Optimal. Leaf size=1442 \[ \frac {a^2 \sqrt {d-c^2 d x^2}}{g}-\frac {2 b^2 \sqrt {d-c^2 d x^2}}{g}-\frac {2 a b c x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}+\frac {2 a b \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{g}-\frac {2 b^2 c x \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{g \sqrt {1-c^2 x^2}}+\frac {b^2 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)^2}{g}+\frac {c x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{3 b g \sqrt {1-c^2 x^2}}-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{3 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{3 b c (f+g x)}-\frac {a^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {ArcTan}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {2 i a b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {ArcSin}(c x) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {i b^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)^2 \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {2 i a b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {ArcSin}(c x) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {i b^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)^2 \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {2 a b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {2 b^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {ArcSin}(c x) \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {2 a b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {2 b^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {ArcSin}(c x) \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {2 i b^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {2 i b^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 2.03, antiderivative size = 1442, normalized size of antiderivative = 1.00, number of steps
used = 38, number of rules used = 23, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.697, Rules used = {4861, 4849,
697, 4841, 4883, 1668, 12, 739, 210, 4881, 4767, 8, 4857, 3404, 2296, 2221, 2317, 2438, 4715, 267,
2611, 2320, 6724} \begin {gather*} \frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{3 b c (f+g x)}+\frac {c x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{3 b g \sqrt {1-c^2 x^2}}-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{3 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {b^2 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)^2}{g}-\frac {2 b^2 c x \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{g \sqrt {1-c^2 x^2}}+\frac {2 a b \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{g}-\frac {a^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {ArcTan}\left (\frac {f x c^2+g}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {i b^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)^2 \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {2 i a b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {ArcSin}(c x) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {i b^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)^2 \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {2 i a b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {ArcSin}(c x) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {2 b^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {ArcSin}(c x) \text {Li}_2\left (\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {2 a b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {Li}_2\left (\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {2 b^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {ArcSin}(c x) \text {Li}_2\left (\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {2 a b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {Li}_2\left (\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {2 i b^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {Li}_3\left (\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {2 i b^2 \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {Li}_3\left (\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {2 a b c x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}+\frac {a^2 \sqrt {d-c^2 d x^2}}{g}-\frac {2 b^2 \sqrt {d-c^2 d x^2}}{g} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 12
Rule 210
Rule 267
Rule 697
Rule 739
Rule 1668
Rule 2221
Rule 2296
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3404
Rule 4715
Rule 4767
Rule 4841
Rule 4849
Rule 4857
Rule 4861
Rule 4881
Rule 4883
Rule 6724
Rubi steps
\begin {align*} \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{f+g x} \, dx &=\frac {\sqrt {d-c^2 d x^2} \int \frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{f+g x} \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (-g-2 c^2 f x-c^2 g x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^3}{(f+g x)^2} \, dx}{3 b c \sqrt {1-c^2 x^2}}\\ &=\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt {1-c^2 x^2}}-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}+\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (\frac {1}{f+g x}-\frac {c^2 \left (g x+\frac {f^2}{f+g x}\right )}{g^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt {1-c^2 x^2}}-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}+\frac {\sqrt {d-c^2 d x^2} \int \left (-\frac {a^2 \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right )}{g^2 (f+g x) \sqrt {1-c^2 x^2}}-\frac {2 a b \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \sin ^{-1}(c x)}{g^2 (f+g x) \sqrt {1-c^2 x^2}}-\frac {b^2 \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \sin ^{-1}(c x)^2}{g^2 (f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt {1-c^2 x^2}}-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac {\left (a^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (2 a b \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \sin ^{-1}(c x)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \sin ^{-1}(c x)^2}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}\\ &=\frac {a^2 \sqrt {d-c^2 d x^2}}{g}+\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt {1-c^2 x^2}}-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac {\left (a^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {c^2 g^2 \left (c^2 f^2-g^2\right )}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{c^2 g^4 \sqrt {1-c^2 x^2}}-\frac {\left (2 a b \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {c^2 g x \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}+\frac {\left (c^2 f^2-g^2\right ) \sin ^{-1}(c x)}{(f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {c^2 g x \sin ^{-1}(c x)^2}{\sqrt {1-c^2 x^2}}+\frac {\left (c^2 f^2-g^2\right ) \sin ^{-1}(c x)^2}{(f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{g^2 \sqrt {1-c^2 x^2}}\\ &=\frac {a^2 \sqrt {d-c^2 d x^2}}{g}+\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt {1-c^2 x^2}}-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac {\left (2 a b c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{g \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \sin ^{-1}(c x)^2}{\sqrt {1-c^2 x^2}} \, dx}{g \sqrt {1-c^2 x^2}}-\frac {\left (a^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (2 a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \int \frac {\sin ^{-1}(c x)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \int \frac {\sin ^{-1}(c x)^2}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}\\ &=\frac {a^2 \sqrt {d-c^2 d x^2}}{g}+\frac {2 a b \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g}+\frac {b^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g}+\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt {1-c^2 x^2}}-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac {\left (2 a b c \sqrt {d-c^2 d x^2}\right ) \int 1 \, dx}{g \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 c \sqrt {d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{g \sqrt {1-c^2 x^2}}+\frac {\left (a^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1}{-c^2 f^2+g^2-x^2} \, dx,x,\frac {g+c^2 f x}{\sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (2 a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x}{c f+g \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x^2}{c f+g \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt {1-c^2 x^2}}\\ &=\frac {a^2 \sqrt {d-c^2 d x^2}}{g}-\frac {2 a b c x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}+\frac {2 a b \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g}-\frac {2 b^2 c x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g \sqrt {1-c^2 x^2}}+\frac {b^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g}+\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt {1-c^2 x^2}}-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac {a^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{g \sqrt {1-c^2 x^2}}-\frac {\left (4 a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c e^{i x} f+i g-i e^{2 i x} g} \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} x^2}{2 c e^{i x} f+i g-i e^{2 i x} g} \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt {1-c^2 x^2}}\\ &=\frac {a^2 \sqrt {d-c^2 d x^2}}{g}-\frac {2 b^2 \sqrt {d-c^2 d x^2}}{g}-\frac {2 a b c x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}+\frac {2 a b \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g}-\frac {2 b^2 c x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g \sqrt {1-c^2 x^2}}+\frac {b^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g}+\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt {1-c^2 x^2}}-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac {a^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (4 i a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c f-2 i e^{i x} g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{g \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (4 i a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c f-2 i e^{i x} g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{g \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (2 i b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} x^2}{2 c f-2 i e^{i x} g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{g \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (2 i b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} x^2}{2 c f-2 i e^{i x} g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{g \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\\ &=\frac {a^2 \sqrt {d-c^2 d x^2}}{g}-\frac {2 b^2 \sqrt {d-c^2 d x^2}}{g}-\frac {2 a b c x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}+\frac {2 a b \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g}-\frac {2 b^2 c x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g \sqrt {1-c^2 x^2}}+\frac {b^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g}+\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt {1-c^2 x^2}}-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac {a^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {2 i a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {2 i a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (2 i a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i e^{i x} g}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (2 i a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i e^{i x} g}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (2 i b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int x \log \left (1-\frac {2 i e^{i x} g}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (2 i b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int x \log \left (1-\frac {2 i e^{i x} g}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\\ &=\frac {a^2 \sqrt {d-c^2 d x^2}}{g}-\frac {2 b^2 \sqrt {d-c^2 d x^2}}{g}-\frac {2 a b c x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}+\frac {2 a b \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g}-\frac {2 b^2 c x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g \sqrt {1-c^2 x^2}}+\frac {b^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g}+\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt {1-c^2 x^2}}-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac {a^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {2 i a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {2 i a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {2 b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {2 b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (2 a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i g x}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (2 a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i g x}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (\frac {2 i e^{i x} g}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (\frac {2 i e^{i x} g}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\\ &=\frac {a^2 \sqrt {d-c^2 d x^2}}{g}-\frac {2 b^2 \sqrt {d-c^2 d x^2}}{g}-\frac {2 a b c x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}+\frac {2 a b \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g}-\frac {2 b^2 c x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g \sqrt {1-c^2 x^2}}+\frac {b^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g}+\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt {1-c^2 x^2}}-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac {a^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {2 i a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {2 i a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {2 a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {2 b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {2 a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {2 b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (2 i b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i g x}{c f-\sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (2 i b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i g x}{c f+\sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\\ &=\frac {a^2 \sqrt {d-c^2 d x^2}}{g}-\frac {2 b^2 \sqrt {d-c^2 d x^2}}{g}-\frac {2 a b c x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}+\frac {2 a b \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g}-\frac {2 b^2 c x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g \sqrt {1-c^2 x^2}}+\frac {b^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g}+\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt {1-c^2 x^2}}-\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac {a^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {2 i a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {2 i a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {2 a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {2 b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {2 a b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {2 b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {2 i b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {2 i b^2 (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \text {Li}_3\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.97, size = 516, normalized size = 0.36 \begin {gather*} \frac {\sqrt {d-c^2 d x^2} \left (\left (c^2 f^2-g^2\right ) (a+b \text {ArcSin}(c x))^3+c^2 g x (f+g x) (a+b \text {ArcSin}(c x))^3+g^2 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^3+3 b c (f+g x) \left (g \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2-2 b g \left (a c x+b \sqrt {1-c^2 x^2}+b c x \text {ArcSin}(c x)\right )+i \sqrt {c^2 f^2-g^2} \left ((a+b \text {ArcSin}(c x))^2 \log \left (1+\frac {i e^{i \text {ArcSin}(c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-(a+b \text {ArcSin}(c x))^2 \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-2 i b (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+2 i b (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )+2 b^2 \text {PolyLog}\left (3,\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-2 b^2 \text {PolyLog}\left (3,\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )\right )\right )}{3 b c g^2 (f+g x) \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2} \sqrt {-c^{2} d \,x^{2}+d}}{g x +f}\, dx\]
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 \begin {gather*} \text {Exception raised: ValueError} \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 {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{f + g x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 )}^2\,\sqrt {d-c^2\,d\,x^2}}{f+g\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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