Optimal. Leaf size=512 \[ \frac{b e \left (c^2 f+g\right ) \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 c^2 g}-\frac{b e \left (c^2 f+g\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}-\sqrt{g}\right )}\right )}{4 c^2 g}-\frac{b e \left (c^2 f+g\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}+\sqrt{g}\right )}\right )}{4 c^2 g}+\frac{1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )+\frac{e \left (f+g x^2\right ) \log \left (f+g x^2\right ) \left (a+b \coth ^{-1}(c x)\right )}{2 g}-\frac{1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}-\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (f+g x^2\right )}{2 c^2 g}-\frac{b e \left (c^2 f+g\right ) \log \left (\frac{2}{c x+1}\right ) \tanh ^{-1}(c x)}{c^2 g}+\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}-\sqrt{g}\right )}\right )}{2 c^2 g}+\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}+\sqrt{g}\right )}\right )}{2 c^2 g}+\frac{b x (d-e)}{2 c}+\frac{b e x \log \left (f+g x^2\right )}{2 c}+\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{c \sqrt{g}}-\frac{b e x}{c} \]
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Rubi [A] time = 0.76723, antiderivative size = 512, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 17, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.773, Rules used = {2454, 2389, 2295, 6084, 321, 207, 517, 2528, 2448, 205, 2470, 12, 5992, 5920, 2402, 2315, 2447} \[ \frac{b e \left (c^2 f+g\right ) \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 c^2 g}-\frac{b e \left (c^2 f+g\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}-\sqrt{g}\right )}\right )}{4 c^2 g}-\frac{b e \left (c^2 f+g\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}+\sqrt{g}\right )}\right )}{4 c^2 g}+\frac{1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )+\frac{e \left (f+g x^2\right ) \log \left (f+g x^2\right ) \left (a+b \coth ^{-1}(c x)\right )}{2 g}-\frac{1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}-\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (f+g x^2\right )}{2 c^2 g}-\frac{b e \left (c^2 f+g\right ) \log \left (\frac{2}{c x+1}\right ) \tanh ^{-1}(c x)}{c^2 g}+\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}-\sqrt{g}\right )}\right )}{2 c^2 g}+\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}+\sqrt{g}\right )}\right )}{2 c^2 g}+\frac{b x (d-e)}{2 c}+\frac{b e x \log \left (f+g x^2\right )}{2 c}+\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{c \sqrt{g}}-\frac{b e x}{c} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2389
Rule 2295
Rule 6084
Rule 321
Rule 207
Rule 517
Rule 2528
Rule 2448
Rule 205
Rule 2470
Rule 12
Rule 5992
Rule 5920
Rule 2402
Rule 2315
Rule 2447
Rubi steps
\begin{align*} \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx &=\frac{1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )+\frac{e \left (f+g x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}-(b c) \int \left (-\frac{(d-e) x^2}{2 \left (-1+c^2 x^2\right )}-\frac{e \left (f+g x^2\right ) \log \left (f+g x^2\right )}{2 g (-1+c x) (1+c x)}\right ) \, dx\\ &=\frac{1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )+\frac{e \left (f+g x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}+\frac{1}{2} (b c (d-e)) \int \frac{x^2}{-1+c^2 x^2} \, dx+\frac{(b c e) \int \frac{\left (f+g x^2\right ) \log \left (f+g x^2\right )}{(-1+c x) (1+c x)} \, dx}{2 g}\\ &=\frac{b (d-e) x}{2 c}+\frac{1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )+\frac{e \left (f+g x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}+\frac{(b (d-e)) \int \frac{1}{-1+c^2 x^2} \, dx}{2 c}+\frac{(b c e) \int \frac{\left (f+g x^2\right ) \log \left (f+g x^2\right )}{-1+c^2 x^2} \, dx}{2 g}\\ &=\frac{b (d-e) x}{2 c}+\frac{1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac{e \left (f+g x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}+\frac{(b c e) \int \left (\frac{g \log \left (f+g x^2\right )}{c^2}+\frac{\left (c^2 f+g\right ) \log \left (f+g x^2\right )}{c^2 \left (-1+c^2 x^2\right )}\right ) \, dx}{2 g}\\ &=\frac{b (d-e) x}{2 c}+\frac{1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac{e \left (f+g x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}+\frac{(b e) \int \log \left (f+g x^2\right ) \, dx}{2 c}+\frac{\left (b e \left (c^2 f+g\right )\right ) \int \frac{\log \left (f+g x^2\right )}{-1+c^2 x^2} \, dx}{2 c g}\\ &=\frac{b (d-e) x}{2 c}+\frac{1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac{b e x \log \left (f+g x^2\right )}{2 c}+\frac{e \left (f+g x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}-\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (f+g x^2\right )}{2 c^2 g}-\frac{(b e g) \int \frac{x^2}{f+g x^2} \, dx}{c}+\frac{\left (b e \left (c^2 f+g\right )\right ) \int \frac{x \tanh ^{-1}(c x)}{c \left (f+g x^2\right )} \, dx}{c}\\ &=\frac{b (d-e) x}{2 c}-\frac{b e x}{c}+\frac{1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac{b e x \log \left (f+g x^2\right )}{2 c}+\frac{e \left (f+g x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}-\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (f+g x^2\right )}{2 c^2 g}+\frac{(b e f) \int \frac{1}{f+g x^2} \, dx}{c}+\frac{\left (b e \left (c^2 f+g\right )\right ) \int \frac{x \tanh ^{-1}(c x)}{f+g x^2} \, dx}{c^2}\\ &=\frac{b (d-e) x}{2 c}-\frac{b e x}{c}+\frac{1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )+\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{c \sqrt{g}}-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac{b e x \log \left (f+g x^2\right )}{2 c}+\frac{e \left (f+g x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}-\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (f+g x^2\right )}{2 c^2 g}+\frac{\left (b e \left (c^2 f+g\right )\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}{c^2}\\ &=\frac{b (d-e) x}{2 c}-\frac{b e x}{c}+\frac{1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )+\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{c \sqrt{g}}-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac{b e x \log \left (f+g x^2\right )}{2 c}+\frac{e \left (f+g x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}-\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (f+g x^2\right )}{2 c^2 g}-\frac{\left (b e \left (c^2 f+g\right )\right ) \int \frac{\tanh ^{-1}(c x)}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 c^2 \sqrt{g}}+\frac{\left (b e \left (c^2 f+g\right )\right ) \int \frac{\tanh ^{-1}(c x)}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 c^2 \sqrt{g}}\\ &=\frac{b (d-e) x}{2 c}-\frac{b e x}{c}+\frac{1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )+\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{c \sqrt{g}}-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}-\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac{2}{1+c x}\right )}{c^2 g}+\frac{b e \left (c^2 f+g\right ) \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 )}{2 c^2 g}+\frac{b e \left (c^2 f+g\right ) \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 )}{2 c^2 g}+\frac{b e x \log \left (f+g x^2\right )}{2 c}+\frac{e \left (f+g x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}-\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (f+g x^2\right )}{2 c^2 g}+2 \frac{\left (b e \left (c^2 f+g\right )\right ) \int \frac{\log \left (\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{2 c g}-\frac{\left (b e \left (c^2 f+g\right )\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}{2 c g}-\frac{\left (b e \left (c^2 f+g\right )\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}{2 c g}\\ &=\frac{b (d-e) x}{2 c}-\frac{b e x}{c}+\frac{1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )+\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{c \sqrt{g}}-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}-\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac{2}{1+c x}\right )}{c^2 g}+\frac{b e \left (c^2 f+g\right ) \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 )}{2 c^2 g}+\frac{b e \left (c^2 f+g\right ) \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 )}{2 c^2 g}+\frac{b e x \log \left (f+g x^2\right )}{2 c}+\frac{e \left (f+g x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}-\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (f+g x^2\right )}{2 c^2 g}-\frac{b e \left (c^2 f+g\right ) \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 c^2 g}-\frac{b e \left (c^2 f+g\right ) \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 c^2 g}+2 \frac{\left (b e \left (c^2 f+g\right )\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+c x}\right )}{2 c^2 g}\\ &=\frac{b (d-e) x}{2 c}-\frac{b e x}{c}+\frac{1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )+\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{c \sqrt{g}}-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}-\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac{2}{1+c x}\right )}{c^2 g}+\frac{b e \left (c^2 f+g\right ) \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 )}{2 c^2 g}+\frac{b e \left (c^2 f+g\right ) \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 )}{2 c^2 g}+\frac{b e x \log \left (f+g x^2\right )}{2 c}+\frac{e \left (f+g x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}-\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (f+g x^2\right )}{2 c^2 g}+\frac{b e \left (c^2 f+g\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 c^2 g}-\frac{b e \left (c^2 f+g\right ) \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 c^2 g}-\frac{b e \left (c^2 f+g\right ) \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 c^2 g}\\ \end{align*}
Mathematica [A] time = 4.25167, size = 677, normalized size = 1.32 \[ \frac{b c^2 e f \text{PolyLog}\left (2,\frac{\left (c^2 f+g\right ) e^{2 \coth ^{-1}(c x)}}{c^2 f+2 c \sqrt{-f} \sqrt{g}-g}\right )+2 b e \left (c^2 f+g\right ) \text{PolyLog}\left (2,e^{-2 \coth ^{-1}(c x)}\right )+b e \left (c^2 f+g\right ) \text{PolyLog}\left (2,\frac{\left (c^2 f+g\right ) e^{2 \coth ^{-1}(c x)}}{c^2 f-2 c \sqrt{-f} \sqrt{g}-g}\right )+b e g \text{PolyLog}\left (2,\frac{\left (c^2 f+g\right ) e^{2 \coth ^{-1}(c x)}}{c^2 f+2 c \sqrt{-f} \sqrt{g}-g}\right )+2 a c^2 d g x^2+2 a c^2 e g x^2 \log \left (f+g x^2\right )+2 a c^2 e f \log \left (f+g x^2\right )-2 a c^2 e g x^2+2 b c^2 d g x^2 \coth ^{-1}(c x)+2 b c^2 e g x^2 \coth ^{-1}(c x) \log \left (f+g x^2\right )+2 b c^2 e f \coth ^{-1}(c x) \log \left (\frac{\left (c^2 f+g\right ) e^{2 \coth ^{-1}(c x)}}{c^2 (-f)-2 c \sqrt{-f} \sqrt{g}+g}+1\right )+2 b c^2 e f \coth ^{-1}(c x) \log \left (\frac{\left (c^2 f+g\right ) e^{2 \coth ^{-1}(c x)}}{c^2 (-f)+2 c \sqrt{-f} \sqrt{g}+g}+1\right )+2 b e g \coth ^{-1}(c x) \log \left (\frac{\left (c^2 f+g\right ) e^{2 \coth ^{-1}(c x)}}{c^2 (-f)-2 c \sqrt{-f} \sqrt{g}+g}+1\right )+2 b e g \coth ^{-1}(c x) \log \left (\frac{\left (c^2 f+g\right ) e^{2 \coth ^{-1}(c x)}}{c^2 (-f)+2 c \sqrt{-f} \sqrt{g}+g}+1\right )-4 b c^2 e f \coth ^{-1}(c x)^2-4 b c^2 e f \coth ^{-1}(c x) \log \left (1-e^{-2 \coth ^{-1}(c x)}\right )-2 b c^2 e g x^2 \coth ^{-1}(c x)+2 b c d g x-2 b d g \coth ^{-1}(c x)+2 b c e g x \log \left (f+g x^2\right )-2 b e g \coth ^{-1}(c x) \log \left (f+g x^2\right )+4 b c e \sqrt{f} \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )-6 b c e g x-4 b e g \coth ^{-1}(c x)^2+2 b e g \coth ^{-1}(c x)-4 b e g \coth ^{-1}(c x) \log \left (1-e^{-2 \coth ^{-1}(c x)}\right )}{4 c^2 g} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 1.996, size = 8491, normalized size = 16.6 \begin{align*} \text{output 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 (b d x \operatorname{arcoth}\left (c x\right ) + a d x +{\left (b e x \operatorname{arcoth}\left (c x\right ) + a e x\right )} \log \left (g x^{2} + f\right ), 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{\left (b \operatorname{arcoth}\left (c x\right ) + a\right )}{\left (e \log \left (g x^{2} + f\right ) + d\right )} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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