Optimal. Leaf size=512 \[ \frac {1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )+\frac {e \left (f+g x^2\right ) \log \left (f+g x^2\right ) \left (a+b \coth ^{-1}(c x)\right )}{2 g}-\frac {1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac {b e \left (c^2 f+g\right ) \text {Li}_2\left (1-\frac {2}{c x+1}\right )}{2 c^2 g}-\frac {b e \left (c^2 f+g\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (c x+1)}\right )}{4 c^2 g}-\frac {b e \left (c^2 f+g\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {g} x+\sqrt {-f}\right )}{\left (\sqrt {-f} c+\sqrt {g}\right ) (c x+1)}\right )}{4 c^2 g}-\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (f+g x^2\right )}{2 c^2 g}-\frac {b e \left (c^2 f+g\right ) \log \left (\frac {2}{c x+1}\right ) \tanh ^{-1}(c x)}{c^2 g}+\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}-\sqrt {g}\right )}\right )}{2 c^2 g}+\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}+\sqrt {g}\right )}\right )}{2 c^2 g}+\frac {b x (d-e)}{2 c}+\frac {b e x \log \left (f+g x^2\right )}{2 c}+\frac {b e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c \sqrt {g}}-\frac {b e x}{c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.77, antiderivative size = 512, normalized size of antiderivative = 1.00, 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.
[In]
[Out]
Rule 12
Rule 205
Rule 207
Rule 321
Rule 517
Rule 2295
Rule 2315
Rule 2389
Rule 2402
Rule 2447
Rule 2448
Rule 2454
Rule 2470
Rule 2528
Rule 5920
Rule 5992
Rule 6084
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 = 4.43, size = 677, normalized size = 1.32 \[ \frac {2 a c^2 d g x^2+2 a c^2 e g x^2 \log \left (f+g x^2\right )+2 a c^2 e f \log \left (f+g x^2\right )-2 a c^2 e g x^2+2 b c^2 d g x^2 \coth ^{-1}(c x)+b c^2 e f \text {Li}_2\left (\frac {e^{2 \coth ^{-1}(c x)} \left (f c^2+g\right )}{f c^2+2 \sqrt {-f} \sqrt {g} c-g}\right )+2 b e \left (c^2 f+g\right ) \text {Li}_2\left (e^{-2 \coth ^{-1}(c x)}\right )+b e \left (c^2 f+g\right ) \text {Li}_2\left (\frac {e^{2 \coth ^{-1}(c x)} \left (f c^2+g\right )}{f c^2-2 \sqrt {-f} \sqrt {g} c-g}\right )+b e g \text {Li}_2\left (\frac {e^{2 \coth ^{-1}(c x)} \left (f c^2+g\right )}{f c^2+2 \sqrt {-f} \sqrt {g} c-g}\right )+2 b c^2 e g x^2 \coth ^{-1}(c x) \log \left (f+g x^2\right )+2 b c^2 e f \coth ^{-1}(c x) \log \left (\frac {\left (c^2 f+g\right ) e^{2 \coth ^{-1}(c x)}}{c^2 (-f)-2 c \sqrt {-f} \sqrt {g}+g}+1\right )+2 b c^2 e f \coth ^{-1}(c x) \log \left (\frac {\left (c^2 f+g\right ) e^{2 \coth ^{-1}(c x)}}{c^2 (-f)+2 c \sqrt {-f} \sqrt {g}+g}+1\right )+2 b e g \coth ^{-1}(c x) \log \left (\frac {\left (c^2 f+g\right ) e^{2 \coth ^{-1}(c x)}}{c^2 (-f)-2 c \sqrt {-f} \sqrt {g}+g}+1\right )+2 b e g \coth ^{-1}(c x) \log \left (\frac {\left (c^2 f+g\right ) e^{2 \coth ^{-1}(c x)}}{c^2 (-f)+2 c \sqrt {-f} \sqrt {g}+g}+1\right )-4 b c^2 e f \coth ^{-1}(c x)^2-4 b c^2 e f \coth ^{-1}(c x) \log \left (1-e^{-2 \coth ^{-1}(c x)}\right )-2 b c^2 e g x^2 \coth ^{-1}(c x)+2 b c d g x-2 b d g \coth ^{-1}(c x)+2 b c e g x \log \left (f+g x^2\right )-2 b e g \coth ^{-1}(c x) \log \left (f+g x^2\right )+4 b c e \sqrt {f} \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )-6 b c e g x-4 b e g \coth ^{-1}(c x)^2+2 b e g \coth ^{-1}(c x)-4 b e g \coth ^{-1}(c x) \log \left (1-e^{-2 \coth ^{-1}(c x)}\right )}{4 c^2 g} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b d x \operatorname {arcoth}\left (c x\right ) + a d x + {\left (b e x \operatorname {arcoth}\left (c x\right ) + a e x\right )} \log \left (g x^{2} + f\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 3.14, size = 8491, normalized size = 16.58 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a d x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcoth}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} b d - \frac {1}{4} \, {\left (2 \, c^{2} g \int \frac {x^{3} \log \left (c x + 1\right )}{c^{2} g x^{2} + c^{2} f}\,{d x} - 2 \, c^{2} g \int \frac {x^{3} \log \left (c x - 1\right )}{c^{2} g x^{2} + c^{2} f}\,{d x} - 2 \, c g {\left (-\frac {i \, f {\left (\log \left (\frac {i \, g x}{\sqrt {f g}} + 1\right ) - \log \left (-\frac {i \, g x}{\sqrt {f g}} + 1\right )\right )}}{\sqrt {f g} c^{2} g} - \frac {2 \, x}{c^{2} g}\right )} - 2 \, g \int \frac {x \log \left (c x + 1\right )}{c^{2} g x^{2} + c^{2} f}\,{d x} + 2 \, g \int \frac {x \log \left (c x - 1\right )}{c^{2} g x^{2} + c^{2} f}\,{d x} - \frac {{\left (2 \, c x + {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) - {\left (c^{2} x^{2} - 1\right )} \log \left (c x - 1\right )\right )} \log \left (g x^{2} + f\right )}{c^{2}}\right )} b e - \frac {{\left (g x^{2} - {\left (g x^{2} + f\right )} \log \left (g x^{2} + f\right ) + f\right )} a e}{2 \, g} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,\left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________