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^2} \, dx=\frac {2 a e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1-\frac {1}{c x}\right )}{\sqrt {f}}+\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {1}{c x}\right )}{\sqrt {f}}+\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} (1-c x)}{\left (i c \sqrt {f}-\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f}}-\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} (1+c x)}{\left (i c \sqrt {f}+\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f}}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} b c \log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c e \operatorname {PolyLog}\left (2,\frac {c^2 \left (f+g x^2\right )}{c^2 f+g}\right )+\frac {1}{2} b c e \operatorname {PolyLog}\left (2,1+\frac {g x^2}{f}\right )-\frac {i b e \sqrt {g} \operatorname {PolyLog}\left (2,1+\frac {2 \sqrt {f} \sqrt {g} (1-c x)}{\left (i c \sqrt {f}-\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f}}+\frac {i b e \sqrt {g} \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} (1+c x)}{\left (i c \sqrt {f}+\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f}} \]
[Out]
Time = 0.85 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6229, 2525, 36, 29, 31, 2463, 2441, 2352, 2440, 2438, 6122, 211, 6120, 2520, 12, 266, 6820, 4996, 4940, 4966, 2449, 2497} \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^2} \, dx=\frac {2 a e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}-\frac {b e \sqrt {g} \log \left (1-\frac {1}{c x}\right ) \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {b e \sqrt {g} \log \left (\frac {1}{c x}+1\right ) \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} (1-c x)}{\left (-\sqrt {g}+i c \sqrt {f}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f}}-\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} (c x+1)}{\left (\sqrt {g}+i c \sqrt {f}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f}}-\frac {1}{2} b c \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )+\frac {1}{2} b c \log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {1}{2} b c e \operatorname {PolyLog}\left (2,\frac {g x^2}{f}+1\right )-\frac {i b e \sqrt {g} \operatorname {PolyLog}\left (2,\frac {2 \sqrt {f} \sqrt {g} (1-c x)}{\left (i c \sqrt {f}-\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+1\right )}{2 \sqrt {f}}+\frac {i b e \sqrt {g} \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} (c x+1)}{\left (i \sqrt {f} c+\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f}} \]
[In]
[Out]
Rule 12
Rule 29
Rule 31
Rule 36
Rule 211
Rule 266
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2449
Rule 2463
Rule 2497
Rule 2520
Rule 2525
Rule 4940
Rule 4966
Rule 4996
Rule 6120
Rule 6122
Rule 6229
Rule 6820
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+(b c) \int \frac {d+e \log \left (f+g x^2\right )}{x \left (1-c^2 x^2\right )} \, dx+(2 e g) \int \frac {a+b \coth ^{-1}(c x)}{f+g x^2} \, dx \\ & = -\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} (b c) \text {Subst}\left (\int \frac {d+e \log (f+g x)}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )+(2 a e g) \int \frac {1}{f+g x^2} \, dx+(2 b e g) \int \frac {\coth ^{-1}(c x)}{f+g x^2} \, dx \\ & = \frac {2 a e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} (b c) \text {Subst}\left (\int \left (\frac {d+e \log (f+g x)}{x}-\frac {c^2 (d+e \log (f+g x))}{-1+c^2 x}\right ) \, dx,x,x^2\right )-(b e g) \int \frac {\log \left (1-\frac {1}{c x}\right )}{f+g x^2} \, dx+(b e g) \int \frac {\log \left (1+\frac {1}{c x}\right )}{f+g x^2} \, dx \\ & = \frac {2 a e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1-\frac {1}{c x}\right )}{\sqrt {f}}+\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {1}{c x}\right )}{\sqrt {f}}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} (b c) \text {Subst}\left (\int \frac {d+e \log (f+g x)}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (b c^3\right ) \text {Subst}\left (\int \frac {d+e \log (f+g x)}{-1+c^2 x} \, dx,x,x^2\right )+\frac {(b e g) \int \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g} \left (1-\frac {1}{c x}\right ) x^2} \, dx}{c}+\frac {(b e g) \int \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g} \left (1+\frac {1}{c x}\right ) x^2} \, dx}{c} \\ & = \frac {2 a e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1-\frac {1}{c x}\right )}{\sqrt {f}}+\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {1}{c x}\right )}{\sqrt {f}}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} b c \log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {\left (b e \sqrt {g}\right ) \int \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\left (1-\frac {1}{c x}\right ) x^2} \, dx}{c \sqrt {f}}+\frac {\left (b e \sqrt {g}\right ) \int \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\left (1+\frac {1}{c x}\right ) x^2} \, dx}{c \sqrt {f}}-\frac {1}{2} (b c e g) \text {Subst}\left (\int \frac {\log \left (-\frac {g x}{f}\right )}{f+g x} \, dx,x,x^2\right )+\frac {1}{2} (b c e g) \text {Subst}\left (\int \frac {\log \left (\frac {g \left (-1+c^2 x\right )}{-c^2 f-g}\right )}{f+g x} \, dx,x,x^2\right ) \\ & = \frac {2 a e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1-\frac {1}{c x}\right )}{\sqrt {f}}+\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {1}{c x}\right )}{\sqrt {f}}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} b c \log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {1}{2} b c e \operatorname {PolyLog}\left (2,1+\frac {g x^2}{f}\right )+\frac {1}{2} (b c e) \text {Subst}\left (\int \frac {\log \left (1+\frac {c^2 x}{-c^2 f-g}\right )}{x} \, dx,x,f+g x^2\right )+\frac {\left (b e \sqrt {g}\right ) \int \frac {c \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x (-1+c x)} \, dx}{c \sqrt {f}}+\frac {\left (b e \sqrt {g}\right ) \int \frac {c \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x (1+c x)} \, dx}{c \sqrt {f}} \\ & = \frac {2 a e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1-\frac {1}{c x}\right )}{\sqrt {f}}+\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {1}{c x}\right )}{\sqrt {f}}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} b c \log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c e \operatorname {PolyLog}\left (2,\frac {c^2 \left (f+g x^2\right )}{c^2 f+g}\right )+\frac {1}{2} b c e \operatorname {PolyLog}\left (2,1+\frac {g x^2}{f}\right )+\frac {\left (b e \sqrt {g}\right ) \int \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x (-1+c x)} \, dx}{\sqrt {f}}+\frac {\left (b e \sqrt {g}\right ) \int \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x (1+c x)} \, dx}{\sqrt {f}} \\ & = \frac {2 a e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1-\frac {1}{c x}\right )}{\sqrt {f}}+\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {1}{c x}\right )}{\sqrt {f}}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} b c \log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c e \operatorname {PolyLog}\left (2,\frac {c^2 \left (f+g x^2\right )}{c^2 f+g}\right )+\frac {1}{2} b c e \operatorname {PolyLog}\left (2,1+\frac {g x^2}{f}\right )+\frac {\left (b e \sqrt {g}\right ) \int \left (-\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x}+\frac {c \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{-1+c x}\right ) \, dx}{\sqrt {f}}+\frac {\left (b e \sqrt {g}\right ) \int \left (\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x}-\frac {c \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{1+c x}\right ) \, dx}{\sqrt {f}} \\ & = \frac {2 a e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1-\frac {1}{c x}\right )}{\sqrt {f}}+\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {1}{c x}\right )}{\sqrt {f}}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} b c \log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c e \operatorname {PolyLog}\left (2,\frac {c^2 \left (f+g x^2\right )}{c^2 f+g}\right )+\frac {1}{2} b c e \operatorname {PolyLog}\left (2,1+\frac {g x^2}{f}\right )+\frac {\left (b c e \sqrt {g}\right ) \int \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{-1+c x} \, dx}{\sqrt {f}}-\frac {\left (b c e \sqrt {g}\right ) \int \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{1+c x} \, dx}{\sqrt {f}} \\ & = \frac {2 a e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1-\frac {1}{c x}\right )}{\sqrt {f}}+\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {1}{c x}\right )}{\sqrt {f}}+\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} (1-c x)}{\left (i c \sqrt {f}-\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f}}-\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} (1+c x)}{\left (i c \sqrt {f}+\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f}}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} b c \log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c e \operatorname {PolyLog}\left (2,\frac {c^2 \left (f+g x^2\right )}{c^2 f+g}\right )+\frac {1}{2} b c e \operatorname {PolyLog}\left (2,1+\frac {g x^2}{f}\right )-\frac {(b e g) \int \frac {\log \left (\frac {2 \sqrt {g} (-1+c x)}{\sqrt {f} \left (i c-\frac {\sqrt {g}}{\sqrt {f}}\right ) \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}\right )}{1+\frac {g x^2}{f}} \, dx}{f}+\frac {(b e g) \int \frac {\log \left (\frac {2 \sqrt {g} (1+c x)}{\sqrt {f} \left (i c+\frac {\sqrt {g}}{\sqrt {f}}\right ) \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}\right )}{1+\frac {g x^2}{f}} \, dx}{f} \\ & = \frac {2 a e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1-\frac {1}{c x}\right )}{\sqrt {f}}+\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {1}{c x}\right )}{\sqrt {f}}+\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} (1-c x)}{\left (i c \sqrt {f}-\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f}}-\frac {b e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} (1+c x)}{\left (i c \sqrt {f}+\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f}}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} b c \log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c e \operatorname {PolyLog}\left (2,\frac {c^2 \left (f+g x^2\right )}{c^2 f+g}\right )+\frac {1}{2} b c e \operatorname {PolyLog}\left (2,1+\frac {g x^2}{f}\right )-\frac {i b e \sqrt {g} \operatorname {PolyLog}\left (2,1+\frac {2 \sqrt {f} \sqrt {g} (1-c x)}{\left (i c \sqrt {f}-\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f}}+\frac {i b e \sqrt {g} \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} (1+c x)}{\left (i c \sqrt {f}+\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f}} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1236\) vs. \(2(560)=1120\).
Time = 2.52 (sec) , antiderivative size = 1236, normalized size of antiderivative = 2.21 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^2} \, dx=-\frac {a d}{x}-\frac {b d \coth ^{-1}(c x)}{x}+b c d \log (x)-\frac {1}{2} b c d \log \left (1-c^2 x^2\right )+a e \left (\frac {2 \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {\log \left (f+g x^2\right )}{x}\right )+\frac {1}{2} b e \left (-\frac {\left (2 \coth ^{-1}(c x)+c x \left (-2 \log (x)+\log \left (1-c^2 x^2\right )\right )\right ) \log \left (f+g x^2\right )}{x}-2 c \left (\log (x) \left (\log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )+\log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right )\right )+\operatorname {PolyLog}\left (2,-\frac {i \sqrt {g} x}{\sqrt {f}}\right )+\operatorname {PolyLog}\left (2,\frac {i \sqrt {g} x}{\sqrt {f}}\right )\right )+c \left (\log \left (-\frac {1}{c}+x\right ) \log \left (\frac {c \left (\sqrt {f}-i \sqrt {g} x\right )}{c \sqrt {f}-i \sqrt {g}}\right )+\log \left (\frac {1}{c}+x\right ) \log \left (\frac {c \left (\sqrt {f}-i \sqrt {g} x\right )}{c \sqrt {f}+i \sqrt {g}}\right )+\log \left (-\frac {1}{c}+x\right ) \log \left (\frac {c \left (\sqrt {f}+i \sqrt {g} x\right )}{c \sqrt {f}+i \sqrt {g}}\right )-\left (\log \left (-\frac {1}{c}+x\right )+\log \left (\frac {1}{c}+x\right )-\log \left (1-c^2 x^2\right )\right ) \log \left (f+g x^2\right )+\log \left (\frac {1}{c}+x\right ) \log \left (1-\frac {\sqrt {g} (1+c x)}{i c \sqrt {f}+\sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {c \sqrt {g} \left (\frac {1}{c}+x\right )}{i c \sqrt {f}+\sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {i \sqrt {g} (-1+c x)}{c \sqrt {f}-i \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,-\frac {i \sqrt {g} (-1+c x)}{c \sqrt {f}+i \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {i \sqrt {g} (1+c x)}{c \sqrt {f}+i \sqrt {g}}\right )\right )-\frac {c g \left (2 i \arccos \left (\frac {c^2 f-g}{c^2 f+g}\right ) \arctan \left (\frac {c f}{\sqrt {c^2 f g} x}\right )-4 \coth ^{-1}(c x) \arctan \left (\frac {c g x}{\sqrt {c^2 f g}}\right )+\left (\arccos \left (\frac {c^2 f-g}{c^2 f+g}\right )+2 \arctan \left (\frac {c f}{\sqrt {c^2 f g} x}\right )\right ) \log \left (\frac {2 g \left (c^2 f-i \sqrt {c^2 f g}\right ) (-1+c x)}{\left (c^2 f+g\right ) \left (i \sqrt {c^2 f g}+c g x\right )}\right )+\left (\arccos \left (\frac {c^2 f-g}{c^2 f+g}\right )-2 \arctan \left (\frac {c f}{\sqrt {c^2 f g} x}\right )\right ) \log \left (\frac {2 g \left (c^2 f+i \sqrt {c^2 f g}\right ) (1+c x)}{\left (c^2 f+g\right ) \left (i \sqrt {c^2 f g}+c g x\right )}\right )-\left (\arccos \left (\frac {c^2 f-g}{c^2 f+g}\right )+2 \left (\arctan \left (\frac {c f}{\sqrt {c^2 f g} x}\right )+\arctan \left (\frac {c g x}{\sqrt {c^2 f g}}\right )\right )\right ) \log \left (\frac {\sqrt {2} e^{-\coth ^{-1}(c x)} \sqrt {c^2 f g}}{\sqrt {c^2 f+g} \sqrt {-c^2 f+g+\left (c^2 f+g\right ) \cosh \left (2 \coth ^{-1}(c x)\right )}}\right )-\left (\arccos \left (\frac {c^2 f-g}{c^2 f+g}\right )-2 \left (\arctan \left (\frac {c f}{\sqrt {c^2 f g} x}\right )+\arctan \left (\frac {c g x}{\sqrt {c^2 f g}}\right )\right )\right ) \log \left (\frac {\sqrt {2} e^{\coth ^{-1}(c x)} \sqrt {c^2 f g}}{\sqrt {c^2 f+g} \sqrt {-c^2 f+g+\left (c^2 f+g\right ) \cosh \left (2 \coth ^{-1}(c x)\right )}}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (c^2 f-g-2 i \sqrt {c^2 f g}\right ) \left (\sqrt {c^2 f g}+i c g x\right )}{\left (c^2 f+g\right ) \left (\sqrt {c^2 f g}-i c g x\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (c^2 f-g+2 i \sqrt {c^2 f g}\right ) \left (\sqrt {c^2 f g}+i c g x\right )}{\left (c^2 f+g\right ) \left (\sqrt {c^2 f g}-i c g x\right )}\right )\right )\right )}{\sqrt {c^2 f g}}\right ) \]
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\[\int \frac {\left (a +b \,\operatorname {arccoth}\left (c x \right )\right ) \left (d +e \ln \left (g \,x^{2}+f \right )\right )}{x^{2}}d x\]
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\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right )}{x^2} \,d x \]
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