Optimal. Leaf size=1323 \[ \frac {i b^2 (2 f g-e h) \sin ^{-1}(c x)^3}{3 h^3}+\frac {i a b (2 f g-e h) \sin ^{-1}(c x)^2}{h^3}+\frac {b^2 f x \sin ^{-1}(c x)^2}{h^2}-\frac {b^2 (2 f g-e h) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)^2}{h^3}-\frac {b^2 (2 f g-e h) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)^2}{h^3}-\frac {b^2 \left (f g^2-e h g+d h^2\right ) \sin ^{-1}(c x)^2}{h^3 (g+h x)}+\frac {2 a b f x \sin ^{-1}(c x)}{h^2}-\frac {2 a b (2 f g-e h) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3}-\frac {2 i b^2 c \left (f g^2-e h g+d h^2\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {2 a b (2 f g-e h) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3}+\frac {2 i b^2 c \left (f g^2-e h g+d h^2\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3 \sqrt {c^2 g^2-h^2}}+\frac {2 i b^2 (2 f g-e h) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3}+\frac {2 i b^2 (2 f g-e h) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3}+\frac {2 b^2 f \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}-\frac {2 a b \left (f g^2-e h g+d h^2\right ) \sin ^{-1}(c x)}{h^3 (g+h x)}+\frac {a^2 f x}{h^2}-\frac {2 b^2 f x}{h^2}+\frac {2 a b c \left (f g^2-e h g+d h^2\right ) \tan ^{-1}\left (\frac {g x c^2+h}{\sqrt {c^2 g^2-h^2} \sqrt {1-c^2 x^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}+\frac {2 i a b (2 f g-e h) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 c \left (f g^2-e h g+d h^2\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}+\frac {2 i a b (2 f g-e h) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 b^2 c \left (f g^2-e h g+d h^2\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {2 b^2 (2 f g-e h) \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 (2 f g-e h) \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 a b f \sqrt {1-c^2 x^2}}{c h^2}-\frac {a^2 \left (f g^2-e h g+d h^2\right )}{h^3 (g+h x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.47, antiderivative size = 1323, normalized size of antiderivative = 1.00, number of steps used = 45, number of rules used = 25, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.893, Rules used = {4759, 698, 4753, 12, 6742, 261, 725, 204, 216, 2404, 4741, 4519, 2190, 2279, 2391, 4619, 4677, 8, 4743, 4773, 3323, 2264, 2531, 2282, 6589} \[ \frac {i b^2 (2 f g-e h) \sin ^{-1}(c x)^3}{3 h^3}+\frac {i a b (2 f g-e h) \sin ^{-1}(c x)^2}{h^3}+\frac {b^2 f x \sin ^{-1}(c x)^2}{h^2}-\frac {b^2 (2 f g-e h) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)^2}{h^3}-\frac {b^2 (2 f g-e h) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)^2}{h^3}-\frac {b^2 \left (f g^2-e h g+d h^2\right ) \sin ^{-1}(c x)^2}{h^3 (g+h x)}+\frac {2 a b f x \sin ^{-1}(c x)}{h^2}-\frac {2 a b (2 f g-e h) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3}-\frac {2 i b^2 c \left (f g^2-e h g+d h^2\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {2 a b (2 f g-e h) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3}+\frac {2 i b^2 c \left (f g^2-e h g+d h^2\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3 \sqrt {c^2 g^2-h^2}}+\frac {2 i b^2 (2 f g-e h) \text {PolyLog}\left (2,\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3}+\frac {2 i b^2 (2 f g-e h) \text {PolyLog}\left (2,\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3}+\frac {2 b^2 f \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}-\frac {2 a b \left (f g^2-e h g+d h^2\right ) \sin ^{-1}(c x)}{h^3 (g+h x)}+\frac {a^2 f x}{h^2}-\frac {2 b^2 f x}{h^2}+\frac {2 a b c \left (f g^2-e h g+d h^2\right ) \tan ^{-1}\left (\frac {g x c^2+h}{\sqrt {c^2 g^2-h^2} \sqrt {1-c^2 x^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}+\frac {2 i a b (2 f g-e h) \text {PolyLog}\left (2,\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 c \left (f g^2-e h g+d h^2\right ) \text {PolyLog}\left (2,\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}+\frac {2 i a b (2 f g-e h) \text {PolyLog}\left (2,\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 b^2 c \left (f g^2-e h g+d h^2\right ) \text {PolyLog}\left (2,\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {2 b^2 (2 f g-e h) \text {PolyLog}\left (3,\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 (2 f g-e h) \text {PolyLog}\left (3,\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 a b f \sqrt {1-c^2 x^2}}{c h^2}-\frac {a^2 \left (f g^2-e h g+d h^2\right )}{h^3 (g+h x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 12
Rule 204
Rule 216
Rule 261
Rule 698
Rule 725
Rule 2190
Rule 2264
Rule 2279
Rule 2282
Rule 2391
Rule 2404
Rule 2531
Rule 3323
Rule 4519
Rule 4619
Rule 4677
Rule 4741
Rule 4743
Rule 4753
Rule 4759
Rule 4773
Rule 6589
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (d+e x+f x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(g+h x)^2} \, dx &=\int \left (\frac {a^2 \left (d+e x+f x^2\right )}{(g+h x)^2}+\frac {2 a b \left (d+e x+f x^2\right ) \sin ^{-1}(c x)}{(g+h x)^2}+\frac {b^2 \left (d+e x+f x^2\right ) \sin ^{-1}(c x)^2}{(g+h x)^2}\right ) \, dx\\ &=a^2 \int \frac {d+e x+f x^2}{(g+h x)^2} \, dx+(2 a b) \int \frac {\left (d+e x+f x^2\right ) \sin ^{-1}(c x)}{(g+h x)^2} \, dx+b^2 \int \frac {\left (d+e x+f x^2\right ) \sin ^{-1}(c x)^2}{(g+h x)^2} \, dx\\ &=\frac {2 a b f x \sin ^{-1}(c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)}{h^3 (g+h x)}-\frac {2 a b (2 f g-e h) \sin ^{-1}(c x) \log (g+h x)}{h^3}+a^2 \int \left (\frac {f}{h^2}+\frac {f g^2-e g h+d h^2}{h^2 (g+h x)^2}+\frac {-2 f g+e h}{h^2 (g+h x)}\right ) \, dx+b^2 \int \left (\frac {f \sin ^{-1}(c x)^2}{h^2}+\frac {\left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^2 (g+h x)^2}+\frac {(-2 f g+e h) \sin ^{-1}(c x)^2}{h^2 (g+h x)}\right ) \, dx-(2 a b c) \int \frac {f h x-\frac {f g^2-e g h+d h^2}{g+h x}-(2 f g-e h) \log (g+h x)}{h^3 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {a^2 f x}{h^2}-\frac {a^2 \left (f g^2-e g h+d h^2\right )}{h^3 (g+h x)}+\frac {2 a b f x \sin ^{-1}(c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)}{h^3 (g+h x)}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}-\frac {2 a b (2 f g-e h) \sin ^{-1}(c x) \log (g+h x)}{h^3}-\frac {(2 a b c) \int \frac {f h x-\frac {f g^2-e g h+d h^2}{g+h x}-(2 f g-e h) \log (g+h x)}{\sqrt {1-c^2 x^2}} \, dx}{h^3}+\frac {\left (b^2 f\right ) \int \sin ^{-1}(c x)^2 \, dx}{h^2}-\frac {\left (b^2 (2 f g-e h)\right ) \int \frac {\sin ^{-1}(c x)^2}{g+h x} \, dx}{h^2}+\frac {\left (b^2 \left (f g^2-e g h+d h^2\right )\right ) \int \frac {\sin ^{-1}(c x)^2}{(g+h x)^2} \, dx}{h^2}\\ &=\frac {a^2 f x}{h^2}-\frac {a^2 \left (f g^2-e g h+d h^2\right )}{h^3 (g+h x)}+\frac {2 a b f x \sin ^{-1}(c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)}{h^3 (g+h x)}+\frac {b^2 f x \sin ^{-1}(c x)^2}{h^2}-\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3 (g+h x)}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}-\frac {2 a b (2 f g-e h) \sin ^{-1}(c x) \log (g+h x)}{h^3}-\frac {(2 a b c) \int \left (\frac {f h x}{\sqrt {1-c^2 x^2}}+\frac {-f g^2+e g h-d h^2}{(g+h x) \sqrt {1-c^2 x^2}}-\frac {(2 f g-e h) \log (g+h x)}{\sqrt {1-c^2 x^2}}\right ) \, dx}{h^3}-\frac {\left (2 b^2 c f\right ) \int \frac {x \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{h^2}-\frac {\left (b^2 (2 f g-e h)\right ) \operatorname {Subst}\left (\int \frac {x^2 \cos (x)}{c g+h \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{h^2}+\frac {\left (2 b^2 c \left (f g^2-e g h+d h^2\right )\right ) \int \frac {\sin ^{-1}(c x)}{(g+h x) \sqrt {1-c^2 x^2}} \, dx}{h^3}\\ &=\frac {a^2 f x}{h^2}-\frac {a^2 \left (f g^2-e g h+d h^2\right )}{h^3 (g+h x)}+\frac {2 a b f x \sin ^{-1}(c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)}{h^3 (g+h x)}+\frac {2 b^2 f \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {b^2 f x \sin ^{-1}(c x)^2}{h^2}-\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3 (g+h x)}+\frac {i b^2 (2 f g-e h) \sin ^{-1}(c x)^3}{3 h^3}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}-\frac {2 a b (2 f g-e h) \sin ^{-1}(c x) \log (g+h x)}{h^3}-\frac {\left (2 b^2 f\right ) \int 1 \, dx}{h^2}-\frac {(2 a b c f) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{h^2}+\frac {(2 a b c (2 f g-e h)) \int \frac {\log (g+h x)}{\sqrt {1-c^2 x^2}} \, dx}{h^3}-\frac {\left (b^2 (2 f g-e h)\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x^2}{c g-i e^{i x} h-\sqrt {c^2 g^2-h^2}} \, dx,x,\sin ^{-1}(c x)\right )}{h^2}-\frac {\left (b^2 (2 f g-e h)\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x^2}{c g-i e^{i x} h+\sqrt {c^2 g^2-h^2}} \, dx,x,\sin ^{-1}(c x)\right )}{h^2}+\frac {\left (2 a b c \left (f g^2-e g h+d h^2\right )\right ) \int \frac {1}{(g+h x) \sqrt {1-c^2 x^2}} \, dx}{h^3}+\frac {\left (2 b^2 c \left (f g^2-e g h+d h^2\right )\right ) \operatorname {Subst}\left (\int \frac {x}{c g+h \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{h^3}\\ &=\frac {a^2 f x}{h^2}-\frac {2 b^2 f x}{h^2}-\frac {a^2 \left (f g^2-e g h+d h^2\right )}{h^3 (g+h x)}+\frac {2 a b f \sqrt {1-c^2 x^2}}{c h^2}+\frac {2 a b f x \sin ^{-1}(c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)}{h^3 (g+h x)}+\frac {2 b^2 f \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {b^2 f x \sin ^{-1}(c x)^2}{h^2}-\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3 (g+h x)}+\frac {i b^2 (2 f g-e h) \sin ^{-1}(c x)^3}{3 h^3}-\frac {b^2 (2 f g-e h) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {b^2 (2 f g-e h) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}+\frac {\left (2 b^2 (2 f g-e h)\right ) \operatorname {Subst}\left (\int x \log \left (1-\frac {i e^{i x} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3}+\frac {\left (2 b^2 (2 f g-e h)\right ) \operatorname {Subst}\left (\int x \log \left (1-\frac {i e^{i x} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3}-\frac {(2 a b c (2 f g-e h)) \int \frac {\sin ^{-1}(c x)}{c g+c h x} \, dx}{h^2}-\frac {\left (2 a b c \left (f g^2-e g h+d h^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-c^2 g^2+h^2-x^2} \, dx,x,\frac {h+c^2 g x}{\sqrt {1-c^2 x^2}}\right )}{h^3}+\frac {\left (4 b^2 c \left (f g^2-e g h+d h^2\right )\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c e^{i x} g+i h-i e^{2 i x} h} \, dx,x,\sin ^{-1}(c x)\right )}{h^3}\\ &=\frac {a^2 f x}{h^2}-\frac {2 b^2 f x}{h^2}-\frac {a^2 \left (f g^2-e g h+d h^2\right )}{h^3 (g+h x)}+\frac {2 a b f \sqrt {1-c^2 x^2}}{c h^2}+\frac {2 a b f x \sin ^{-1}(c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)}{h^3 (g+h x)}+\frac {2 b^2 f \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {b^2 f x \sin ^{-1}(c x)^2}{h^2}-\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3 (g+h x)}+\frac {i b^2 (2 f g-e h) \sin ^{-1}(c x)^3}{3 h^3}+\frac {2 a b c \left (f g^2-e g h+d h^2\right ) \tan ^{-1}\left (\frac {h+c^2 g x}{\sqrt {c^2 g^2-h^2} \sqrt {1-c^2 x^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {b^2 (2 f g-e h) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}+\frac {2 i b^2 (2 f g-e h) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 i b^2 (2 f g-e h) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {\left (2 i b^2 (2 f g-e h)\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (\frac {i e^{i x} h}{c g-\sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3}-\frac {\left (2 i b^2 (2 f g-e h)\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (\frac {i e^{i x} h}{c g+\sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3}-\frac {(2 a b c (2 f g-e h)) \operatorname {Subst}\left (\int \frac {x \cos (x)}{c^2 g+c h \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{h^2}-\frac {\left (4 i b^2 c \left (f g^2-e g h+d h^2\right )\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c g-2 i e^{i x} h-2 \sqrt {c^2 g^2-h^2}} \, dx,x,\sin ^{-1}(c x)\right )}{h^2 \sqrt {c^2 g^2-h^2}}+\frac {\left (4 i b^2 c \left (f g^2-e g h+d h^2\right )\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c g-2 i e^{i x} h+2 \sqrt {c^2 g^2-h^2}} \, dx,x,\sin ^{-1}(c x)\right )}{h^2 \sqrt {c^2 g^2-h^2}}\\ &=\frac {a^2 f x}{h^2}-\frac {2 b^2 f x}{h^2}-\frac {a^2 \left (f g^2-e g h+d h^2\right )}{h^3 (g+h x)}+\frac {2 a b f \sqrt {1-c^2 x^2}}{c h^2}+\frac {2 a b f x \sin ^{-1}(c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)}{h^3 (g+h x)}+\frac {2 b^2 f \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {i a b (2 f g-e h) \sin ^{-1}(c x)^2}{h^3}+\frac {b^2 f x \sin ^{-1}(c x)^2}{h^2}-\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3 (g+h x)}+\frac {i b^2 (2 f g-e h) \sin ^{-1}(c x)^3}{3 h^3}+\frac {2 a b c \left (f g^2-e g h+d h^2\right ) \tan ^{-1}\left (\frac {h+c^2 g x}{\sqrt {c^2 g^2-h^2} \sqrt {1-c^2 x^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {2 i b^2 c \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 i b^2 c \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}+\frac {2 i b^2 (2 f g-e h) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 i b^2 (2 f g-e h) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {\left (2 b^2 (2 f g-e h)\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i h x}{c g-\sqrt {c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{h^3}-\frac {\left (2 b^2 (2 f g-e h)\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i h x}{c g+\sqrt {c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{h^3}-\frac {(2 a b c (2 f g-e h)) \operatorname {Subst}\left (\int \frac {e^{i x} x}{c^2 g-i c e^{i x} h-c \sqrt {c^2 g^2-h^2}} \, dx,x,\sin ^{-1}(c x)\right )}{h^2}-\frac {(2 a b c (2 f g-e h)) \operatorname {Subst}\left (\int \frac {e^{i x} x}{c^2 g-i c e^{i x} h+c \sqrt {c^2 g^2-h^2}} \, dx,x,\sin ^{-1}(c x)\right )}{h^2}+\frac {\left (2 i b^2 c \left (f g^2-e g h+d h^2\right )\right ) \operatorname {Subst}\left (\int \log \left (1-\frac {2 i e^{i x} h}{2 c g-2 \sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {\left (2 i b^2 c \left (f g^2-e g h+d h^2\right )\right ) \operatorname {Subst}\left (\int \log \left (1-\frac {2 i e^{i x} h}{2 c g+2 \sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3 \sqrt {c^2 g^2-h^2}}\\ &=\frac {a^2 f x}{h^2}-\frac {2 b^2 f x}{h^2}-\frac {a^2 \left (f g^2-e g h+d h^2\right )}{h^3 (g+h x)}+\frac {2 a b f \sqrt {1-c^2 x^2}}{c h^2}+\frac {2 a b f x \sin ^{-1}(c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)}{h^3 (g+h x)}+\frac {2 b^2 f \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {i a b (2 f g-e h) \sin ^{-1}(c x)^2}{h^3}+\frac {b^2 f x \sin ^{-1}(c x)^2}{h^2}-\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3 (g+h x)}+\frac {i b^2 (2 f g-e h) \sin ^{-1}(c x)^3}{3 h^3}+\frac {2 a b c \left (f g^2-e g h+d h^2\right ) \tan ^{-1}\left (\frac {h+c^2 g x}{\sqrt {c^2 g^2-h^2} \sqrt {1-c^2 x^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {2 a b (2 f g-e h) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 i b^2 c \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 a b (2 f g-e h) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 i b^2 c \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}+\frac {2 i b^2 (2 f g-e h) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 i b^2 (2 f g-e h) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 (2 f g-e h) \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 (2 f g-e h) \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {(2 a b (2 f g-e h)) \operatorname {Subst}\left (\int \log \left (1-\frac {i c e^{i x} h}{c^2 g-c \sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3}+\frac {(2 a b (2 f g-e h)) \operatorname {Subst}\left (\int \log \left (1-\frac {i c e^{i x} h}{c^2 g+c \sqrt {c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3}+\frac {\left (2 b^2 c \left (f g^2-e g h+d h^2\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 i h x}{2 c g-2 \sqrt {c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {\left (2 b^2 c \left (f g^2-e g h+d h^2\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 i h x}{2 c g+2 \sqrt {c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{h^3 \sqrt {c^2 g^2-h^2}}\\ &=\frac {a^2 f x}{h^2}-\frac {2 b^2 f x}{h^2}-\frac {a^2 \left (f g^2-e g h+d h^2\right )}{h^3 (g+h x)}+\frac {2 a b f \sqrt {1-c^2 x^2}}{c h^2}+\frac {2 a b f x \sin ^{-1}(c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)}{h^3 (g+h x)}+\frac {2 b^2 f \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {i a b (2 f g-e h) \sin ^{-1}(c x)^2}{h^3}+\frac {b^2 f x \sin ^{-1}(c x)^2}{h^2}-\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3 (g+h x)}+\frac {i b^2 (2 f g-e h) \sin ^{-1}(c x)^3}{3 h^3}+\frac {2 a b c \left (f g^2-e g h+d h^2\right ) \tan ^{-1}\left (\frac {h+c^2 g x}{\sqrt {c^2 g^2-h^2} \sqrt {1-c^2 x^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {2 a b (2 f g-e h) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 i b^2 c \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 a b (2 f g-e h) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 i b^2 c \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}-\frac {2 b^2 c \left (f g^2-e g h+d h^2\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}+\frac {2 i b^2 (2 f g-e h) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 b^2 c \left (f g^2-e g h+d h^2\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}+\frac {2 i b^2 (2 f g-e h) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 (2 f g-e h) \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 (2 f g-e h) \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {(2 i a b (2 f g-e h)) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {i c h x}{c^2 g-c \sqrt {c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{h^3}-\frac {(2 i a b (2 f g-e h)) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {i c h x}{c^2 g+c \sqrt {c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{h^3}\\ &=\frac {a^2 f x}{h^2}-\frac {2 b^2 f x}{h^2}-\frac {a^2 \left (f g^2-e g h+d h^2\right )}{h^3 (g+h x)}+\frac {2 a b f \sqrt {1-c^2 x^2}}{c h^2}+\frac {2 a b f x \sin ^{-1}(c x)}{h^2}-\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)}{h^3 (g+h x)}+\frac {2 b^2 f \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {i a b (2 f g-e h) \sin ^{-1}(c x)^2}{h^3}+\frac {b^2 f x \sin ^{-1}(c x)^2}{h^2}-\frac {b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3 (g+h x)}+\frac {i b^2 (2 f g-e h) \sin ^{-1}(c x)^3}{3 h^3}+\frac {2 a b c \left (f g^2-e g h+d h^2\right ) \tan ^{-1}\left (\frac {h+c^2 g x}{\sqrt {c^2 g^2-h^2} \sqrt {1-c^2 x^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {2 a b (2 f g-e h) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 i b^2 c \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 a b (2 f g-e h) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 i b^2 c \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}-\frac {b^2 (2 f g-e h) \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {a^2 (2 f g-e h) \log (g+h x)}{h^3}+\frac {2 i a b (2 f g-e h) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 c \left (f g^2-e g h+d h^2\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}+\frac {2 i b^2 (2 f g-e h) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 i a b (2 f g-e h) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}+\frac {2 b^2 c \left (f g^2-e g h+d h^2\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3 \sqrt {c^2 g^2-h^2}}+\frac {2 i b^2 (2 f g-e h) \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 (2 f g-e h) \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )}{h^3}-\frac {2 b^2 (2 f g-e h) \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )}{h^3}\\ \end {align*}
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Mathematica [A] time = 1.39, size = 688, normalized size = 0.52 \[ \frac {\frac {6 b c \left (h (d h-e g)+f g^2\right ) \left (-i \left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (1+\frac {i h e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 g^2-h^2}-c g}\right )-\log \left (1-\frac {i h e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 g^2-h^2}+c g}\right )\right )-b \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )+b \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )\right )}{\sqrt {c^2 g^2-h^2}}+6 b (2 f g-e h) \left (i \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )-b \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )\right )+6 b (2 f g-e h) \left (i \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )-b \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )\right )-3 (2 f g-e h) \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1+\frac {i h e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 g^2-h^2}-c g}\right )-3 (2 f g-e h) \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac {i h e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 g^2-h^2}+c g}\right )-6 b f h \left (b x-\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}\right )-\frac {3 \left (a+b \sin ^{-1}(c x)\right )^2 \left (h (d h-e g)+f g^2\right )}{g+h x}+\frac {i (2 f g-e h) \left (a+b \sin ^{-1}(c x)\right )^3}{b}+3 f h x \left (a+b \sin ^{-1}(c x)\right )^2}{3 h^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} f x^{2} + a^{2} e x + a^{2} d + {\left (b^{2} f x^{2} + b^{2} e x + b^{2} d\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b f x^{2} + a b e x + a b d\right )} \arcsin \left (c x\right )}{h^{2} x^{2} + 2 \, g h x + g^{2}}, x\right ) \]
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 \[ \int \frac {{\left (f x^{2} + e x + d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 5.76, size = 0, normalized size = 0.00 \[ \int \frac {\left (f \,x^{2}+e x +d \right ) \left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (h x +g \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^2} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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