Optimal. Leaf size=512 \[ \frac {b (d-e) x}{2 c}-\frac {b e x}{c}+\frac {b e \sqrt {f} \text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c \sqrt {g}}-\frac {b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac {2}{1+c x}\right )}{c^2 g}+\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{2 c^2 g}+\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{2 c^2 g}+\frac {b e x \log \left (f+g x^2\right )}{2 c}-\frac {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (f+g x^2\right )}{2 c^2 g}+\frac {e \left (f+g x^2\right ) \left (a+b \tanh ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}+\frac {b e \left (c^2 f+g\right ) \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 c^2 g}-\frac {b e \left (c^2 f+g\right ) \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{4 c^2 g}-\frac {b e \left (c^2 f+g\right ) \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{4 c^2 g} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.54, 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 = {2504, 2436,
2332, 6230, 327, 213, 531, 2608, 2498, 211, 2520, 12, 6139, 6057, 2449, 2352, 2497}
\begin {gather*} \frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {e \left (f+g x^2\right ) \log \left (f+g x^2\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 g}-\frac {1}{2} e x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {b e \sqrt {f} \text {ArcTan}\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 ) \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 x}{c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 211
Rule 213
Rule 327
Rule 531
Rule 2332
Rule 2352
Rule 2436
Rule 2449
Rule 2497
Rule 2498
Rule 2504
Rule 2520
Rule 2608
Rule 6057
Rule 6139
Rule 6230
Rubi steps
\begin {align*} \int x \left (a+b \tanh ^{-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 \tanh ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {e \left (f+g x^2\right ) \left (a+b \tanh ^{-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 \tanh ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {e \left (f+g x^2\right ) \left (a+b \tanh ^{-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 \tanh ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {e \left (f+g x^2\right ) \left (a+b \tanh ^{-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 {b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {e \left (f+g x^2\right ) \left (a+b \tanh ^{-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 {b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {e \left (f+g x^2\right ) \left (a+b \tanh ^{-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 {b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {b e x \log \left (f+g x^2\right )}{2 c}-\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 {e \left (f+g x^2\right ) \left (a+b \tanh ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{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 {b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {b e x \log \left (f+g x^2\right )}{2 c}-\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 {e \left (f+g x^2\right ) \left (a+b \tanh ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{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 {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 {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {b e x \log \left (f+g x^2\right )}{2 c}-\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 {e \left (f+g x^2\right ) \left (a+b \tanh ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{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 {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 {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {b e x \log \left (f+g x^2\right )}{2 c}-\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 {e \left (f+g x^2\right ) \left (a+b \tanh ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{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 {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 {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \tanh ^{-1}(c x)\right )-\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 {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (f+g x^2\right )}{2 c^2 g}+\frac {e \left (f+g x^2\right ) \left (a+b \tanh ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{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 {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 {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \tanh ^{-1}(c x)\right )-\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 {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (f+g x^2\right )}{2 c^2 g}+\frac {e \left (f+g x^2\right ) \left (a+b \tanh ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{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 ) \text {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 {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 {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \tanh ^{-1}(c x)\right )-\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 {b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (f+g x^2\right )}{2 c^2 g}+\frac {e \left (f+g x^2\right ) \left (a+b \tanh ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 4.46, size = 1376, normalized size = 2.69 \begin {gather*} \frac {-4 b c e g x+2 a c^2 (d-e) g x^2+4 b c e \sqrt {f} \sqrt {g} \text {ArcTan}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )+2 a c^2 e f \log \left (f+g x^2\right )+2 e g \left (c x (b+a c x)+b \left (-1+c^2 x^2\right ) \tanh ^{-1}(c x)\right ) \log \left (f+g x^2\right )+b d \left (2 \left (c g x+c^2 f \tanh ^{-1}(c x)^2+\tanh ^{-1}(c x) \left (-g+c^2 g x^2+2 c^2 f \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )-c^2 f \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )\right )-c^2 f \left (2 \tanh ^{-1}(c x) \left (-\tanh ^{-1}(c x)+\log \left (1+\frac {e^{2 \tanh ^{-1}(c x)} \left (c^2 f+g\right )}{c^2 f-2 c \sqrt {-f} \sqrt {g}-g}\right )+\log \left (1+\frac {e^{2 \tanh ^{-1}(c x)} \left (c^2 f+g\right )}{c^2 f+2 c \sqrt {-f} \sqrt {g}-g}\right )\right )+\text {PolyLog}\left (2,-\frac {e^{2 \tanh ^{-1}(c x)} \left (c^2 f+g\right )}{c^2 f-2 c \sqrt {-f} \sqrt {g}-g}\right )+\text {PolyLog}\left (2,-\frac {e^{2 \tanh ^{-1}(c x)} \left (c^2 f+g\right )}{c^2 f+2 c \sqrt {-f} \sqrt {g}-g}\right )\right )\right )+b e \left (-2 \left (c g x+c^2 f \tanh ^{-1}(c x)^2+\tanh ^{-1}(c x) \left (-g+c^2 g x^2+2 c^2 f \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )-c^2 f \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )\right )+c^2 f \left (2 \tanh ^{-1}(c x) \left (-\tanh ^{-1}(c x)+\log \left (1+\frac {e^{2 \tanh ^{-1}(c x)} \left (c^2 f+g\right )}{c^2 f-2 c \sqrt {-f} \sqrt {g}-g}\right )+\log \left (1+\frac {e^{2 \tanh ^{-1}(c x)} \left (c^2 f+g\right )}{c^2 f+2 c \sqrt {-f} \sqrt {g}-g}\right )\right )+\text {PolyLog}\left (2,-\frac {e^{2 \tanh ^{-1}(c x)} \left (c^2 f+g\right )}{c^2 f-2 c \sqrt {-f} \sqrt {g}-g}\right )+\text {PolyLog}\left (2,-\frac {e^{2 \tanh ^{-1}(c x)} \left (c^2 f+g\right )}{c^2 f+2 c \sqrt {-f} \sqrt {g}-g}\right )\right )\right )+b c^2 d f \left (2 \tanh ^{-1}(c x)^2-4 i \text {ArcSin}\left (\sqrt {\frac {c^2 f}{c^2 f+g}}\right ) \tanh ^{-1}\left (\frac {c g x}{\sqrt {-c^2 f g}}\right )-2 \tanh ^{-1}(c x) \left (\tanh ^{-1}(c x)+2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+2 \left (-i \text {ArcSin}\left (\sqrt {\frac {c^2 f}{c^2 f+g}}\right )+\tanh ^{-1}(c x)\right ) \log \left (\frac {e^{-2 \tanh ^{-1}(c x)} \left (c^2 \left (1+e^{2 \tanh ^{-1}(c x)}\right ) f+\left (-1+e^{2 \tanh ^{-1}(c x)}\right ) g-2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )+2 \left (i \text {ArcSin}\left (\sqrt {\frac {c^2 f}{c^2 f+g}}\right )+\tanh ^{-1}(c x)\right ) \log \left (\frac {e^{-2 \tanh ^{-1}(c x)} \left (c^2 \left (1+e^{2 \tanh ^{-1}(c x)}\right ) f+\left (-1+e^{2 \tanh ^{-1}(c x)}\right ) g+2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )+2 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )-\text {PolyLog}\left (2,\frac {e^{-2 \tanh ^{-1}(c x)} \left (-c^2 f+g-2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )-\text {PolyLog}\left (2,\frac {e^{-2 \tanh ^{-1}(c x)} \left (-c^2 f+g+2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )\right )+b e g \left (2 \tanh ^{-1}(c x)^2-4 i \text {ArcSin}\left (\sqrt {\frac {c^2 f}{c^2 f+g}}\right ) \tanh ^{-1}\left (\frac {c g x}{\sqrt {-c^2 f g}}\right )-2 \tanh ^{-1}(c x) \left (\tanh ^{-1}(c x)+2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+2 \left (-i \text {ArcSin}\left (\sqrt {\frac {c^2 f}{c^2 f+g}}\right )+\tanh ^{-1}(c x)\right ) \log \left (\frac {e^{-2 \tanh ^{-1}(c x)} \left (c^2 \left (1+e^{2 \tanh ^{-1}(c x)}\right ) f+\left (-1+e^{2 \tanh ^{-1}(c x)}\right ) g-2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )+2 \left (i \text {ArcSin}\left (\sqrt {\frac {c^2 f}{c^2 f+g}}\right )+\tanh ^{-1}(c x)\right ) \log \left (\frac {e^{-2 \tanh ^{-1}(c x)} \left (c^2 \left (1+e^{2 \tanh ^{-1}(c x)}\right ) f+\left (-1+e^{2 \tanh ^{-1}(c x)}\right ) g+2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )+2 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )-\text {PolyLog}\left (2,\frac {e^{-2 \tanh ^{-1}(c x)} \left (-c^2 f+g-2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )-\text {PolyLog}\left (2,\frac {e^{-2 \tanh ^{-1}(c x)} \left (-c^2 f+g+2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )\right )}{4 c^2 g} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 7.21, size = 10161, normalized size = 19.85
method | result | size |
risch | \(-\frac {3 b e x}{2 c}+\frac {b d x}{2 c}-\frac {a e \,x^{2}}{2}+\frac {b d \ln \left (-c x +1\right )}{4 c^{2}}-\frac {b d \ln \left (c x +1\right )}{4 c^{2}}+\frac {a d \,x^{2}}{2}+\frac {b e \ln \left (c x +1\right )}{4 c^{2}}-\frac {e b \dilog \left (\frac {c \sqrt {-g f}-\left (-c x +1\right ) g +g}{c \sqrt {-g f}+g}\right ) f}{4 g}+\left (\frac {b e \,x^{2} \ln \left (c x +1\right )}{4}-\frac {e \left (b \,x^{2} \ln \left (-c x +1\right ) c^{2}-2 a \,c^{2} x^{2}-2 b c x +b \ln \left (c x +1\right )-b \ln \left (-c x +1\right )\right )}{4 c^{2}}\right ) \ln \left (g \,x^{2}+f \right )+\frac {e b \dilog \left (\frac {c \sqrt {-g f}+\left (c x +1\right ) g -g}{c \sqrt {-g f}-g}\right ) f}{4 g}-\frac {e b \dilog \left (\frac {c \sqrt {-g f}+\left (-c x +1\right ) g -g}{c \sqrt {-g f}-g}\right ) f}{4 g}+\frac {e b \dilog \left (\frac {c \sqrt {-g f}-\left (c x +1\right ) g +g}{c \sqrt {-g f}+g}\right ) f}{4 g}+\frac {e b \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-g f}-\left (c x +1\right ) g +g}{c \sqrt {-g f}+g}\right )}{4 c^{2}}+\frac {e b \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-g f}+\left (c x +1\right ) g -g}{c \sqrt {-g f}-g}\right )}{4 c^{2}}-\frac {e b \ln \left (-c x +1\right ) \ln \left (\frac {c \sqrt {-g f}-\left (-c x +1\right ) g +g}{c \sqrt {-g f}+g}\right )}{4 c^{2}}-\frac {e b \ln \left (-c x +1\right ) \ln \left (\frac {c \sqrt {-g f}+\left (-c x +1\right ) g -g}{c \sqrt {-g f}-g}\right )}{4 c^{2}}+\frac {e b \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-g f}-\left (c x +1\right ) g +g}{c \sqrt {-g f}+g}\right ) f}{4 g}+\frac {e b \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-g f}+\left (c x +1\right ) g -g}{c \sqrt {-g f}-g}\right ) f}{4 g}-\frac {e b \ln \left (-c x +1\right ) \ln \left (\frac {c \sqrt {-g f}-\left (-c x +1\right ) g +g}{c \sqrt {-g f}+g}\right ) f}{4 g}-\frac {e b \ln \left (-c x +1\right ) \ln \left (\frac {c \sqrt {-g f}+\left (-c x +1\right ) g -g}{c \sqrt {-g f}-g}\right ) f}{4 g}+\frac {e f b \arctan \left (\frac {x g}{\sqrt {g f}}\right )}{c \sqrt {g f}}+\frac {e b \ln \left (-c x +1\right ) x^{2}}{4}+\frac {d b \ln \left (c x +1\right ) x^{2}}{4}-\frac {d b \ln \left (-c x +1\right ) x^{2}}{4}+\frac {a e \ln \left (g \,x^{2}+f \right ) f}{2 g}-\frac {b e \,x^{2} \ln \left (c x +1\right )}{4}-\frac {b e \ln \left (-c x +1\right )}{4 c^{2}}-\frac {e b \dilog \left (\frac {c \sqrt {-g f}-\left (-c x +1\right ) g +g}{c \sqrt {-g f}+g}\right )}{4 c^{2}}-\frac {e b \dilog \left (\frac {c \sqrt {-g f}+\left (-c x +1\right ) g -g}{c \sqrt {-g f}-g}\right )}{4 c^{2}}+\frac {e b \dilog \left (\frac {c \sqrt {-g f}-\left (c x +1\right ) g +g}{c \sqrt {-g f}+g}\right )}{4 c^{2}}+\frac {e b \dilog \left (\frac {c \sqrt {-g f}+\left (c x +1\right ) g -g}{c \sqrt {-g f}-g}\right )}{4 c^{2}}\) | \(970\) |
default | \(\text {Expression too large to display}\) | \(10161\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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