Optimal. Leaf size=1067 \[ -\frac {i b^2 \left (f g^2-e h g+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac {b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac {i a b \left (f g^2-e h g+d h^2\right ) \sin ^{-1}(c x)^2}{h^3}-\frac {b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac {b^2 \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)^2}{h^3}+\frac {b^2 \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)^2}{h^3}-\frac {b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {2 a b \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}+\frac {2 a b \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}-\frac {2 i b^2 \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 ) \sin ^{-1}(c x)}{h^3}-\frac {2 i b^2 \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 ) \sin ^{-1}(c x)}{h^3}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac {a b f \sin ^{-1}(c x)}{2 c^2 h}+\frac {a^2 f x^2}{2 h}-\frac {b^2 f x^2}{4 h}-\frac {a^2 (f g-e h) x}{h^2}+\frac {2 b^2 (f g-e h) x}{h^2}+\frac {a^2 \left (f g^2-e h g+d h^2\right ) \log (g+h x)}{h^3}-\frac {2 i a b \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}-\frac {2 i a b \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}+\frac {2 b^2 \left (f g^2-e h g+d h^2\right ) \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 \left (f g^2-e h g+d h^2\right ) \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 {a b (4 (f g-e h)-f h x) \sqrt {1-c^2 x^2}}{2 c h^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.95, antiderivative size = 1067, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 23, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.821, Rules used = {4759, 698, 4753, 12, 6742, 780, 216, 2404, 4741, 4519, 2190, 2279, 2391, 4619, 4677, 8, 4627, 4707, 4641, 30, 2531, 2282, 6589} \[ -\frac {i b^2 \left (f g^2-e h g+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac {b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac {i a b \left (f g^2-e h g+d h^2\right ) \sin ^{-1}(c x)^2}{h^3}-\frac {b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac {b^2 \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)^2}{h^3}+\frac {b^2 \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)^2}{h^3}-\frac {b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {2 a b \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}+\frac {2 a b \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}-\frac {2 i b^2 \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 ) \sin ^{-1}(c x)}{h^3}-\frac {2 i b^2 \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 ) \sin ^{-1}(c x)}{h^3}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac {a b f \sin ^{-1}(c x)}{2 c^2 h}+\frac {a^2 f x^2}{2 h}-\frac {b^2 f x^2}{4 h}-\frac {a^2 (f g-e h) x}{h^2}+\frac {2 b^2 (f g-e h) x}{h^2}+\frac {a^2 \left (f g^2-e h g+d h^2\right ) \log (g+h x)}{h^3}-\frac {2 i a b \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}-\frac {2 i a b \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}+\frac {2 b^2 \left (f g^2-e h g+d h^2\right ) \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 \left (f g^2-e h g+d h^2\right ) \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 {a b (4 (f g-e h)-f h x) \sqrt {1-c^2 x^2}}{2 c h^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 12
Rule 30
Rule 216
Rule 698
Rule 780
Rule 2190
Rule 2279
Rule 2282
Rule 2391
Rule 2404
Rule 2531
Rule 4519
Rule 4619
Rule 4627
Rule 4641
Rule 4677
Rule 4707
Rule 4741
Rule 4753
Rule 4759
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} \, dx &=\int \left (\frac {a^2 \left (d+e x+f x^2\right )}{g+h x}+\frac {2 a b \left (d+e x+f x^2\right ) \sin ^{-1}(c x)}{g+h x}+\frac {b^2 \left (d+e x+f x^2\right ) \sin ^{-1}(c x)^2}{g+h x}\right ) \, dx\\ &=a^2 \int \frac {d+e x+f x^2}{g+h x} \, dx+(2 a b) \int \frac {\left (d+e x+f x^2\right ) \sin ^{-1}(c x)}{g+h x} \, dx+b^2 \int \frac {\left (d+e x+f x^2\right ) \sin ^{-1}(c x)^2}{g+h x} \, dx\\ &=-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}+\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log (g+h x)}{h^3}+a^2 \int \left (\frac {-f g+e h}{h^2}+\frac {f x}{h}+\frac {f g^2-e g h+d h^2}{h^2 (g+h x)}\right ) \, dx+b^2 \int \left (\frac {(-f g+e h) \sin ^{-1}(c x)^2}{h^2}+\frac {f x \sin ^{-1}(c x)^2}{h}+\frac {\left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^2 (g+h x)}\right ) \, dx-(2 a b c) \int \frac {h x (-2 f g+2 e h+f h x)+2 \left (f g^2+h (-e g+d h)\right ) \log (g+h x)}{2 h^3 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {a^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}+\frac {a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}+\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log (g+h x)}{h^3}-\frac {(a b c) \int \frac {h x (-2 f g+2 e h+f h x)+2 \left (f g^2+h (-e g+d h)\right ) \log (g+h x)}{\sqrt {1-c^2 x^2}} \, dx}{h^3}+\frac {\left (b^2 f\right ) \int x \sin ^{-1}(c x)^2 \, dx}{h}-\frac {\left (b^2 (f g-e h)\right ) \int \sin ^{-1}(c x)^2 \, 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} \, dx}{h^2}\\ &=-\frac {a^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}-\frac {b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac {b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}+\frac {a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}+\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log (g+h x)}{h^3}-\frac {(a b c) \int \left (\frac {h x (-2 f g+2 e h+f h x)}{\sqrt {1-c^2 x^2}}+\frac {2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{\sqrt {1-c^2 x^2}}\right ) \, dx}{h^3}-\frac {\left (b^2 c f\right ) \int \frac {x^2 \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{h}+\frac {\left (2 b^2 c (f g-e h)\right ) \int \frac {x \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{h^2}+\frac {\left (b^2 \left (f g^2-e g h+d h^2\right )\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 {a^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac {b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac {b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac {i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac {a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}+\frac {2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log (g+h x)}{h^3}-\frac {(a b c) \int \frac {x (-2 f g+2 e h+f h x)}{\sqrt {1-c^2 x^2}} \, dx}{h^2}-\frac {\left (b^2 f\right ) \int x \, dx}{2 h}-\frac {\left (b^2 f\right ) \int \frac {\sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 c h}+\frac {\left (2 b^2 (f g-e h)\right ) \int 1 \, dx}{h^2}-\frac {\left (2 a b c \left (f g^2-e g h+d h^2\right )\right ) \int \frac {\log (g+h x)}{\sqrt {1-c^2 x^2}} \, dx}{h^3}+\frac {\left (b^2 \left (f g^2-e g h+d h^2\right )\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 \left (f g^2-e g h+d h^2\right )\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 {a^2 (f g-e h) x}{h^2}+\frac {2 b^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {b^2 f x^2}{4 h}-\frac {a b (4 (f g-e h)-f h x) \sqrt {1-c^2 x^2}}{2 c h^2}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac {b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}-\frac {b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac {b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac {i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac {(a b f) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{2 c h}-\frac {\left (2 b^2 \left (f g^2-e g h+d h^2\right )\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 \left (f g^2-e g h+d h^2\right )\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 a b c \left (f g^2-e g h+d h^2\right )\right ) \int \frac {\sin ^{-1}(c x)}{c g+c h x} \, dx}{h^2}\\ &=-\frac {a^2 (f g-e h) x}{h^2}+\frac {2 b^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {b^2 f x^2}{4 h}-\frac {a b (4 (f g-e h)-f h x) \sqrt {1-c^2 x^2}}{2 c h^2}-\frac {a b f \sin ^{-1}(c x)}{2 c^2 h}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac {b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}-\frac {b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac {b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac {i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right )\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 \left (f g^2-e g h+d h^2\right )\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 a b c \left (f g^2-e g h+d h^2\right )\right ) \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 {a^2 (f g-e h) x}{h^2}+\frac {2 b^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {b^2 f x^2}{4 h}-\frac {a b (4 (f g-e h)-f h x) \sqrt {1-c^2 x^2}}{2 c h^2}-\frac {a b f \sin ^{-1}(c x)}{2 c^2 h}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac {b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}-\frac {i a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3}-\frac {b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac {b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac {i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right )\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 \left (f g^2-e g h+d h^2\right )\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 a b c \left (f g^2-e g h+d h^2\right )\right ) \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 a b c \left (f g^2-e g h+d h^2\right )\right ) \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 {a^2 (f g-e h) x}{h^2}+\frac {2 b^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {b^2 f x^2}{4 h}-\frac {a b (4 (f g-e h)-f h x) \sqrt {1-c^2 x^2}}{2 c h^2}-\frac {a b f \sin ^{-1}(c x)}{2 c^2 h}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac {b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}-\frac {i a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3}-\frac {b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac {b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac {i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac {2 a b \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}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \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 \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}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \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 {\left (2 a b \left (f g^2-e g h+d h^2\right )\right ) \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 a b \left (f g^2-e g h+d h^2\right )\right ) \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 {a^2 (f g-e h) x}{h^2}+\frac {2 b^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {b^2 f x^2}{4 h}-\frac {a b (4 (f g-e h)-f h x) \sqrt {1-c^2 x^2}}{2 c h^2}-\frac {a b f \sin ^{-1}(c x)}{2 c^2 h}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac {b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}-\frac {i a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3}-\frac {b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac {b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac {i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac {2 a b \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}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \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 \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}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \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 {\left (2 i a b \left (f g^2-e g h+d h^2\right )\right ) \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 {\left (2 i a b \left (f g^2-e g h+d h^2\right )\right ) \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 g-e h) x}{h^2}+\frac {2 b^2 (f g-e h) x}{h^2}+\frac {a^2 f x^2}{2 h}-\frac {b^2 f x^2}{4 h}-\frac {a b (4 (f g-e h)-f h x) \sqrt {1-c^2 x^2}}{2 c h^2}-\frac {a b f \sin ^{-1}(c x)}{2 c^2 h}-\frac {2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac {a b f x^2 \sin ^{-1}(c x)}{h}-\frac {2 b^2 (f g-e h) \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac {b^2 f x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac {b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}-\frac {i a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3}-\frac {b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac {b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac {i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac {2 a b \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}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \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 \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}+\frac {b^2 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac {2 i a b \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}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \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 \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}-\frac {2 i b^2 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \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 \left (f g^2-e g h+d h^2\right ) \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 = 0.75, size = 556, normalized size = 0.52 \[ \frac {-24 b \left (h (d h-e g)+f g^2\right ) \left (i \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )-b \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt {c^2 g^2-h^2}}\right )\right )-24 b \left (h (d h-e g)+f g^2\right ) \left (i \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )-b \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt {c^2 g^2-h^2}}\right )\right )+12 \left (a+b \sin ^{-1}(c x)\right )^2 \left (h (d h-e g)+f g^2\right ) \log \left (1+\frac {i h e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 g^2-h^2}-c g}\right )+12 \left (a+b \sin ^{-1}(c x)\right )^2 \left (h (d h-e g)+f g^2\right ) \log \left (1-\frac {i h e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 g^2-h^2}+c g}\right )+24 b h (f g-e h) \left (b x-\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}\right )-3 b f h^2 \left (-\frac {2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b c^2}+b x^2\right )-\frac {4 i \left (a+b \sin ^{-1}(c x)\right )^3 \left (h (d h-e g)+f g^2\right )}{b}+12 h x (e h-f g) \left (a+b \sin ^{-1}(c x)\right )^2+6 f h^2 x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{12 h^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} f x^{2} + a^{2} e x + a^{2} d + {\left (b^{2} f x^{2} + b^{2} e x + b^{2} d\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b f x^{2} + a b e x + a b d\right )} \arcsin \left (c x\right )}{h x + g}, x\right ) \]
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 \[ \int \frac {{\left (f x^{2} + e x + d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{h x + g}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.55, size = 0, normalized size = 0.00 \[ \int \frac {\left (f \,x^{2}+e x +d \right ) \left (a +b \arcsin \left (c x \right )\right )^{2}}{h x +g}\, dx \]
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 \[ a^{2} e {\left (\frac {x}{h} - \frac {g \log \left (h x + g\right )}{h^{2}}\right )} + \frac {1}{2} \, a^{2} f {\left (\frac {2 \, g^{2} \log \left (h x + g\right )}{h^{3}} + \frac {h x^{2} - 2 \, g x}{h^{2}}\right )} + \frac {a^{2} d \log \left (h x + g\right )}{h} + \int \frac {{\left (b^{2} f x^{2} + b^{2} e x + b^{2} d\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b f x^{2} + a b e x + a b d\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{h x + g}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (f\,x^2+e\,x+d\right )}{g+h\,x} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (d + e x + f x^{2}\right )}{g + h x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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