Integrand size = 31, antiderivative size = 725 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{f+g x} \, dx=\frac {a \sqrt {d-c^2 d x^2}}{g}+\frac {b c x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}+\frac {b \sqrt {d-c^2 d x^2} \arccos (c x)}{g}-\frac {c x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b g \sqrt {1-c^2 x^2}}+\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x)}-\frac {a \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {i b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {i b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}} \]
[Out]
Time = 1.23 (sec) , antiderivative size = 725, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.613, Rules used = {4862, 4850, 697, 4842, 6874, 739, 210, 1668, 12, 4884, 4882, 4768, 8, 4858, 3402, 2296, 2221, 2317, 2438} \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{f+g x} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (1-\frac {c^2 f^2}{g^2}\right ) (a+b \arccos (c x))^2}{2 b c \sqrt {1-c^2 x^2} (f+g x)}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x)}-\frac {c x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {a \sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2} \arctan \left (\frac {c^2 f x+g}{\sqrt {1-c^2 x^2} \sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {a \sqrt {d-c^2 d x^2}}{g}-\frac {b \sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {b \sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {i b \arccos (c x) \sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2} \log \left (1+\frac {g e^{i \arccos (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {i b \arccos (c x) \sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2} \log \left (1+\frac {g e^{i \arccos (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {b \arccos (c x) \sqrt {d-c^2 d x^2}}{g}+\frac {b c x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}} \]
[In]
[Out]
Rule 8
Rule 12
Rule 210
Rule 697
Rule 739
Rule 1668
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3402
Rule 4768
Rule 4842
Rule 4850
Rule 4858
Rule 4862
Rule 4882
Rule 4884
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {d-c^2 d x^2} \int \frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{f+g x} \, dx}{\sqrt {1-c^2 x^2}} \\ & = -\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x)}+\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (-g-2 c^2 f x-c^2 g x^2\right ) (a+b \arccos (c x))^2}{(f+g x)^2} \, dx}{2 b c \sqrt {1-c^2 x^2}} \\ & = -\frac {c x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b g \sqrt {1-c^2 x^2}}+\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x)}+\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (\frac {1}{f+g x}-\frac {c^2 \left (g x+\frac {f^2}{f+g x}\right )}{g^2}\right ) (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}} \\ & = -\frac {c x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b g \sqrt {1-c^2 x^2}}+\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x)}+\frac {\sqrt {d-c^2 d x^2} \int \left (-\frac {a \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right )}{g^2 (f+g x) \sqrt {1-c^2 x^2}}-\frac {b \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \arccos (c x)}{g^2 (f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{\sqrt {1-c^2 x^2}} \\ & = -\frac {c x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b g \sqrt {1-c^2 x^2}}+\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x)}-\frac {\left (a \sqrt {d-c^2 d x^2}\right ) \int \frac {c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \arccos (c x)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}} \\ & = \frac {a \sqrt {d-c^2 d x^2}}{g}-\frac {c x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b g \sqrt {1-c^2 x^2}}+\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x)}-\frac {\left (a \sqrt {d-c^2 d x^2}\right ) \int \frac {c^2 g^2 \left (c^2 f^2-g^2\right )}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{c^2 g^4 \sqrt {1-c^2 x^2}}-\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {c^2 g x \arccos (c x)}{\sqrt {1-c^2 x^2}}+\frac {\left (c^2 f^2-g^2\right ) \arccos (c x)}{(f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{g^2 \sqrt {1-c^2 x^2}} \\ & = \frac {a \sqrt {d-c^2 d x^2}}{g}-\frac {c x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b g \sqrt {1-c^2 x^2}}+\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x)}-\frac {\left (b c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \arccos (c x)}{\sqrt {1-c^2 x^2}} \, dx}{g \sqrt {1-c^2 x^2}}-\frac {\left (a (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \int \frac {\arccos (c x)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}} \\ & = \frac {a \sqrt {d-c^2 d x^2}}{g}+\frac {b \sqrt {d-c^2 d x^2} \arccos (c x)}{g}-\frac {c x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b g \sqrt {1-c^2 x^2}}+\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x)}+\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int 1 \, dx}{g \sqrt {1-c^2 x^2}}+\frac {\left (a (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1}{-c^2 f^2+g^2-x^2} \, dx,x,\frac {g+c^2 f x}{\sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {\left (b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x}{c f+g \cos (x)} \, dx,x,\arccos (c x)\right )}{g^2 \sqrt {1-c^2 x^2}} \\ & = \frac {a \sqrt {d-c^2 d x^2}}{g}+\frac {b c x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}+\frac {b \sqrt {d-c^2 d x^2} \arccos (c x)}{g}-\frac {c x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b g \sqrt {1-c^2 x^2}}+\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x)}-\frac {a (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (2 b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c e^{i x} f+g+e^{2 i x} g} \, dx,x,\arccos (c x)\right )}{g^2 \sqrt {1-c^2 x^2}} \\ & = \frac {a \sqrt {d-c^2 d x^2}}{g}+\frac {b c x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}+\frac {b \sqrt {d-c^2 d x^2} \arccos (c x)}{g}-\frac {c x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b g \sqrt {1-c^2 x^2}}+\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x)}-\frac {a (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (2 b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c f+2 e^{i x} g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\arccos (c x)\right )}{g \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (2 b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c f+2 e^{i x} g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\arccos (c x)\right )}{g \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}} \\ & = \frac {a \sqrt {d-c^2 d x^2}}{g}+\frac {b c x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}+\frac {b \sqrt {d-c^2 d x^2} \arccos (c x)}{g}-\frac {c x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b g \sqrt {1-c^2 x^2}}+\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x)}-\frac {a (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (i b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e^{i x} g}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\arccos (c x)\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (i b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e^{i x} g}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\arccos (c x)\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}} \\ & = \frac {a \sqrt {d-c^2 d x^2}}{g}+\frac {b c x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}+\frac {b \sqrt {d-c^2 d x^2} \arccos (c x)}{g}-\frac {c x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b g \sqrt {1-c^2 x^2}}+\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x)}-\frac {a (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \arccos (c x)}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \arccos (c x)}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}} \\ & = \frac {a \sqrt {d-c^2 d x^2}}{g}+\frac {b c x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}+\frac {b \sqrt {d-c^2 d x^2} \arccos (c x)}{g}-\frac {c x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b g \sqrt {1-c^2 x^2}}+\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x)}-\frac {a (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}} \\ \end{align*}
Time = 4.19 (sec) , antiderivative size = 1095, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{f+g x} \, dx=-\frac {-2 a g \sqrt {d-c^2 d x^2}+2 a c \sqrt {d} f \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-2 a \sqrt {d} \sqrt {-c^2 f^2+g^2} \log (f+g x)+2 a \sqrt {d} \sqrt {-c^2 f^2+g^2} \log \left (d \left (g+c^2 f x\right )+\sqrt {d} \sqrt {-c^2 f^2+g^2} \sqrt {d-c^2 d x^2}\right )+b \sqrt {d-c^2 d x^2} \left (-\frac {2 c g x}{\sqrt {1-c^2 x^2}}-2 g \arccos (c x)+\frac {c f \arccos (c x)^2}{\sqrt {1-c^2 x^2}}+\frac {2 (-c f+g) (c f+g) \left (2 \arccos (c x) \text {arctanh}\left (\frac {(c f+g) \cot \left (\frac {1}{2} \arccos (c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )-2 \arccos \left (-\frac {c f}{g}\right ) \text {arctanh}\left (\frac {(-c f+g) \tan \left (\frac {1}{2} \arccos (c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )+\left (\arccos \left (-\frac {c f}{g}\right )-2 i \text {arctanh}\left (\frac {(c f+g) \cot \left (\frac {1}{2} \arccos (c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )+2 i \text {arctanh}\left (\frac {(-c f+g) \tan \left (\frac {1}{2} \arccos (c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right ) \log \left (\frac {e^{-\frac {1}{2} i \arccos (c x)} \sqrt {-c^2 f^2+g^2}}{\sqrt {2} \sqrt {g} \sqrt {c (f+g x)}}\right )+\left (\arccos \left (-\frac {c f}{g}\right )+2 i \left (\text {arctanh}\left (\frac {(c f+g) \cot \left (\frac {1}{2} \arccos (c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )-\text {arctanh}\left (\frac {(-c f+g) \tan \left (\frac {1}{2} \arccos (c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right )\right ) \log \left (\frac {e^{\frac {1}{2} i \arccos (c x)} \sqrt {-c^2 f^2+g^2}}{\sqrt {2} \sqrt {g} \sqrt {c (f+g x)}}\right )-\left (\arccos \left (-\frac {c f}{g}\right )-2 i \text {arctanh}\left (\frac {(-c f+g) \tan \left (\frac {1}{2} \arccos (c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right ) \log \left (\frac {(c f+g) \left (-i c f+i g+\sqrt {-c^2 f^2+g^2}\right ) \left (-i+\tan \left (\frac {1}{2} \arccos (c x)\right )\right )}{g \left (c f+g+\sqrt {-c^2 f^2+g^2} \tan \left (\frac {1}{2} \arccos (c x)\right )\right )}\right )-\left (\arccos \left (-\frac {c f}{g}\right )+2 i \text {arctanh}\left (\frac {(-c f+g) \tan \left (\frac {1}{2} \arccos (c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right ) \log \left (\frac {(c f+g) \left (i c f-i g+\sqrt {-c^2 f^2+g^2}\right ) \left (i+\tan \left (\frac {1}{2} \arccos (c x)\right )\right )}{g \left (c f+g+\sqrt {-c^2 f^2+g^2} \tan \left (\frac {1}{2} \arccos (c x)\right )\right )}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (c f-i \sqrt {-c^2 f^2+g^2}\right ) \left (c f+g-\sqrt {-c^2 f^2+g^2} \tan \left (\frac {1}{2} \arccos (c x)\right )\right )}{g \left (c f+g+\sqrt {-c^2 f^2+g^2} \tan \left (\frac {1}{2} \arccos (c x)\right )\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (c f+i \sqrt {-c^2 f^2+g^2}\right ) \left (c f+g-\sqrt {-c^2 f^2+g^2} \tan \left (\frac {1}{2} \arccos (c x)\right )\right )}{g \left (c f+g+\sqrt {-c^2 f^2+g^2} \tan \left (\frac {1}{2} \arccos (c x)\right )\right )}\right )\right )\right )}{\sqrt {-c^2 f^2+g^2} \sqrt {1-c^2 x^2}}\right )}{2 g^2} \]
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Time = 2.64 (sec) , antiderivative size = 816, normalized size of antiderivative = 1.13
method | result | size |
default | \(\frac {a \left (\sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}+\frac {c^{2} d f \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}\right )}{g \sqrt {c^{2} d}}+\frac {d \left (c^{2} f^{2}-g^{2}\right ) \ln \left (\frac {-\frac {2 d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}+\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}\, \sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g^{2} \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}\right )}{g}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} f c}{2 \left (c^{2} x^{2}-1\right ) g^{2}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )+i\right )}{2 \left (c^{2} x^{2}-1\right ) g}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arccos \left (c x \right )-i\right )}{2 \left (c^{2} x^{2}-1\right ) g}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arccos \left (c x \right ) \ln \left (\frac {-\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )-i \arccos \left (c x \right ) \ln \left (\frac {\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )+\operatorname {dilog}\left (\frac {-\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )-\operatorname {dilog}\left (\frac {\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )\right )}{\left (c^{2} x^{2}-1\right ) g^{2}}\right )\) | \(816\) |
parts | \(\frac {a \left (\sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}+\frac {c^{2} d f \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}\right )}{g \sqrt {c^{2} d}}+\frac {d \left (c^{2} f^{2}-g^{2}\right ) \ln \left (\frac {-\frac {2 d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}+\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}\, \sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g^{2} \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}\right )}{g}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} f c}{2 \left (c^{2} x^{2}-1\right ) g^{2}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )+i\right )}{2 \left (c^{2} x^{2}-1\right ) g}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arccos \left (c x \right )-i\right )}{2 \left (c^{2} x^{2}-1\right ) g}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arccos \left (c x \right ) \ln \left (\frac {-\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )-i \arccos \left (c x \right ) \ln \left (\frac {\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )+\operatorname {dilog}\left (\frac {-\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )-\operatorname {dilog}\left (\frac {\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )\right )}{\left (c^{2} x^{2}-1\right ) g^{2}}\right )\) | \(816\) |
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\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{f+g x} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arccos \left (c x\right ) + a\right )}}{g x + f} \,d x } \]
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\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{f+g x} \, dx=\int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acos}{\left (c x \right )}\right )}{f + g x}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{f+g x} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{f+g x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{f+g x} \, dx=\int \frac {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{f+g\,x} \,d x \]
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