Integrand size = 31, antiderivative size = 851 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx=-\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d-c^2 d x^2} \arccos (c x)}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \arccos (c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x)^2}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {a c^2 f \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b c^2 f \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b c^2 f \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1-c^2 x^2}}+\frac {b c^2 f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {b c^2 f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}} \]
[Out]
Time = 1.73 (sec) , antiderivative size = 851, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.710, Rules used = {4862, 4850, 37, 4840, 12, 1665, 858, 222, 739, 210, 4884, 4882, 4738, 4858, 3405, 3402, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx=\frac {b f^2 \sqrt {d-c^2 d x^2} \arccos (c x)^2 c^3}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {a f^2 \sqrt {d-c^2 d x^2} \arcsin (c x) c^3}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {a f \sqrt {d-c^2 d x^2} \arctan \left (\frac {f x c^2+g}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right ) c^2}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b f \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right ) c^2}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b f \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right ) c^2}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {b f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {b f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {b \sqrt {d-c^2 d x^2} \log (f+g x) c}{g^2 \sqrt {1-c^2 x^2}}-\frac {b \sqrt {d-c^2 d x^2} \arccos (c x)}{g (f+g x)}-\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b (f+g x)^2 c}-\frac {\left (f x c^2+g\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt {1-c^2 x^2} c} \]
[In]
[Out]
Rule 12
Rule 31
Rule 37
Rule 210
Rule 222
Rule 739
Rule 858
Rule 1665
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2747
Rule 3402
Rule 3405
Rule 4738
Rule 4840
Rule 4850
Rule 4858
Rule 4862
Rule 4882
Rule 4884
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)^2} \, 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)^2}+\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (-2 g-2 c^2 f x\right ) (a+b \arccos (c x))^2}{(f+g x)^3} \, dx}{2 b c \sqrt {1-c^2 x^2}} \\ & = -\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (g+c^2 f x\right )^2 (a+b \arccos (c x))}{\left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}} \\ & = -\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (g+c^2 f x\right )^2 (a+b \arccos (c x))}{(f+g x)^2 \sqrt {1-c^2 x^2}} \, dx}{\left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}} \\ & = -\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {\sqrt {d-c^2 d x^2} \int \left (\frac {a \left (g+c^2 f x\right )^2}{(f+g x)^2 \sqrt {1-c^2 x^2}}+\frac {b \left (g+c^2 f x\right )^2 \arccos (c x)}{(f+g x)^2 \sqrt {1-c^2 x^2}}\right ) \, dx}{\left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}} \\ & = -\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {\left (a \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (g+c^2 f x\right )^2}{(f+g x)^2 \sqrt {1-c^2 x^2}} \, dx}{\left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (g+c^2 f x\right )^2 \arccos (c x)}{(f+g x)^2 \sqrt {1-c^2 x^2}} \, dx}{\left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}} \\ & = -\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {\left (a \sqrt {d-c^2 d x^2}\right ) \int \frac {c^2 f \left (c^2 f^2-g^2\right )+c^4 f^2 \left (\frac {c^2 f^2}{g}-g\right ) x}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{\left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2}}-\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {c^4 f^2 \arccos (c x)}{g^2 \sqrt {1-c^2 x^2}}+\frac {\left (-c^2 f^2+g^2\right )^2 \arccos (c x)}{g^2 (f+g x)^2 \sqrt {1-c^2 x^2}}+\frac {2 c^2 f \left (-c^2 f^2+g^2\right ) \arccos (c x)}{g^2 (f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{\left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}} \\ & = -\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}+\frac {\left (a c^2 f \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 (a c^4 f^2 \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{\left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2}}-\frac {\left (b c^4 f^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\arccos (c x)}{\sqrt {1-c^2 x^2}} \, dx}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (b \left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {\arccos (c x)}{(f+g x)^2 \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \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 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}} \\ & = -\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \arccos (c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (a c^2 f \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 \left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x}{(c f+g \cos (x))^2} \, dx,x,\arccos (c x)\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \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 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}} \\ & = -\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d-c^2 d x^2} \arccos (c x)}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \arccos (c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {a c^2 f \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 (b c^2 f \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 {\left (b c \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\sin (x)}{c f+g \cos (x)} \, dx,x,\arccos (c x)\right )}{g \sqrt {1-c^2 x^2}}+\frac {\left (4 b c^2 f \left (-c^2 f^2+g^2\right ) \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 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}} \\ & = -\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d-c^2 d x^2} \arccos (c x)}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \arccos (c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {a c^2 f \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 (b c \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1}{c f+x} \, dx,x,c g x\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {\left (2 b c^2 f \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 {\left (4 b c^2 f \left (-c^2 f^2+g^2\right ) \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 \left (c^2 f^2-g^2\right )^{3/2} \sqrt {1-c^2 x^2}}-\frac {\left (4 b c^2 f \left (-c^2 f^2+g^2\right ) \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 \left (c^2 f^2-g^2\right )^{3/2} \sqrt {1-c^2 x^2}} \\ & = -\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d-c^2 d x^2} \arccos (c x)}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \arccos (c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {a c^2 f \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 {2 i b c^2 f \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 {2 i b c^2 f \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 \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1-c^2 x^2}}+\frac {\left (2 b c^2 f \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^2 f \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 i b c^2 f \left (-c^2 f^2+g^2\right ) \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 \left (c^2 f^2-g^2\right )^{3/2} \sqrt {1-c^2 x^2}}-\frac {\left (2 i b c^2 f \left (-c^2 f^2+g^2\right ) \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 \left (c^2 f^2-g^2\right )^{3/2} \sqrt {1-c^2 x^2}} \\ & = -\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d-c^2 d x^2} \arccos (c x)}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \arccos (c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {a c^2 f \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^2 f \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^2 f \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 \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1-c^2 x^2}}+\frac {\left (i b c^2 f \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^2 f \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 (2 b c^2 f \left (-c^2 f^2+g^2\right ) \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 \left (c^2 f^2-g^2\right )^{3/2} \sqrt {1-c^2 x^2}}-\frac {\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \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 \left (c^2 f^2-g^2\right )^{3/2} \sqrt {1-c^2 x^2}} \\ & = -\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d-c^2 d x^2} \arccos (c x)}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \arccos (c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {a c^2 f \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^2 f \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^2 f \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 \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1-c^2 x^2}}+\frac {2 b c^2 f \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 {2 b c^2 f \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 {\left (b c^2 f \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^2 f \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 (f+g x)}-\frac {b \sqrt {d-c^2 d x^2} \arccos (c x)}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \arccos (c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {a c^2 f \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^2 f \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^2 f \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 \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1-c^2 x^2}}+\frac {b c^2 f \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^2 f \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 = 10.57 (sec) , antiderivative size = 1130, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx=-\frac {a \sqrt {-d \left (-1+c^2 x^2\right )}}{g (f+g x)}+\frac {a c \sqrt {d} \arctan \left (\frac {c x \sqrt {-d \left (-1+c^2 x^2\right )}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{g^2}+\frac {a c^2 \sqrt {d} f \log (f+g x)}{g^2 \sqrt {-c^2 f^2+g^2}}-\frac {a c^2 \sqrt {d} f \log \left (d g+c^2 d f x+\sqrt {d} \sqrt {-c^2 f^2+g^2} \sqrt {-d \left (-1+c^2 x^2\right )}\right )}{g^2 \sqrt {-c^2 f^2+g^2}}-\frac {b c \sqrt {d \left (1-c^2 x^2\right )} \left (\frac {2 g \arccos (c x)}{c f+c g x}-\frac {\arccos (c x)^2}{\sqrt {1-c^2 x^2}}+\frac {2 \log \left (1+\frac {g x}{f}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 c f \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+c 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+c 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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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1938 vs. \(2 (813 ) = 1626\).
Time = 2.81 (sec) , antiderivative size = 1939, normalized size of antiderivative = 2.28
method | result | size |
default | \(\text {Expression too large to display}\) | \(1939\) |
parts | \(\text {Expression too large to display}\) | \(1939\) |
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\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arccos \left (c x\right ) + a\right )}}{{\left (g x + f\right )}^{2}} \,d x } \]
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\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx=\int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acos}{\left (c x \right )}\right )}{\left (f + g x\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx=\int \frac {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{{\left (f+g\,x\right )}^2} \,d x \]
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