Integrand size = 31, antiderivative size = 1016 \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\frac {b i \sqrt {1-c^2 x^2}}{c e^3}+\frac {5 b c d^3 i \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b c d^2 (3 e h+4 d i) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c d \left (e^2 g+4 d e h-4 d^2 i\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c \left (e^3 f-2 d e^2 g+2 d^3 i\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i b (e h-3 d i) \arcsin (c x)^2}{2 e^4}+\frac {i x (a+b \arcsin (c x))}{e^3}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{2 e^4 (d+e x)^2}-\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {5 b c^3 d^4 i \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c d^2 \left (3 c^2 d h+4 e i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b c d \left (4 e^2 (e h-2 d i)+c^2 \left (d e^2 g+4 d^3 i\right )\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c \left (2 e^4 g-6 d^2 e^2 i-c^2 \left (d e^3 f-4 d^4 i\right )\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b (e h-3 d i) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b (e h-3 d i) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b (e h-3 d i) \arcsin (c x) \log (d+e x)}{e^4}+\frac {(e h-3 d i) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {i b (e h-3 d i) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b (e h-3 d i) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4} \]
[Out]
Time = 1.73 (sec) , antiderivative size = 1016, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.581, Rules used = {1864, 4837, 12, 6874, 745, 739, 210, 821, 1665, 858, 222, 1668, 2451, 4825, 4615, 2221, 2317, 2438} \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\frac {5 b c^3 i \arctan \left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) d^4}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {5 b c i \sqrt {1-c^2 x^2} d^3}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b c \left (3 d h c^2+4 e i\right ) \arctan \left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) d^2}{2 e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c (3 e h+4 d i) \sqrt {1-c^2 x^2} d^2}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c \left (\left (4 i d^3+e^2 g d\right ) c^2+4 e^2 (e h-2 d i)\right ) \arctan \left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) d}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b c \left (-4 i d^2+4 e h d+e^2 g\right ) \sqrt {1-c^2 x^2} d}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i b (e h-3 d i) \arcsin (c x)^2}{2 e^4}+\frac {i x (a+b \arcsin (c x))}{e^3}-\frac {\left (3 i d^2-2 e h d+e^2 g\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}-\frac {\left (-i d^3+e h d^2-e^2 g d+e^3 f\right ) (a+b \arcsin (c x))}{2 e^4 (d+e x)^2}-\frac {b c \left (2 g e^4-6 d^2 i e^2-c^2 \left (d e^3 f-4 d^4 i\right )\right ) \arctan \left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b (e h-3 d i) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b (e h-3 d i) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b (e h-3 d i) \arcsin (c x) \log (d+e x)}{e^4}+\frac {(e h-3 d i) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {i b (e h-3 d i) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b (e h-3 d i) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b i \sqrt {1-c^2 x^2}}{c e^3}+\frac {b c \left (2 i d^3-2 e^2 g d+e^3 f\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)} \]
[In]
[Out]
Rule 12
Rule 210
Rule 222
Rule 739
Rule 745
Rule 821
Rule 858
Rule 1665
Rule 1668
Rule 1864
Rule 2221
Rule 2317
Rule 2438
Rule 2451
Rule 4615
Rule 4825
Rule 4837
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {i x (a+b \arcsin (c x))}{e^3}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{2 e^4 (d+e x)^2}-\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {(e h-3 d i) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-(b c) \int \frac {-5 d^3 i+d^2 e (3 h-4 i x)-e^3 \left (f+2 g x-2 i x^3\right )+d e^2 (-g+4 x (h+i x))+2 (e h-3 d i) (d+e x)^2 \log (d+e x)}{2 e^4 (d+e x)^2 \sqrt {1-c^2 x^2}} \, dx \\ & = \frac {i x (a+b \arcsin (c x))}{e^3}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{2 e^4 (d+e x)^2}-\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {(e h-3 d i) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {(b c) \int \frac {-5 d^3 i+d^2 e (3 h-4 i x)-e^3 \left (f+2 g x-2 i x^3\right )+d e^2 (-g+4 x (h+i x))+2 (e h-3 d i) (d+e x)^2 \log (d+e x)}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{2 e^4} \\ & = \frac {i x (a+b \arcsin (c x))}{e^3}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{2 e^4 (d+e x)^2}-\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {(e h-3 d i) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {(b c) \int \left (-\frac {5 d^3 i}{(d+e x)^2 \sqrt {1-c^2 x^2}}-\frac {d^2 e (-3 h+4 i x)}{(d+e x)^2 \sqrt {1-c^2 x^2}}+\frac {d e^2 \left (-g+4 h x+4 i x^2\right )}{(d+e x)^2 \sqrt {1-c^2 x^2}}+\frac {e^3 \left (-f-2 g x+2 i x^3\right )}{(d+e x)^2 \sqrt {1-c^2 x^2}}+\frac {2 (e h-3 d i) \log (d+e x)}{\sqrt {1-c^2 x^2}}\right ) \, dx}{2 e^4} \\ & = \frac {i x (a+b \arcsin (c x))}{e^3}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{2 e^4 (d+e x)^2}-\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {(e h-3 d i) (a+b \arcsin (c x)) \log (d+e x)}{e^4}+\frac {\left (b c d^2\right ) \int \frac {-3 h+4 i x}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{2 e^3}-\frac {(b c d) \int \frac {-g+4 h x+4 i x^2}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{2 e^2}-\frac {(b c) \int \frac {-f-2 g x+2 i x^3}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{2 e}+\frac {\left (5 b c d^3 i\right ) \int \frac {1}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{2 e^4}-\frac {(b c (e h-3 d i)) \int \frac {\log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{e^4} \\ & = \frac {5 b c d^3 i \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b c d^2 (3 e h+4 d i) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c d \left (e^2 g+4 d e h-4 d^2 i\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c \left (e^3 f-2 d e^2 g+2 d^3 i\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {i x (a+b \arcsin (c x))}{e^3}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{2 e^4 (d+e x)^2}-\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}-\frac {b (e h-3 d i) \arcsin (c x) \log (d+e x)}{e^4}+\frac {(e h-3 d i) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {(b c d) \int \frac {-c^2 d g-4 e h+4 d i+4 \left (\frac {c^2 d^2}{e}-e\right ) i x}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{2 e^2 \left (c^2 d^2-e^2\right )}-\frac {(b c) \int \frac {-c^2 d f+2 e g-\frac {2 d^2 i}{e}+2 d \left (1-\frac {c^2 d^2}{e^2}\right ) i x+2 \left (\frac {c^2 d^2}{e}-e\right ) i x^2}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{2 e \left (c^2 d^2-e^2\right )}+\frac {\left (5 b c^3 d^4 i\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{2 e^4 \left (c^2 d^2-e^2\right )}+\frac {(b c (e h-3 d i)) \int \frac {\arcsin (c x)}{c d+c e x} \, dx}{e^3}-\frac {\left (b c d^2 \left (3 c^2 d h+4 e i\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{2 e^3 \left (c^2 d^2-e^2\right )} \\ & = \frac {b i \sqrt {1-c^2 x^2}}{c e^3}+\frac {5 b c d^3 i \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b c d^2 (3 e h+4 d i) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c d \left (e^2 g+4 d e h-4 d^2 i\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c \left (e^3 f-2 d e^2 g+2 d^3 i\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {i x (a+b \arcsin (c x))}{e^3}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{2 e^4 (d+e x)^2}-\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}-\frac {b (e h-3 d i) \arcsin (c x) \log (d+e x)}{e^4}+\frac {(e h-3 d i) (a+b \arcsin (c x)) \log (d+e x)}{e^4}+\frac {b \int \frac {c^2 e \left (c^2 d e f-2 e^2 g+2 d^2 i\right )+4 c^2 d (c d-e) (c d+e) i x}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{2 c e^3 \left (c^2 d^2-e^2\right )}-\frac {(2 b c d i) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{e^4}-\frac {\left (5 b c^3 d^4 i\right ) \text {Subst}\left (\int \frac {1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac {e+c^2 d x}{\sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )}+\frac {(b c (e h-3 d i)) \text {Subst}\left (\int \frac {x \cos (x)}{c^2 d+c e \sin (x)} \, dx,x,\arcsin (c x)\right )}{e^3}+\frac {\left (b c d^2 \left (3 c^2 d h+4 e i\right )\right ) \text {Subst}\left (\int \frac {1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac {e+c^2 d x}{\sqrt {1-c^2 x^2}}\right )}{2 e^3 \left (c^2 d^2-e^2\right )}+\frac {\left (b c d \left (4 e^2 (e h-2 d i)+c^2 \left (d e^2 g+4 d^3 i\right )\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{2 e^4 \left (c^2 d^2-e^2\right )} \\ & = \frac {b i \sqrt {1-c^2 x^2}}{c e^3}+\frac {5 b c d^3 i \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b c d^2 (3 e h+4 d i) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c d \left (e^2 g+4 d e h-4 d^2 i\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c \left (e^3 f-2 d e^2 g+2 d^3 i\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {2 b d i \arcsin (c x)}{e^4}-\frac {i b (e h-3 d i) \arcsin (c x)^2}{2 e^4}+\frac {i x (a+b \arcsin (c x))}{e^3}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{2 e^4 (d+e x)^2}-\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {5 b c^3 d^4 i \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c d^2 \left (3 c^2 d h+4 e i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b (e h-3 d i) \arcsin (c x) \log (d+e x)}{e^4}+\frac {(e h-3 d i) (a+b \arcsin (c x)) \log (d+e x)}{e^4}+\frac {(2 b c d i) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{e^4}+\frac {(b c (e h-3 d i)) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 d-c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\arcsin (c x)\right )}{e^3}+\frac {(b c (e h-3 d i)) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 d+c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\arcsin (c x)\right )}{e^3}-\frac {\left (b c d \left (4 e^2 (e h-2 d i)+c^2 \left (d e^2 g+4 d^3 i\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac {e+c^2 d x}{\sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )}-\frac {\left (b c \left (2 e^4 g-6 d^2 e^2 i-c^2 \left (d e^3 f-4 d^4 i\right )\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{2 e^4 \left (c^2 d^2-e^2\right )} \\ & = \frac {b i \sqrt {1-c^2 x^2}}{c e^3}+\frac {5 b c d^3 i \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b c d^2 (3 e h+4 d i) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c d \left (e^2 g+4 d e h-4 d^2 i\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c \left (e^3 f-2 d e^2 g+2 d^3 i\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i b (e h-3 d i) \arcsin (c x)^2}{2 e^4}+\frac {i x (a+b \arcsin (c x))}{e^3}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{2 e^4 (d+e x)^2}-\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {5 b c^3 d^4 i \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c d^2 \left (3 c^2 d h+4 e i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b c d \left (4 e^2 (e h-2 d i)+c^2 \left (d e^2 g+4 d^3 i\right )\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b (e h-3 d i) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b (e h-3 d i) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b (e h-3 d i) \arcsin (c x) \log (d+e x)}{e^4}+\frac {(e h-3 d i) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {(b (e h-3 d i)) \text {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\arcsin (c x)\right )}{e^4}-\frac {(b (e h-3 d i)) \text {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\arcsin (c x)\right )}{e^4}+\frac {\left (b c \left (2 e^4 g-6 d^2 e^2 i-c^2 \left (d e^3 f-4 d^4 i\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac {e+c^2 d x}{\sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )} \\ & = \frac {b i \sqrt {1-c^2 x^2}}{c e^3}+\frac {5 b c d^3 i \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b c d^2 (3 e h+4 d i) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c d \left (e^2 g+4 d e h-4 d^2 i\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c \left (e^3 f-2 d e^2 g+2 d^3 i\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i b (e h-3 d i) \arcsin (c x)^2}{2 e^4}+\frac {i x (a+b \arcsin (c x))}{e^3}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{2 e^4 (d+e x)^2}-\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {5 b c^3 d^4 i \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c d^2 \left (3 c^2 d h+4 e i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b c d \left (4 e^2 (e h-2 d i)+c^2 \left (d e^2 g+4 d^3 i\right )\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c \left (2 e^4 g-6 d^2 e^2 i-c^2 \left (d e^3 f-4 d^4 i\right )\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b (e h-3 d i) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b (e h-3 d i) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b (e h-3 d i) \arcsin (c x) \log (d+e x)}{e^4}+\frac {(e h-3 d i) (a+b \arcsin (c x)) \log (d+e x)}{e^4}+\frac {(i b (e h-3 d i)) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{e^4}+\frac {(i b (e h-3 d i)) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{e^4} \\ & = \frac {b i \sqrt {1-c^2 x^2}}{c e^3}+\frac {5 b c d^3 i \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b c d^2 (3 e h+4 d i) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c d \left (e^2 g+4 d e h-4 d^2 i\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c \left (e^3 f-2 d e^2 g+2 d^3 i\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i b (e h-3 d i) \arcsin (c x)^2}{2 e^4}+\frac {i x (a+b \arcsin (c x))}{e^3}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \arcsin (c x))}{2 e^4 (d+e x)^2}-\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \arcsin (c x))}{e^4 (d+e x)}+\frac {5 b c^3 d^4 i \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c d^2 \left (3 c^2 d h+4 e i\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b c d \left (4 e^2 (e h-2 d i)+c^2 \left (d e^2 g+4 d^3 i\right )\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c \left (2 e^4 g-6 d^2 e^2 i-c^2 \left (d e^3 f-4 d^4 i\right )\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^4 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b (e h-3 d i) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b (e h-3 d i) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b (e h-3 d i) \arcsin (c x) \log (d+e x)}{e^4}+\frac {(e h-3 d i) (a+b \arcsin (c x)) \log (d+e x)}{e^4}-\frac {i b (e h-3 d i) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b (e h-3 d i) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.06 (sec) , antiderivative size = 1556, normalized size of antiderivative = 1.53 \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\frac {a i x}{e^3}+\frac {-a e^3 f+a d e^2 g-a d^2 e h+a d^3 i}{2 e^4 (d+e x)^2}+\frac {-a e^2 g+2 a d e h-3 a d^2 i}{e^4 (d+e x)}+b f \left (-\frac {c \sqrt {1+\frac {-d-\sqrt {\frac {1}{c^2}} e}{d+e x}} \sqrt {1+\frac {-d+\sqrt {\frac {1}{c^2}} e}{d+e x}} \operatorname {AppellF1}\left (2,\frac {1}{2},\frac {1}{2},3,-\frac {-d+\sqrt {\frac {1}{c^2}} e}{d+e x},-\frac {-d-\sqrt {\frac {1}{c^2}} e}{d+e x}\right )}{4 e^2 (d+e x) \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{2 e (d+e x)^2}\right )+\frac {(a e h-3 a d i) \log (d+e x)}{e^4}+b g \left (\frac {-\frac {\arcsin (c x)}{d+e x}+\frac {c \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d^2-e^2}}}{e^2}-\frac {d \left (\frac {c \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\arcsin (c x)}{e (d+e x)^2}-\frac {i c^3 d \left (\log (4)+\log \left (\frac {e^2 \sqrt {c^2 d^2-e^2} \left (i e+i c^2 d x+\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}\right )}{c^3 d (d+e x)}\right )\right )}{(c d-e) e (c d+e) \sqrt {c^2 d^2-e^2}}\right )}{2 e}\right )+b i \left (\frac {\sqrt {1-c^2 x^2}+c x \arcsin (c x)}{c e^3}+\frac {3 d^2 \left (-\frac {\arcsin (c x)}{d+e x}+\frac {c \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {d^3 \left (\frac {c \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\arcsin (c x)}{e (d+e x)^2}-\frac {i c^3 d \left (\log (4)+\log \left (\frac {e^2 \sqrt {c^2 d^2-e^2} \left (i e+i c^2 d x+\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}\right )}{c^3 d (d+e x)}\right )\right )}{(c d-e) e (c d+e) \sqrt {c^2 d^2-e^2}}\right )}{2 e^3}-\frac {3 d \left (-\frac {i \arcsin (c x)^2}{2 e}+\frac {\arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {i \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}\right )}{e^3}\right )+b h \left (-\frac {2 d \left (-\frac {\arcsin (c x)}{d+e x}+\frac {c \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {d^2 \left (\frac {c \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\arcsin (c x)}{e (d+e x)^2}-\frac {i c^3 d \left (\log (4)+\log \left (\frac {e^2 \sqrt {c^2 d^2-e^2} \left (i e+i c^2 d x+\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}\right )}{c^3 d (d+e x)}\right )\right )}{(c d-e) e (c d+e) \sqrt {c^2 d^2-e^2}}\right )}{2 e^2}+\frac {-\frac {i \arcsin (c x)^2}{2 e}+\frac {\arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {i \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}}{e^2}\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3609 vs. \(2 (979 ) = 1958\).
Time = 7.05 (sec) , antiderivative size = 3610, normalized size of antiderivative = 3.55
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(3610\) |
default | \(\text {Expression too large to display}\) | \(3610\) |
parts | \(\text {Expression too large to display}\) | \(3617\) |
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\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\int { \frac {{\left (i x^{3} + h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{3}} \,d x } \]
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\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x + h x^{2} + i x^{3}\right )}{\left (d + e x\right )^{3}}\, dx \]
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Exception generated. \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\int { \frac {{\left (i x^{3} + h x^{2} + g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (f+g x+h x^2+i x^3\right ) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (i\,x^3+h\,x^2+g\,x+f\right )}{{\left (d+e\,x\right )}^3} \,d x \]
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