Optimal. Leaf size=1589 \[ -\frac {i \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 (c f-g)^3}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) (c f-g)^3}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \csc ^2\left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) (c f-g)^3}{24 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) (c f-g)^3}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {1-c^2 x^2} \cot \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) (c f-g)^3}{6 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right ) (c f-g)^3}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right ) (c f-g)^3}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {i (c f+2 g) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 (c f-g)^2}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {(c f+2 g) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) (c f-g)^2}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f+2 g) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right ) (c f-g)^2}{c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {i b^2 (c f+2 g) \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right ) (c f-g)^2}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {i (c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {i (c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^2\left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {i b^2 (c f+g)^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{-i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {i b^2 (c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{-i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right )}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \tan \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right )}{24 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 (c f+g)^3 \sqrt {1-c^2 x^2} \tan \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right )}{6 c^4 d^2 \sqrt {d-c^2 d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.11, antiderivative size = 1589, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 12, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4777, 4775, 4773, 3318, 4186, 3767, 8, 4184, 3717, 2190, 2279, 2391} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2190
Rule 2279
Rule 2391
Rule 3318
Rule 3717
Rule 3767
Rule 4184
Rule 4186
Rule 4773
Rule 4775
Rule 4777
Rubi steps
\begin {align*} \int \frac {(f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \left (\frac {(c f+g)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^3 (-1+c x)^2 \sqrt {1-c^2 x^2}}-\frac {(c f-2 g) (c f+g)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^3 (-1+c x) \sqrt {1-c^2 x^2}}+\frac {(c f-g)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^3 (1+c x)^2 \sqrt {1-c^2 x^2}}+\frac {(c f-g)^2 (c f+2 g) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^3 (1+c x) \sqrt {1-c^2 x^2}}\right ) \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {\left ((c f-g)^3 \sqrt {1-c^2 x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{(1+c x)^2 \sqrt {1-c^2 x^2}} \, dx}{4 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left ((c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{(-1+c x) \sqrt {1-c^2 x^2}} \, dx}{4 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left ((c f+g)^3 \sqrt {1-c^2 x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{(-1+c x)^2 \sqrt {1-c^2 x^2}} \, dx}{4 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left ((c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{(1+c x) \sqrt {1-c^2 x^2}} \, dx}{4 c^3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {\left ((c f-g)^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^2}{(c+c \sin (x))^2} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left ((c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^2}{-c+c \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left ((c f+g)^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^2}{(-c+c \sin (x))^2} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left ((c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^2}{c+c \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {\left ((c f-g)^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \csc ^4\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{16 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left ((c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{8 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left ((c f+g)^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \csc ^4\left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{16 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left ((c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{8 c^4 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {(c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left ((c f-g)^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 (c f-g)^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \csc ^2\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \cot \left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left ((c f+g)^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 (c f+g)^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \csc ^2\left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b (c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \cot \left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 c^4 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {i (c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {i (c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b (c f-g)^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \cot \left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 (c f-g)^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int 1 \, dx,x,\cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )\right )}{6 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{-i x} (a+b x)}{1-i e^{-i x}} \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f+g)^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \cot \left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 (c f+g)^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int 1 \, dx,x,\cot \left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right )\right )}{6 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b (c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} (a+b x)}{1-i e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {i (c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {i (c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {i (c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {i (c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 (c f+g)^3 \sqrt {1-c^2 x^2} \cot \left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 (c f-g)^3 \sqrt {1-c^2 x^2} \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b (c f-g)^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} (a+b x)}{1-i e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 (c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^{-i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f+g)^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{-i x} (a+b x)}{1-i e^{-i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 (c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {i (c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {i (c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {i (c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {i (c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 (c f+g)^3 \sqrt {1-c^2 x^2} \cot \left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 (c f-g)^3 \sqrt {1-c^2 x^2} \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 (c f-g)^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (i b^2 (c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{-i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 (c f+g)^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^{-i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (i b^2 (c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {i (c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {i (c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {i (c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {i (c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 (c f+g)^3 \sqrt {1-c^2 x^2} \cot \left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 (c f-g)^3 \sqrt {1-c^2 x^2} \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {i b^2 (c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{-i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {i b^2 (c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (i b^2 (c f-g)^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (i b^2 (c f+g)^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{-i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {i (c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {i (c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {i (c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {i (c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 (c f+g)^3 \sqrt {1-c^2 x^2} \cot \left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 (c f-g)^3 \sqrt {1-c^2 x^2} \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {i b^2 (c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{-i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {i b^2 (c f+g)^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{-i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {i b^2 (c f-g)^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {i b^2 (c f-g)^2 (c f+2 g) \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f-2 g) (c f+g)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right ) \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 6.28, size = 715, normalized size = 0.45 \[ \frac {\sqrt {1-c^2 x^2} \left (-\frac {(c f-g)^3 \left (-2 \left (-\tan \left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b}-4 \left (i \log \left (1+e^{\frac {1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right ) \left (a+b \sin ^{-1}(c x)\right )-b \text {Li}_2\left (-e^{\frac {1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right )\right )\right )\right )+2 b \sec ^2\left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )+\tan \left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right ) \sec ^2\left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )^2+4 b^2 \tan \left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right )\right )}{24 c^4}-\frac {(c f+g)^3 \left (2 \left (-\tan \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b}+4 \left (i \log \left (1+e^{\frac {1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right ) \left (a+b \sin ^{-1}(c x)\right )+b \text {Li}_2\left (-e^{\frac {1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right )\right )\right )\right )+2 b \sec ^2\left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )-\tan \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \sec ^2\left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2-4 b^2 \tan \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right )\right )}{24 c^4}+\frac {(c f+2 g) (c f-g)^2 \left (-\tan \left (\frac {\pi }{4}-\frac {1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b}-4 \left (i \log \left (1+e^{\frac {1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right ) \left (a+b \sin ^{-1}(c x)\right )-b \text {Li}_2\left (-e^{\frac {1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right )\right )\right )\right )}{4 c^4}-\frac {(c f-2 g) (c f+g)^2 \left (-\tan \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b}+4 \left (i \log \left (1+e^{\frac {1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right ) \left (a+b \sin ^{-1}(c x)\right )+b \text {Li}_2\left (-e^{\frac {1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right )\right )\right )\right )}{4 c^4}\right )}{d^2 \sqrt {d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 2.27, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a^{2} g^{3} x^{3} + 3 \, a^{2} f g^{2} x^{2} + 3 \, a^{2} f^{2} g x + a^{2} f^{3} + {\left (b^{2} g^{3} x^{3} + 3 \, b^{2} f g^{2} x^{2} + 3 \, b^{2} f^{2} g x + b^{2} f^{3}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b g^{3} x^{3} + 3 \, a b f g^{2} x^{2} + 3 \, a b f^{2} g x + a b f^{3}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.60, size = 13136, normalized size = 8.27 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, a b c f^{3} {\left (\frac {1}{c^{4} d^{\frac {5}{2}} x^{2} - c^{2} d^{\frac {5}{2}}} + \frac {2 \, \log \left (c x + 1\right )}{c^{2} d^{\frac {5}{2}}} + \frac {2 \, \log \left (c x - 1\right )}{c^{2} d^{\frac {5}{2}}}\right )} + \frac {2}{3} \, a b f^{3} {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} \arcsin \left (c x\right ) + \frac {1}{3} \, a^{2} f^{3} {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} + \frac {1}{3} \, a^{2} g^{3} {\left (\frac {3 \, x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} - \frac {2}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4} d}\right )} - a^{2} f g^{2} {\left (\frac {x}{\sqrt {-c^{2} d x^{2} + d} c^{2} d^{2}} - \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d}\right )} + \sqrt {d} \int \frac {{\left (b^{2} g^{3} x^{3} + 3 \, b^{2} f g^{2} x^{2} + 3 \, b^{2} f^{2} g x + b^{2} f^{3}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b g^{3} x^{3} + 3 \, a b f g^{2} x^{2} + 3 \, a b f^{2} g x\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{{\left (c^{4} d^{3} x^{4} - 2 \, c^{2} d^{3} x^{2} + d^{3}\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}\,{d x} + \frac {a^{2} f^{2} g}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (f+g\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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