Optimal. Leaf size=2989 \[ \text {result too large to display} \]
[Out]
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Rubi [A]
time = 3.36, antiderivative size = 2989, normalized size of antiderivative = 1.00, number of steps
used = 74, number of rules used = 35, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.061, Rules used = {4861, 4851,
4741, 4737, 4723, 327, 222, 4767, 4739, 455, 45, 4783, 4795, 4715, 267, 272, 4849, 697, 4841, 4883,
1668, 12, 739, 210, 4881, 8, 4857, 3404, 2296, 2221, 2317, 2438, 2611, 2320, 6724}
\begin {gather*} \text {Too large to display} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 12
Rule 45
Rule 210
Rule 222
Rule 267
Rule 272
Rule 327
Rule 455
Rule 697
Rule 739
Rule 1668
Rule 2221
Rule 2296
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3404
Rule 4715
Rule 4723
Rule 4737
Rule 4739
Rule 4741
Rule 4767
Rule 4783
Rule 4795
Rule 4841
Rule 4849
Rule 4851
Rule 4857
Rule 4861
Rule 4881
Rule 4883
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{f+g x} \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{f+g x} \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-\frac {c^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{g^4}+\frac {c^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{g^3}-\frac {c^4 f x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{g^2}+\frac {c^4 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{g}+\frac {\left (-c^2 f^2+g^2\right )^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{g^4 (f+g x)}\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {\left (c^4 d^2 f \sqrt {d-c^2 d x^2}\right ) \int x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{g^2 \sqrt {1-c^2 x^2}}+\frac {\left (c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{g \sqrt {1-c^2 x^2}}-\frac {\left (c^2 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{g^4 \sqrt {1-c^2 x^2}}+\frac {\left (c^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{g^3 \sqrt {1-c^2 x^2}}+\frac {\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{f+g x} \, dx}{g^4 \sqrt {1-c^2 x^2}}\\ &=-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \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 g^4 (f+g x)}-\frac {\left (c^4 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {\left (b c^5 d^2 f \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{2 g^2 \sqrt {1-c^2 x^2}}+\frac {\left (c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{5 g \sqrt {1-c^2 x^2}}-\frac {\left (2 b c^5 d^2 \sqrt {d-c^2 d x^2}\right ) \int x^4 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{5 g \sqrt {1-c^2 x^2}}-\frac {\left (c^2 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {\left (b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{g^4 \sqrt {1-c^2 x^2}}+\frac {\left (2 b c d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \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 g^4 \sqrt {1-c^2 x^2}}\\ &=\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \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 g^4 (f+g x)}-\frac {\left (c^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b c^3 d^2 f \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^6 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {\left (2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{15 g \sqrt {1-c^2 x^2}}+\frac {\left (2 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{15 g \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^6 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5}{\sqrt {1-c^2 x^2}} \, dx}{25 g \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^4 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2 g^4 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 c^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (1-\frac {c^2 x^2}{3}\right )}{\sqrt {1-c^2 x^2}} \, dx}{3 g^3 \sqrt {1-c^2 x^2}}+\frac {\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \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}{g^4 \sqrt {1-c^2 x^2}}\\ &=\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \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 g^4 (f+g x)}-\frac {\left (3 b^2 c^4 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{32 g^2 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^4 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {\left (4 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{15 g \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3}{\sqrt {1-c^2 x^2}} \, dx}{45 g \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^6 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 g^4 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1-\frac {c^2 x}{3}}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{3 g^3 \sqrt {1-c^2 x^2}}+\frac {\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \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}{g^4 \sqrt {1-c^2 x^2}}\\ &=-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \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 g^4 (f+g x)}-\frac {\left (3 b^2 c^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{64 g^2 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{16 g^2 \sqrt {1-c^2 x^2}}+\frac {\left (4 b^2 c d^2 \sqrt {d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{15 g \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{45 g \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^6 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^4 \sqrt {1-c^2 x}}-\frac {2 \sqrt {1-c^2 x}}{c^4}+\frac {\left (1-c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {2}{3 \sqrt {1-c^2 x}}+\frac {1}{3} \sqrt {1-c^2 x}\right ) \, dx,x,x^2\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {\left (a^2 d^2 \left (c^2 f^2-g^2\right )^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^6 \sqrt {1-c^2 x^2}}-\frac {\left (2 a b d^2 \left (c^2 f^2-g^2\right )^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)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 \left (c^2 f^2-g^2\right )^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^6 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 b^2 d^2 \sqrt {d-c^2 d x^2}}{25 g}+\frac {4 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}}{9 g^3}+\frac {a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}+\frac {4 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 g}+\frac {2 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 g^3}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 g}+\frac {b^2 c d^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 g^2 \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {4 b^2 c d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 g \sqrt {1-c^2 x^2}}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \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 g^4 (f+g x)}-\frac {\left (4 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{15 g \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^2 \sqrt {1-c^2 x}}-\frac {\sqrt {1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {\left (a^2 d^2 \left (c^2 f^2-g^2\right )^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^8 \sqrt {1-c^2 x^2}}-\frac {\left (2 a b d^2 \left (c^2 f^2-g^2\right )^2 \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^6 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 \left (c^2 f^2-g^2\right )^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^6 \sqrt {1-c^2 x^2}}\\ &=\frac {52 b^2 d^2 \sqrt {d-c^2 d x^2}}{225 g}+\frac {4 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}}{9 g^3}+\frac {a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}+\frac {26 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 g}+\frac {2 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 g^3}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 g}+\frac {b^2 c d^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 g^2 \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {4 b^2 c d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 g \sqrt {1-c^2 x^2}}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \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 g^4 (f+g x)}-\frac {\left (2 a b c^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{g^5 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^2 d^2 \left (c^2 f^2-g^2\right )^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^5 \sqrt {1-c^2 x^2}}-\frac {\left (a^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (2 a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \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^6 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \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^6 \sqrt {1-c^2 x^2}}\\ &=\frac {52 b^2 d^2 \sqrt {d-c^2 d x^2}}{225 g}+\frac {4 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}}{9 g^3}+\frac {a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}+\frac {26 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 g}+\frac {2 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 g^3}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 g}+\frac {2 a b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac {b^2 c d^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 g^2 \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {4 b^2 c d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 g \sqrt {1-c^2 x^2}}+\frac {b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g^5}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \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 g^4 (f+g x)}-\frac {\left (2 a b c d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int 1 \, dx}{g^5 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 c d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{g^5 \sqrt {1-c^2 x^2}}+\frac {\left (a^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \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^6 \sqrt {1-c^2 x^2}}-\frac {\left (2 a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \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^6 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \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^6 \sqrt {1-c^2 x^2}}\\ &=\frac {52 b^2 d^2 \sqrt {d-c^2 d x^2}}{225 g}+\frac {4 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}}{9 g^3}+\frac {a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {2 a b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {1-c^2 x^2}}+\frac {26 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 g}+\frac {2 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 g^3}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 g}+\frac {2 a b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac {b^2 c d^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 g^2 \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {4 b^2 c d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 g \sqrt {1-c^2 x^2}}-\frac {2 b^2 c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5 \sqrt {1-c^2 x^2}}+\frac {b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g^5}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \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 g^4 (f+g x)}-\frac {a^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{g^5 \sqrt {1-c^2 x^2}}-\frac {\left (4 a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \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^6 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \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^6 \sqrt {1-c^2 x^2}}\\ &=\frac {52 b^2 d^2 \sqrt {d-c^2 d x^2}}{225 g}+\frac {4 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}}{9 g^3}+\frac {a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {2 b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {2 a b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {1-c^2 x^2}}+\frac {26 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 g}+\frac {2 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 g^3}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 g}+\frac {2 a b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac {b^2 c d^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 g^2 \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {4 b^2 c d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 g \sqrt {1-c^2 x^2}}-\frac {2 b^2 c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5 \sqrt {1-c^2 x^2}}+\frac {b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g^5}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \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 g^4 (f+g x)}-\frac {a^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {\left (4 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^5 \sqrt {1-c^2 x^2}}-\frac {\left (4 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^5 \sqrt {1-c^2 x^2}}+\frac {\left (2 i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^5 \sqrt {1-c^2 x^2}}-\frac {\left (2 i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^5 \sqrt {1-c^2 x^2}}\\ &=\frac {52 b^2 d^2 \sqrt {d-c^2 d x^2}}{225 g}+\frac {4 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}}{9 g^3}+\frac {a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {2 b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {2 a b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {1-c^2 x^2}}+\frac {26 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 g}+\frac {2 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 g^3}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 g}+\frac {2 a b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac {b^2 c d^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 g^2 \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {4 b^2 c d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 g \sqrt {1-c^2 x^2}}-\frac {2 b^2 c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5 \sqrt {1-c^2 x^2}}+\frac {b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g^5}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \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 g^4 (f+g x)}-\frac {a^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}-\frac {2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}-\frac {i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}-\frac {\left (2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {\left (2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}-\frac {\left (2 i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {\left (2 i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}\\ &=\frac {52 b^2 d^2 \sqrt {d-c^2 d x^2}}{225 g}+\frac {4 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}}{9 g^3}+\frac {a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {2 b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {2 a b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {1-c^2 x^2}}+\frac {26 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 g}+\frac {2 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 g^3}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 g}+\frac {2 a b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac {b^2 c d^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 g^2 \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {4 b^2 c d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 g \sqrt {1-c^2 x^2}}-\frac {2 b^2 c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5 \sqrt {1-c^2 x^2}}+\frac {b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g^5}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \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 g^4 (f+g x)}-\frac {a^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}-\frac {2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}-\frac {i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {2 b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}-\frac {2 b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}-\frac {\left (2 a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {\left (2 a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}\\ &=\frac {52 b^2 d^2 \sqrt {d-c^2 d x^2}}{225 g}+\frac {4 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}}{9 g^3}+\frac {a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {2 b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {2 a b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {1-c^2 x^2}}+\frac {26 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 g}+\frac {2 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 g^3}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 g}+\frac {2 a b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac {b^2 c d^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 g^2 \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {4 b^2 c d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 g \sqrt {1-c^2 x^2}}-\frac {2 b^2 c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5 \sqrt {1-c^2 x^2}}+\frac {b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g^5}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \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 g^4 (f+g x)}-\frac {a^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}-\frac {2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}-\frac {i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {2 a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {2 b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}-\frac {2 a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}-\frac {2 b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {\left (2 i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}-\frac {\left (2 i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}\\ &=\frac {52 b^2 d^2 \sqrt {d-c^2 d x^2}}{225 g}+\frac {4 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}}{9 g^3}+\frac {a^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {2 b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {b^2 c^2 d^2 f x \sqrt {d-c^2 d x^2}}{64 g^2}+\frac {b^2 c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{4 g^4}+\frac {b^2 c^4 d^2 f x^3 \sqrt {d-c^2 d x^2}}{32 g^2}+\frac {4 a b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {2 a b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {1-c^2 x^2}}+\frac {26 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{675 g}+\frac {2 b^2 d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 g^3}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 g}+\frac {2 a b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}+\frac {b^2 c d^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 g^2 \sqrt {1-c^2 x^2}}-\frac {b^2 c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {4 b^2 c d^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{15 g \sqrt {1-c^2 x^2}}-\frac {2 b^2 c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^5 \sqrt {1-c^2 x^2}}+\frac {b^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{g^5}+\frac {2 b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^4 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 g \sqrt {1-c^2 x^2}}-\frac {2 b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 g^2 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 g \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 g^4}-\frac {c^2 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 g}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 g^2}+\frac {c^4 d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 g^3}-\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b g^4 \sqrt {1-c^2 x^2}}+\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 \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 g^4 (f+g x)}-\frac {a^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}-\frac {2 i a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}-\frac {i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {2 a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {2 b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}-\frac {2 a b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}-\frac {2 b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}+\frac {2 i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}-\frac {2 i b^2 d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \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^6 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 3.09, size = 1277, normalized size = 0.43 \begin {gather*} \frac {d^2 \sqrt {d-c^2 d x^2} \left (-\frac {c^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{2 g^4}-\frac {c^4 f x^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{4 g^2}+\frac {c^4 x^4 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{5 g}-\frac {\left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2}{3 g^3}-\frac {c f \left (c^2 f^2-2 g^2\right ) (a+b \text {ArcSin}(c x))^3}{6 b g^4}-\frac {\left (-c^2 f^2+g^2\right )^2 \left (-1+c^2 x^2\right ) (a+b \text {ArcSin}(c x))^3}{3 b c g^4 (f+g x)}+\frac {b c f \left (c^2 f^2-2 g^2\right ) \left (c x \left (2 a c x+b \sqrt {1-c^2 x^2}\right )+b \left (-1+2 c^2 x^2\right ) \text {ArcSin}(c x)\right )}{4 g^4}+\frac {b c f \left (8 a c^4 x^4+b c x \sqrt {1-c^2 x^2} \left (3+2 c^2 x^2\right )+b \left (-3+8 c^4 x^4\right ) \text {ArcSin}(c x)\right )}{64 g^2}+\frac {2 b \left (c^2 f^2-2 g^2\right ) \left (-b \sqrt {1-c^2 x^2} \left (-7+c^2 x^2\right )+9 c x (a+b \text {ArcSin}(c x))-3 c^3 x^3 (a+b \text {ArcSin}(c x))\right )}{27 g^3}-\frac {2 b \left (b \sqrt {1-c^2 x^2} \left (8+4 c^2 x^2+3 c^4 x^4\right )+15 c^5 x^5 (a+b \text {ArcSin}(c x))\right )}{375 g}+\frac {c f \left (6 b c x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2-2 (a+b \text {ArcSin}(c x))^3-3 b^2 \left (c x \left (2 a c x+b \sqrt {1-c^2 x^2}\right )+b \left (-1+2 c^2 x^2\right ) \text {ArcSin}(c x)\right )\right )}{48 b g^2}-\frac {9 c^2 x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2-2 b \left (b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+3 c^3 x^3 (a+b \text {ArcSin}(c x))\right )+18 \left (\sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2-2 b \left (a c x+b \sqrt {1-c^2 x^2}+b c x \text {ArcSin}(c x)\right )\right )}{135 g}+\frac {\left (-c^2 f^2+g^2\right )^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+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^6 (f+g x)}\right )}{\sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{g x +f}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{f+g\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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