Integrand size = 22, antiderivative size = 512 \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\frac {b (d-e) x}{2 c}-\frac {b e x}{c}+\frac {1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )+\frac {b e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c \sqrt {g}}-\frac {b (d-e) \text {arctanh}(c x)}{2 c^2}-\frac {b e \left (c^2 f+g\right ) \text {arctanh}(c x) \log \left (\frac {2}{1+c x}\right )}{c^2 g}+\frac {b e \left (c^2 f+g\right ) \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{2 c^2 g}+\frac {b e \left (c^2 f+g\right ) \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{2 c^2 g}+\frac {b e x \log \left (f+g x^2\right )}{2 c}+\frac {e \left (f+g x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}-\frac {b e \left (c^2 f+g\right ) \text {arctanh}(c x) \log \left (f+g x^2\right )}{2 c^2 g}+\frac {b e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 c^2 g}-\frac {b e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{4 c^2 g}-\frac {b e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{4 c^2 g} \]
[Out]
Time = 0.56 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, Rules used = {2504, 2436, 2332, 6231, 327, 213, 531, 2608, 2498, 211, 2520, 12, 6139, 6057, 2449, 2352, 2497} \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\frac {1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )+\frac {e \left (f+g x^2\right ) \log \left (f+g x^2\right ) \left (a+b \coth ^{-1}(c x)\right )}{2 g}-\frac {1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )+\frac {b e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c \sqrt {g}}-\frac {b (d-e) \text {arctanh}(c x)}{2 c^2}-\frac {b e \text {arctanh}(c x) \left (c^2 f+g\right ) \log \left (f+g x^2\right )}{2 c^2 g}-\frac {b e \text {arctanh}(c x) \left (c^2 f+g\right ) \log \left (\frac {2}{c x+1}\right )}{c^2 g}+\frac {b e \text {arctanh}(c x) \left (c^2 f+g\right ) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}-\sqrt {g}\right )}\right )}{2 c^2 g}+\frac {b e \text {arctanh}(c x) \left (c^2 f+g\right ) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}+\sqrt {g}\right )}\right )}{2 c^2 g}+\frac {b e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 c^2 g}-\frac {b e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (c x+1)}\right )}{4 c^2 g}-\frac {b e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {g} x+\sqrt {-f}\right )}{\left (\sqrt {-f} c+\sqrt {g}\right ) (c x+1)}\right )}{4 c^2 g}+\frac {b x (d-e)}{2 c}+\frac {b e x \log \left (f+g x^2\right )}{2 c}-\frac {b e x}{c} \]
[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 6231
Rubi steps \begin{align*} \text {integral}& = \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) \text {arctanh}(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) \text {arctanh}(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) \text {arctanh}(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 ) \text {arctanh}(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 \text {arctanh}(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) \text {arctanh}(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 ) \text {arctanh}(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 \text {arctanh}(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} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c \sqrt {g}}-\frac {b (d-e) \text {arctanh}(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 ) \text {arctanh}(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 {\text {arctanh}(c x)}{2 \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\text {arctanh}(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} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c \sqrt {g}}-\frac {b (d-e) \text {arctanh}(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 ) \text {arctanh}(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 {\text {arctanh}(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 {\text {arctanh}(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} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c \sqrt {g}}-\frac {b (d-e) \text {arctanh}(c x)}{2 c^2}-\frac {b e \left (c^2 f+g\right ) \text {arctanh}(c x) \log \left (\frac {2}{1+c x}\right )}{c^2 g}+\frac {b e \left (c^2 f+g\right ) \text {arctanh}(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 ) \text {arctanh}(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 ) \text {arctanh}(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} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c \sqrt {g}}-\frac {b (d-e) \text {arctanh}(c x)}{2 c^2}-\frac {b e \left (c^2 f+g\right ) \text {arctanh}(c x) \log \left (\frac {2}{1+c x}\right )}{c^2 g}+\frac {b e \left (c^2 f+g\right ) \text {arctanh}(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 ) \text {arctanh}(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 ) \text {arctanh}(c x) \log \left (f+g x^2\right )}{2 c^2 g}-\frac {b e \left (c^2 f+g\right ) \operatorname {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 ) \operatorname {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}+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 {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} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c \sqrt {g}}-\frac {b (d-e) \text {arctanh}(c x)}{2 c^2}-\frac {b e \left (c^2 f+g\right ) \text {arctanh}(c x) \log \left (\frac {2}{1+c x}\right )}{c^2 g}+\frac {b e \left (c^2 f+g\right ) \text {arctanh}(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 ) \text {arctanh}(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 ) \text {arctanh}(c x) \log \left (f+g x^2\right )}{2 c^2 g}+\frac {b e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 c^2 g}-\frac {b e \left (c^2 f+g\right ) \operatorname {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 ) \operatorname {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} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.10 (sec) , antiderivative size = 1128, normalized size of antiderivative = 2.20 \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\frac {2 b c d g x-6 b c e g x+2 a c^2 d g x^2-2 a c^2 e g x^2-2 b d g \coth ^{-1}(c x)+2 b e g \coth ^{-1}(c x)+2 b c^2 d g x^2 \coth ^{-1}(c x)-2 b c^2 e g x^2 \coth ^{-1}(c x)+4 b c e \sqrt {f} \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )-4 i b c^2 e f \arcsin \left (\sqrt {\frac {g}{c^2 f+g}}\right ) \text {arctanh}\left (\frac {c f}{\sqrt {-c^2 f g} x}\right )-4 i b e g \arcsin \left (\sqrt {\frac {g}{c^2 f+g}}\right ) \text {arctanh}\left (\frac {c f}{\sqrt {-c^2 f g} x}\right )-4 b c^2 e f \coth ^{-1}(c x) \log \left (1-e^{-2 \coth ^{-1}(c x)}\right )-4 b e g \coth ^{-1}(c x) \log \left (1-e^{-2 \coth ^{-1}(c x)}\right )+2 b c^2 e f \coth ^{-1}(c x) \log \left (\frac {e^{-2 \coth ^{-1}(c x)} \left (c^2 \left (-1+e^{2 \coth ^{-1}(c x)}\right ) f+g+e^{2 \coth ^{-1}(c x)} g-2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )+2 b e g \coth ^{-1}(c x) \log \left (\frac {e^{-2 \coth ^{-1}(c x)} \left (c^2 \left (-1+e^{2 \coth ^{-1}(c x)}\right ) f+g+e^{2 \coth ^{-1}(c x)} g-2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )-2 i b c^2 e f \arcsin \left (\sqrt {\frac {g}{c^2 f+g}}\right ) \log \left (\frac {e^{-2 \coth ^{-1}(c x)} \left (c^2 \left (-1+e^{2 \coth ^{-1}(c x)}\right ) f+g+e^{2 \coth ^{-1}(c x)} g-2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )-2 i b e g \arcsin \left (\sqrt {\frac {g}{c^2 f+g}}\right ) \log \left (\frac {e^{-2 \coth ^{-1}(c x)} \left (c^2 \left (-1+e^{2 \coth ^{-1}(c x)}\right ) f+g+e^{2 \coth ^{-1}(c x)} g-2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )+2 b c^2 e f \coth ^{-1}(c x) \log \left (\frac {e^{-2 \coth ^{-1}(c x)} \left (c^2 \left (-1+e^{2 \coth ^{-1}(c x)}\right ) f+g+e^{2 \coth ^{-1}(c x)} g+2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )+2 b e g \coth ^{-1}(c x) \log \left (\frac {e^{-2 \coth ^{-1}(c x)} \left (c^2 \left (-1+e^{2 \coth ^{-1}(c x)}\right ) f+g+e^{2 \coth ^{-1}(c x)} g+2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )+2 i b c^2 e f \arcsin \left (\sqrt {\frac {g}{c^2 f+g}}\right ) \log \left (\frac {e^{-2 \coth ^{-1}(c x)} \left (c^2 \left (-1+e^{2 \coth ^{-1}(c x)}\right ) f+g+e^{2 \coth ^{-1}(c x)} g+2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )+2 i b e g \arcsin \left (\sqrt {\frac {g}{c^2 f+g}}\right ) \log \left (\frac {e^{-2 \coth ^{-1}(c x)} \left (c^2 \left (-1+e^{2 \coth ^{-1}(c x)}\right ) f+g+e^{2 \coth ^{-1}(c x)} g+2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )+2 a c^2 e f \log \left (f+g x^2\right )+2 b c e g x \log \left (f+g x^2\right )+2 a c^2 e g x^2 \log \left (f+g x^2\right )-2 b e g \coth ^{-1}(c x) \log \left (f+g x^2\right )+2 b c^2 e g x^2 \coth ^{-1}(c x) \log \left (f+g x^2\right )+2 b e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(c x)}\right )-b e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,\frac {e^{-2 \coth ^{-1}(c x)} \left (c^2 f-g+2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )-b c^2 e f \operatorname {PolyLog}\left (2,-\frac {e^{-2 \coth ^{-1}(c x)} \left (-c^2 f+g+2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )-b e g \operatorname {PolyLog}\left (2,-\frac {e^{-2 \coth ^{-1}(c x)} \left (-c^2 f+g+2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )}{4 c^2 g} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(951\) vs. \(2(448)=896\).
Time = 6.95 (sec) , antiderivative size = 952, normalized size of antiderivative = 1.86
method | result | size |
risch | \(\frac {d b \ln \left (c x +1\right ) x^{2}}{4}-\frac {d b \ln \left (c x +1\right )}{4 c^{2}}+\frac {d b \ln \left (c x -1\right )}{4 c^{2}}-\frac {e b \ln \left (c x -1\right ) \ln \left (\frac {c \sqrt {-f g}-g \left (c x -1\right )-g}{c \sqrt {-f g}-g}\right ) f}{4 g}-\frac {e b \ln \left (c x -1\right ) \ln \left (\frac {c \sqrt {-f g}+g \left (c x -1\right )+g}{c \sqrt {-f g}+g}\right ) f}{4 g}+\frac {e b \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right ) f}{4 g}+\frac {e b \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right ) f}{4 g}+\frac {e f b \arctan \left (\frac {x g}{\sqrt {f g}}\right )}{c \sqrt {f g}}+\frac {a d \,x^{2}}{2}-\frac {3 b e x}{2 c}+\frac {b d x}{2 c}-\frac {b d \,x^{2} \ln \left (c x -1\right )}{4}+\frac {b e \,x^{2} \ln \left (c x -1\right )}{4}-\frac {a e \,x^{2}}{2}+\left (\frac {b \,x^{2} e \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 {b e \ln \left (c x +1\right )}{4 c^{2}}+\frac {e b \operatorname {dilog}\left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right )}{4 c^{2}}+\frac {e b \operatorname {dilog}\left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right )}{4 c^{2}}-\frac {e b \ln \left (c x -1\right )}{4 c^{2}}-\frac {e b \operatorname {dilog}\left (\frac {c \sqrt {-f g}-g \left (c x -1\right )-g}{c \sqrt {-f g}-g}\right )}{4 c^{2}}-\frac {e b \operatorname {dilog}\left (\frac {c \sqrt {-f g}+g \left (c x -1\right )+g}{c \sqrt {-f g}+g}\right )}{4 c^{2}}-\frac {e b \ln \left (c x -1\right ) \ln \left (\frac {c \sqrt {-f g}-g \left (c x -1\right )-g}{c \sqrt {-f g}-g}\right )}{4 c^{2}}-\frac {e b \ln \left (c x -1\right ) \ln \left (\frac {c \sqrt {-f g}+g \left (c x -1\right )+g}{c \sqrt {-f g}+g}\right )}{4 c^{2}}+\frac {e b \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right )}{4 c^{2}}+\frac {e b \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right )}{4 c^{2}}+\frac {e f a \ln \left (g \,x^{2}+f \right )}{2 g}+\frac {e b \operatorname {dilog}\left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right ) f}{4 g}+\frac {e b \operatorname {dilog}\left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right ) f}{4 g}-\frac {e b \operatorname {dilog}\left (\frac {c \sqrt {-f g}-g \left (c x -1\right )-g}{c \sqrt {-f g}-g}\right ) f}{4 g}-\frac {e b \operatorname {dilog}\left (\frac {c \sqrt {-f g}+g \left (c x -1\right )+g}{c \sqrt {-f g}+g}\right ) f}{4 g}-\frac {b \,x^{2} e \ln \left (c x +1\right )}{4}\) | \(952\) |
default | \(\text {Expression too large to display}\) | \(8478\) |
parts | \(\text {Expression too large to display}\) | \(8478\) |
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\[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int { {\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )} x \,d x } \]
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Timed out. \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\text {Timed out} \]
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\[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int { {\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )} x \,d x } \]
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\[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int { {\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )} x \,d x } \]
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Timed out. \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int x\,\left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right ) \,d x \]
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