Optimal. Leaf size=712 \[ \frac{1}{4} b c^2 e \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}-\sqrt{g}\right )}\right )+\frac{1}{4} b c^2 e \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}+\sqrt{g}\right )}\right )-\frac{1}{2} b c^2 e \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )+\frac{b e g \text{PolyLog}\left (2,-\frac{1}{c x}\right )}{2 f}-\frac{b e g \text{PolyLog}\left (2,\frac{1}{c x}\right )}{2 f}-\frac{b e g \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 f}+\frac{b e g \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}-\sqrt{g}\right )}\right )}{4 f}+\frac{b e g \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}+\sqrt{g}\right )}\right )}{4 f}-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac{a e g \log \left (f+g x^2\right )}{2 f}+\frac{a e g \log (x)}{f}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}-\sqrt{g}\right )}\right )-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}+\sqrt{g}\right )}\right )+b c^2 e \log \left (\frac{2}{c x+1}\right ) \tanh ^{-1}(c x)-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac{b c e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}+\frac{b e g \log \left (\frac{2}{c x+1}\right ) \coth ^{-1}(c x)}{f}-\frac{b e g \coth ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}-\sqrt{g}\right )}\right )}{2 f}-\frac{b e g \coth ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}+\sqrt{g}\right )}\right )}{2 f} \]
[Out]
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Rubi [A] time = 1.11314, antiderivative size = 712, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 17, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.708, Rules used = {5917, 325, 206, 6086, 6725, 801, 635, 205, 260, 5993, 5913, 5921, 2402, 2315, 2447, 5992, 5920} \[ \frac{1}{4} b c^2 e \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}-\sqrt{g}\right )}\right )+\frac{1}{4} b c^2 e \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}+\sqrt{g}\right )}\right )-\frac{1}{2} b c^2 e \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )+\frac{b e g \text{PolyLog}\left (2,-\frac{1}{c x}\right )}{2 f}-\frac{b e g \text{PolyLog}\left (2,\frac{1}{c x}\right )}{2 f}-\frac{b e g \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 f}+\frac{b e g \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}-\sqrt{g}\right )}\right )}{4 f}+\frac{b e g \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}+\sqrt{g}\right )}\right )}{4 f}-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac{a e g \log \left (f+g x^2\right )}{2 f}+\frac{a e g \log (x)}{f}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}-\sqrt{g}\right )}\right )-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}+\sqrt{g}\right )}\right )+b c^2 e \log \left (\frac{2}{c x+1}\right ) \tanh ^{-1}(c x)-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac{b c e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}+\frac{b e g \log \left (\frac{2}{c x+1}\right ) \coth ^{-1}(c x)}{f}-\frac{b e g \coth ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}-\sqrt{g}\right )}\right )}{2 f}-\frac{b e g \coth ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}+\sqrt{g}\right )}\right )}{2 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5917
Rule 325
Rule 206
Rule 6086
Rule 6725
Rule 801
Rule 635
Rule 205
Rule 260
Rule 5993
Rule 5913
Rule 5921
Rule 2402
Rule 2315
Rule 2447
Rule 5992
Rule 5920
Rubi steps
\begin{align*} \int \frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx &=-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-(2 e g) \int \left (\frac{-a-b c x-b \coth ^{-1}(c x)}{2 x \left (f+g x^2\right )}+\frac{b c^2 x \tanh ^{-1}(c x)}{2 \left (f+g x^2\right )}\right ) \, dx\\ &=-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-(e g) \int \frac{-a-b c x-b \coth ^{-1}(c x)}{x \left (f+g x^2\right )} \, dx-\left (b c^2 e g\right ) \int \frac{x \tanh ^{-1}(c x)}{f+g x^2} \, dx\\ &=-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-(e g) \int \left (\frac{-a-b c x}{x \left (f+g x^2\right )}-\frac{b \coth ^{-1}(c x)}{x \left (f+g x^2\right )}\right ) \, dx-\left (b c^2 e g\right ) \int \left (-\frac{\tanh ^{-1}(c x)}{2 \sqrt{g} \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{\tanh ^{-1}(c x)}{2 \sqrt{g} \left (\sqrt{-f}+\sqrt{g} x\right )}\right ) \, dx\\ &=-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac{1}{2} \left (b c^2 e \sqrt{g}\right ) \int \frac{\tanh ^{-1}(c x)}{\sqrt{-f}-\sqrt{g} x} \, dx-\frac{1}{2} \left (b c^2 e \sqrt{g}\right ) \int \frac{\tanh ^{-1}(c x)}{\sqrt{-f}+\sqrt{g} x} \, dx-(e g) \int \frac{-a-b c x}{x \left (f+g x^2\right )} \, dx+(b e g) \int \frac{\coth ^{-1}(c x)}{x \left (f+g x^2\right )} \, dx\\ &=b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2}{1+c x}\right )-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-2 \left (\frac{1}{2} \left (b c^3 e\right ) \int \frac{\log \left (\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx\right )+\frac{1}{2} \left (b c^3 e\right ) \int \frac{\log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx+\frac{1}{2} \left (b c^3 e\right ) \int \frac{\log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx-(e g) \int \left (-\frac{a}{f x}+\frac{-b c f+a g x}{f \left (f+g x^2\right )}\right ) \, dx+(b e g) \int \left (\frac{\coth ^{-1}(c x)}{f x}-\frac{g x \coth ^{-1}(c x)}{f \left (f+g x^2\right )}\right ) \, dx\\ &=\frac{a e g \log (x)}{f}+b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2}{1+c x}\right )-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac{1}{4} b c^2 e \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )+\frac{1}{4} b c^2 e \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )-2 \left (\frac{1}{2} \left (b c^2 e\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+c x}\right )\right )-\frac{(e g) \int \frac{-b c f+a g x}{f+g x^2} \, dx}{f}+\frac{(b e g) \int \frac{\coth ^{-1}(c x)}{x} \, dx}{f}-\frac{\left (b e g^2\right ) \int \frac{x \coth ^{-1}(c x)}{f+g x^2} \, dx}{f}\\ &=\frac{a e g \log (x)}{f}+b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2}{1+c x}\right )-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac{b e g \text{Li}_2\left (-\frac{1}{c x}\right )}{2 f}-\frac{b e g \text{Li}_2\left (\frac{1}{c x}\right )}{2 f}-\frac{1}{2} b c^2 e \text{Li}_2\left (1-\frac{2}{1+c x}\right )+\frac{1}{4} b c^2 e \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )+\frac{1}{4} b c^2 e \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )+(b c e g) \int \frac{1}{f+g x^2} \, dx-\frac{\left (a e g^2\right ) \int \frac{x}{f+g x^2} \, dx}{f}-\frac{\left (b e g^2\right ) \int \left (-\frac{\coth ^{-1}(c x)}{2 \sqrt{g} \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{\coth ^{-1}(c x)}{2 \sqrt{g} \left (\sqrt{-f}+\sqrt{g} x\right )}\right ) \, dx}{f}\\ &=\frac{b c e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}+\frac{a e g \log (x)}{f}+b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2}{1+c x}\right )-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )-\frac{a e g \log \left (f+g x^2\right )}{2 f}-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac{b e g \text{Li}_2\left (-\frac{1}{c x}\right )}{2 f}-\frac{b e g \text{Li}_2\left (\frac{1}{c x}\right )}{2 f}-\frac{1}{2} b c^2 e \text{Li}_2\left (1-\frac{2}{1+c x}\right )+\frac{1}{4} b c^2 e \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )+\frac{1}{4} b c^2 e \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )+\frac{\left (b e g^{3/2}\right ) \int \frac{\coth ^{-1}(c x)}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 f}-\frac{\left (b e g^{3/2}\right ) \int \frac{\coth ^{-1}(c x)}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 f}\\ &=\frac{b c e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}+\frac{a e g \log (x)}{f}+\frac{b e g \coth ^{-1}(c x) \log \left (\frac{2}{1+c x}\right )}{f}+b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2}{1+c x}\right )-\frac{b e g \coth ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )}{2 f}-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )-\frac{b e g \coth ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )}{2 f}-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )-\frac{a e g \log \left (f+g x^2\right )}{2 f}-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac{b e g \text{Li}_2\left (-\frac{1}{c x}\right )}{2 f}-\frac{b e g \text{Li}_2\left (\frac{1}{c x}\right )}{2 f}-\frac{1}{2} b c^2 e \text{Li}_2\left (1-\frac{2}{1+c x}\right )+\frac{1}{4} b c^2 e \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )+\frac{1}{4} b c^2 e \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )-2 \frac{(b c e g) \int \frac{\log \left (\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{2 f}+\frac{(b c e g) \int \frac{\log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx}{2 f}+\frac{(b c e g) \int \frac{\log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx}{2 f}\\ &=\frac{b c e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}+\frac{a e g \log (x)}{f}+\frac{b e g \coth ^{-1}(c x) \log \left (\frac{2}{1+c x}\right )}{f}+b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2}{1+c x}\right )-\frac{b e g \coth ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )}{2 f}-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )-\frac{b e g \coth ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )}{2 f}-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )-\frac{a e g \log \left (f+g x^2\right )}{2 f}-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac{b e g \text{Li}_2\left (-\frac{1}{c x}\right )}{2 f}-\frac{b e g \text{Li}_2\left (\frac{1}{c x}\right )}{2 f}-\frac{1}{2} b c^2 e \text{Li}_2\left (1-\frac{2}{1+c x}\right )+\frac{1}{4} b c^2 e \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )+\frac{b e g \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )}{4 f}+\frac{1}{4} b c^2 e \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )+\frac{b e g \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )}{4 f}-2 \frac{(b e g) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+c x}\right )}{2 f}\\ &=\frac{b c e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}+\frac{a e g \log (x)}{f}+\frac{b e g \coth ^{-1}(c x) \log \left (\frac{2}{1+c x}\right )}{f}+b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2}{1+c x}\right )-\frac{b e g \coth ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )}{2 f}-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )-\frac{b e g \coth ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )}{2 f}-\frac{1}{2} b c^2 e \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )-\frac{a e g \log \left (f+g x^2\right )}{2 f}-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac{b e g \text{Li}_2\left (-\frac{1}{c x}\right )}{2 f}-\frac{b e g \text{Li}_2\left (\frac{1}{c x}\right )}{2 f}-\frac{1}{2} b c^2 e \text{Li}_2\left (1-\frac{2}{1+c x}\right )-\frac{b e g \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 f}+\frac{1}{4} b c^2 e \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )+\frac{b e g \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-\sqrt{g}\right ) (1+c x)}\right )}{4 f}+\frac{1}{4} b c^2 e \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )+\frac{b e g \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+\sqrt{g}\right ) (1+c x)}\right )}{4 f}\\ \end{align*}
Mathematica [C] time = 5.57142, size = 1318, normalized size = 1.85 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 3.198, size = 937, normalized size = 1.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b d \operatorname{arcoth}\left (c x\right ) + a d +{\left (b e \operatorname{arcoth}\left (c x\right ) + a e\right )} \log \left (g x^{2} + f\right )}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcoth}\left (c x\right ) + a\right )}{\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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