Optimal. Leaf size=1278 \[ \frac {b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {11 b c d^3 i \sqrt {1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c d^2 (2 e h+27 d i) \sqrt {1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c d \left (e^2 g-6 d e h-18 d^2 i\right ) \sqrt {1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c \left (2 e^2 (e g-4 d h)-c^2 d \left (2 e^2 f-d e g-2 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {11 b c^3 d^4 i \sqrt {1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c d^2 \left (18 e^2 i+c^2 d (2 e h+9 d i)\right ) \sqrt {1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c d \left (4 e^2 (e h+6 d i)-c^2 d \left (e^2 g-2 d e h+6 d^2 i\right )\right ) \sqrt {1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {i b i \text {ArcSin}(c x)^2}{2 e^4}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \text {ArcSin}(c x))}{3 e^4 (d+e x)^3}-\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \text {ArcSin}(c x))}{2 e^4 (d+e x)^2}-\frac {(e h-3 d i) (a+b \text {ArcSin}(c x))}{e^4 (d+e x)}+\frac {b c \left (4 c^4 d^2 f+12 e^2 h+c^2 \left (2 e^2 f-9 d e g+6 d^2 h\right )\right ) \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{12 e \left (c^2 d^2-e^2\right )^{5/2}}-\frac {11 b c^3 d^3 \left (2 c^2 d^2+e^2\right ) i \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{12 e^4 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c^3 d^2 \left (4 c^2 d^2 h+e (2 e h+81 d i)\right ) \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{12 e^3 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c d \left (2 c^4 d^2 g-36 e^2 i+c^2 \left (e^2 g-18 d e h-18 d^2 i\right )\right ) \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{12 e^2 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b i \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b i \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b i \text {ArcSin}(c x) \log (d+e x)}{e^4}+\frac {i (a+b \text {ArcSin}(c x)) \log (d+e x)}{e^4}-\frac {i b i \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b i \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 2.05, antiderivative size = 1278, normalized size of antiderivative = 1.00, number of steps
used = 29, number of rules used = 17, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.548, Rules used = {1864,
4837, 12, 6874, 759, 821, 739, 210, 849, 1665, 222, 2451, 4825, 4615, 2221, 2317, 2438}
\begin {gather*} -\frac {11 b c^3 i \sqrt {1-c^2 x^2} d^4}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {11 b c^3 \left (2 c^2 d^2+e^2\right ) i \text {ArcTan}\left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) d^3}{12 e^4 \left (c^2 d^2-e^2\right )^{5/2}}-\frac {11 b c i \sqrt {1-c^2 x^2} d^3}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c^3 \left (4 c^2 h d^2+e (2 e h+81 d i)\right ) \text {ArcTan}\left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) d^2}{12 e^3 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c \left (d (2 e h+9 d i) c^2+18 e^2 i\right ) \sqrt {1-c^2 x^2} d^2}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c (2 e h+27 d i) \sqrt {1-c^2 x^2} d^2}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (2 d^2 g c^4+\left (-18 i d^2-18 e h d+e^2 g\right ) c^2-36 e^2 i\right ) \text {ArcTan}\left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) d}{12 e^2 \left (c^2 d^2-e^2\right )^{5/2}}-\frac {b c \left (4 e^2 (e h+6 d i)-c^2 d \left (6 i d^2-2 e h d+e^2 g\right )\right ) \sqrt {1-c^2 x^2} d}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c \left (-18 i d^2-6 e h d+e^2 g\right ) \sqrt {1-c^2 x^2} d}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {i b i \text {ArcSin}(c x)^2}{2 e^4}-\frac {(e h-3 d i) (a+b \text {ArcSin}(c x))}{e^4 (d+e x)}-\frac {\left (3 i d^2-2 e h d+e^2 g\right ) (a+b \text {ArcSin}(c x))}{2 e^4 (d+e x)^2}-\frac {\left (-i d^3+e h d^2-e^2 g d+e^3 f\right ) (a+b \text {ArcSin}(c x))}{3 e^4 (d+e x)^3}+\frac {b c \left (4 d^2 f c^4+\left (6 h d^2-9 e g d+2 e^2 f\right ) c^2+12 e^2 h\right ) \text {ArcTan}\left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{12 e \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b i \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b i \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b i \text {ArcSin}(c x) \log (d+e x)}{e^4}+\frac {i (a+b \text {ArcSin}(c x)) \log (d+e x)}{e^4}-\frac {i b i \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b i \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b c \left (2 e^2 (e g-4 d h)-c^2 d \left (-2 h d^2-e g d+2 e^2 f\right )\right ) \sqrt {1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c \left (6 h d^2-3 e g d+2 e^2 f\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 210
Rule 222
Rule 739
Rule 759
Rule 821
Rule 849
Rule 1665
Rule 1864
Rule 2221
Rule 2317
Rule 2438
Rule 2451
Rule 4615
Rule 4825
Rule 4837
Rule 6874
Rubi steps
\begin {align*} \int \frac {\left (f+g x+h x^2+112 x^3\right ) \left (a+b \sin ^{-1}(c x)\right )}{(d+e x)^4} \, dx &=\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-(b c) \int \frac {1232 d^3-2 d^2 e (h-1512 x)-d e^2 (g+6 (h-336 x) x)-e^3 (2 f+3 x (g+2 h x))+672 (d+e x)^3 \log (d+e x)}{6 e^4 (d+e x)^3 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {(b c) \int \frac {1232 d^3-2 d^2 e (h-1512 x)-d e^2 (g+6 (h-336 x) x)-e^3 (2 f+3 x (g+2 h x))+672 (d+e x)^3 \log (d+e x)}{(d+e x)^3 \sqrt {1-c^2 x^2}} \, dx}{6 e^4}\\ &=\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {(b c) \int \left (\frac {1232 d^3}{(d+e x)^3 \sqrt {1-c^2 x^2}}-\frac {2 d^2 e (h-1512 x)}{(d+e x)^3 \sqrt {1-c^2 x^2}}+\frac {d e^2 \left (-g-6 h x+2016 x^2\right )}{(d+e x)^3 \sqrt {1-c^2 x^2}}-\frac {e^3 \left (2 f+3 g x+6 h x^2\right )}{(d+e x)^3 \sqrt {1-c^2 x^2}}+\frac {672 \log (d+e x)}{\sqrt {1-c^2 x^2}}\right ) \, dx}{6 e^4}\\ &=\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {(112 b c) \int \frac {\log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{e^4}-\frac {\left (616 b c d^3\right ) \int \frac {1}{(d+e x)^3 \sqrt {1-c^2 x^2}} \, dx}{3 e^4}+\frac {\left (b c d^2\right ) \int \frac {h-1512 x}{(d+e x)^3 \sqrt {1-c^2 x^2}} \, dx}{3 e^3}-\frac {(b c d) \int \frac {-g-6 h x+2016 x^2}{(d+e x)^3 \sqrt {1-c^2 x^2}} \, dx}{6 e^2}+\frac {(b c) \int \frac {2 f+3 g x+6 h x^2}{(d+e x)^3 \sqrt {1-c^2 x^2}} \, dx}{6 e}\\ &=-\frac {308 b c d^3 \sqrt {1-c^2 x^2}}{3 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c d^2 (1512 d+e h) \sqrt {1-c^2 x^2}}{6 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c d \left (2016 d^2-e^2 g+6 d e h\right ) \sqrt {1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac {112 b \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac {(112 b c) \int \frac {\sin ^{-1}(c x)}{c d+c e x} \, dx}{e^3}+\frac {\left (308 b c^3 d^3\right ) \int \frac {-2 d+e x}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{3 e^4 \left (c^2 d^2-e^2\right )}+\frac {\left (b c d^2\right ) \int \frac {2 \left (1512 e+c^2 d h\right )-c^2 (1512 d+e h) x}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{6 e^3 \left (c^2 d^2-e^2\right )}-\frac {(b c d) \int \frac {2 \left (\frac {1}{2} d \left (4032-2 c^2 g\right )+6 e h\right )-\left (4032 e-c^2 \left (\frac {2016 d^2}{e}+e g-6 d h\right )\right ) x}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{12 e^2 \left (c^2 d^2-e^2\right )}+\frac {(b c) \int \frac {2 \left (2 c^2 d f-3 e g+6 d h\right )-\left (12 e h+c^2 \left (2 e f-3 d g-\frac {6 d^2 h}{e}\right )\right ) x}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{12 e \left (c^2 d^2-e^2\right )}\\ &=-\frac {308 b c d^3 \sqrt {1-c^2 x^2}}{3 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c d^2 (1512 d+e h) \sqrt {1-c^2 x^2}}{6 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c d \left (2016 d^2-e^2 g+6 d e h\right ) \sqrt {1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {308 b c^3 d^4 \sqrt {1-c^2 x^2}}{e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c \left (2 e^2 (e g-4 d h)-c^2 d \left (2 e^2 f-d e g-2 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c d^2 \left (1008 e^2+c^2 d (504 d+e h)\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c d \left (4 e^2 (672 d+e h)-c^2 d \left (672 d^2+e^2 g-2 d e h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac {112 b \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac {(112 b c) \text {Subst}\left (\int \frac {x \cos (x)}{c^2 d+c e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}-\frac {\left (308 b c^3 d^3 \left (2 c^2 d^2+e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{3 e^4 \left (c^2 d^2-e^2\right )^2}+\frac {\left (b c^3 d^2 \left (4536 d e+2 c^2 d^2 h+e^2 h\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{6 e^3 \left (c^2 d^2-e^2\right )^2}+\frac {\left (b c \left (4 c^4 d^2 f+12 e^2 h+c^2 \left (2 e^2 f-9 d e g+6 d^2 h\right )\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{12 e \left (c^2 d^2-e^2\right )^2}-\frac {\left (b c d \left (4032 e^2-2 c^4 d^2 g+c^2 \left (2016 d^2-e^2 g+18 d e h\right )\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{12 e^2 \left (c^2 d^2-e^2\right )^2}\\ &=-\frac {308 b c d^3 \sqrt {1-c^2 x^2}}{3 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c d^2 (1512 d+e h) \sqrt {1-c^2 x^2}}{6 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c d \left (2016 d^2-e^2 g+6 d e h\right ) \sqrt {1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {308 b c^3 d^4 \sqrt {1-c^2 x^2}}{e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c \left (2 e^2 (e g-4 d h)-c^2 d \left (2 e^2 f-d e g-2 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c d^2 \left (1008 e^2+c^2 d (504 d+e h)\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c d \left (4 e^2 (672 d+e h)-c^2 d \left (672 d^2+e^2 g-2 d e h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {56 i b \sin ^{-1}(c x)^2}{e^4}+\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac {112 b \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac {(112 b c) \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,\sin ^{-1}(c x)\right )}{e^3}+\frac {(112 b c) \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,\sin ^{-1}(c x)\right )}{e^3}+\frac {\left (308 b c^3 d^3 \left (2 c^2 d^2+e^2\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 )}{3 e^4 \left (c^2 d^2-e^2\right )^2}-\frac {\left (b c^3 d^2 \left (4536 d e+2 c^2 d^2 h+e^2 h\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 )}{6 e^3 \left (c^2 d^2-e^2\right )^2}-\frac {\left (b c \left (4 c^4 d^2 f+12 e^2 h+c^2 \left (2 e^2 f-9 d e g+6 d^2 h\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 )}{12 e \left (c^2 d^2-e^2\right )^2}+\frac {\left (b c d \left (4032 e^2-2 c^4 d^2 g+c^2 \left (2016 d^2-e^2 g+18 d e h\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 )}{12 e^2 \left (c^2 d^2-e^2\right )^2}\\ &=-\frac {308 b c d^3 \sqrt {1-c^2 x^2}}{3 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c d^2 (1512 d+e h) \sqrt {1-c^2 x^2}}{6 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c d \left (2016 d^2-e^2 g+6 d e h\right ) \sqrt {1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {308 b c^3 d^4 \sqrt {1-c^2 x^2}}{e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c \left (2 e^2 (e g-4 d h)-c^2 d \left (2 e^2 f-d e g-2 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c d^2 \left (1008 e^2+c^2 d (504 d+e h)\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c d \left (4 e^2 (672 d+e h)-c^2 d \left (672 d^2+e^2 g-2 d e h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {56 i b \sin ^{-1}(c x)^2}{e^4}+\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac {308 b c^3 d^3 \left (2 c^2 d^2+e^2\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{3 e^4 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c^3 d^2 \left (4536 d e+2 c^2 d^2 h+e^2 h\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{6 e^3 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c \left (4 c^4 d^2 f+12 e^2 h+c^2 \left (2 e^2 f-9 d e g+6 d^2 h\right )\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{12 e \left (c^2 d^2-e^2\right )^{5/2}}-\frac {b c d \left (4032 e^2-2 c^4 d^2 g+c^2 \left (2016 d^2-e^2 g+18 d e h\right )\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{12 e^2 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {112 b \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {112 b \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {112 b \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {(112 b) \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,\sin ^{-1}(c x)\right )}{e^4}-\frac {(112 b) \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,\sin ^{-1}(c x)\right )}{e^4}\\ &=-\frac {308 b c d^3 \sqrt {1-c^2 x^2}}{3 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c d^2 (1512 d+e h) \sqrt {1-c^2 x^2}}{6 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c d \left (2016 d^2-e^2 g+6 d e h\right ) \sqrt {1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {308 b c^3 d^4 \sqrt {1-c^2 x^2}}{e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c \left (2 e^2 (e g-4 d h)-c^2 d \left (2 e^2 f-d e g-2 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c d^2 \left (1008 e^2+c^2 d (504 d+e h)\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c d \left (4 e^2 (672 d+e h)-c^2 d \left (672 d^2+e^2 g-2 d e h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {56 i b \sin ^{-1}(c x)^2}{e^4}+\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac {308 b c^3 d^3 \left (2 c^2 d^2+e^2\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{3 e^4 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c^3 d^2 \left (4536 d e+2 c^2 d^2 h+e^2 h\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{6 e^3 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c \left (4 c^4 d^2 f+12 e^2 h+c^2 \left (2 e^2 f-9 d e g+6 d^2 h\right )\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{12 e \left (c^2 d^2-e^2\right )^{5/2}}-\frac {b c d \left (4032 e^2-2 c^4 d^2 g+c^2 \left (2016 d^2-e^2 g+18 d e h\right )\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{12 e^2 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {112 b \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {112 b \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {112 b \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac {(112 i b) \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 \sin ^{-1}(c x)}\right )}{e^4}+\frac {(112 i b) \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 \sin ^{-1}(c x)}\right )}{e^4}\\ &=-\frac {308 b c d^3 \sqrt {1-c^2 x^2}}{3 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (2 e^2 f-3 d e g+6 d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c d^2 (1512 d+e h) \sqrt {1-c^2 x^2}}{6 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c d \left (2016 d^2-e^2 g+6 d e h\right ) \sqrt {1-c^2 x^2}}{12 e^3 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {308 b c^3 d^4 \sqrt {1-c^2 x^2}}{e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c \left (2 e^2 (e g-4 d h)-c^2 d \left (2 e^2 f-d e g-2 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c d^2 \left (1008 e^2+c^2 d (504 d+e h)\right ) \sqrt {1-c^2 x^2}}{2 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c d \left (4 e^2 (672 d+e h)-c^2 d \left (672 d^2+e^2 g-2 d e h\right )\right ) \sqrt {1-c^2 x^2}}{4 e^3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {56 i b \sin ^{-1}(c x)^2}{e^4}+\frac {\left (112 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 e^4 (d+e x)^3}-\frac {\left (336 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 e^4 (d+e x)^2}+\frac {(336 d-e h) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac {308 b c^3 d^3 \left (2 c^2 d^2+e^2\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{3 e^4 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c^3 d^2 \left (4536 d e+2 c^2 d^2 h+e^2 h\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{6 e^3 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c \left (4 c^4 d^2 f+12 e^2 h+c^2 \left (2 e^2 f-9 d e g+6 d^2 h\right )\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{12 e \left (c^2 d^2-e^2\right )^{5/2}}-\frac {b c d \left (4032 e^2-2 c^4 d^2 g+c^2 \left (2016 d^2-e^2 g+18 d e h\right )\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{12 e^2 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {112 b \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {112 b \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {112 b \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {112 \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {112 i b \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {112 i b \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 6.40, size = 1921, normalized size = 1.50 \begin {gather*} \frac {-a e^3 f+a d e^2 g-a d^2 e h+a d^3 i}{3 e^4 (d+e x)^3}+\frac {-a e^2 g+2 a d e h-3 a d^2 i}{2 e^4 (d+e x)^2}+\frac {-a e h+3 a d 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}} F_1\left (3;\frac {1}{2},\frac {1}{2};4;-\frac {-d+\sqrt {\frac {1}{c^2}} e}{d+e x},-\frac {-d-\sqrt {\frac {1}{c^2}} e}{d+e x}\right )}{9 e^2 (d+e x)^2 \sqrt {1-c^2 x^2}}-\frac {\text {ArcSin}(c x)}{3 e (d+e x)^3}\right )+\frac {a i \log (d+e x)}{e^4}+b h \left (\frac {-\frac {\text {ArcSin}(c x)}{d+e x}+\frac {c \text {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^3}-\frac {d \left (\frac {c \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\text {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 )}{e^2}+\frac {d^2 \left (\frac {\sqrt {1-c^2 x^2} \left (-c e^2+c^3 d (4 d+3 e x)\right )}{\left (-c^2 d^2+e^2\right )^2 (d+e x)^2}-\frac {2 \text {ArcSin}(c x)}{e (d+e x)^3}+\frac {c^3 \left (2 c^2 d^2+e^2\right ) \log (d+e x)}{e (-c d+e)^2 (c d+e)^2 \sqrt {-c^2 d^2+e^2}}-\frac {c^3 \left (2 c^2 d^2+e^2\right ) \log \left (e+c^2 d x+\sqrt {-c^2 d^2+e^2} \sqrt {1-c^2 x^2}\right )}{e (-c d+e)^2 (c d+e)^2 \sqrt {-c^2 d^2+e^2}}\right )}{6 e^2}\right )+b g \left (\frac {\frac {c \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\text {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}}}{2 e}-\frac {d \left (\frac {\sqrt {1-c^2 x^2} \left (-c e^2+c^3 d (4 d+3 e x)\right )}{\left (-c^2 d^2+e^2\right )^2 (d+e x)^2}-\frac {2 \text {ArcSin}(c x)}{e (d+e x)^3}+\frac {c^3 \left (2 c^2 d^2+e^2\right ) \log (d+e x)}{e (-c d+e)^2 (c d+e)^2 \sqrt {-c^2 d^2+e^2}}-\frac {c^3 \left (2 c^2 d^2+e^2\right ) \log \left (e+c^2 d x+\sqrt {-c^2 d^2+e^2} \sqrt {1-c^2 x^2}\right )}{e (-c d+e)^2 (c d+e)^2 \sqrt {-c^2 d^2+e^2}}\right )}{6 e}\right )+b i \left (-\frac {3 d \left (-\frac {\text {ArcSin}(c x)}{d+e x}+\frac {c \text {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 {3 d^2 \left (\frac {c \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\text {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 {d^3 \left (\frac {\sqrt {1-c^2 x^2} \left (-c e^2+c^3 d (4 d+3 e x)\right )}{\left (-c^2 d^2+e^2\right )^2 (d+e x)^2}-\frac {2 \text {ArcSin}(c x)}{e (d+e x)^3}+\frac {c^3 \left (2 c^2 d^2+e^2\right ) \log (d+e x)}{e (-c d+e)^2 (c d+e)^2 \sqrt {-c^2 d^2+e^2}}-\frac {c^3 \left (2 c^2 d^2+e^2\right ) \log \left (e+c^2 d x+\sqrt {-c^2 d^2+e^2} \sqrt {1-c^2 x^2}\right )}{e (-c d+e)^2 (c d+e)^2 \sqrt {-c^2 d^2+e^2}}\right )}{6 e^3}+\frac {-\frac {i \text {ArcSin}(c x)^2}{2 e}+\frac {\text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {i \text {PolyLog}\left (2,-\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {i \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}}{e^3}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 5710 vs. \(2 (1225 ) = 2450\).
time = 6.75, size = 5711, normalized size = 4.47
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(5711\) |
default | \(\text {Expression too large to display}\) | \(5711\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 )^{4}}\, dx \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 \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 )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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