3.1.79 \(\int \frac {(f+g x)^3 (a+b \text {ArcSin}(c x))^2}{(d-c^2 d x^2)^{5/2}} \, dx\) [79]

Optimal. Leaf size=1589 \[ -\frac {i (c f-g)^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^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} (a+b \text {ArcSin}(c x))^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} (a+b \text {ArcSin}(c x))^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} (a+b \text {ArcSin}(c x))^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} \text {ArcSin}(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} (a+b \text {ArcSin}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(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} (a+b \text {ArcSin}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(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} (a+b \text {ArcSin}(c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(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} (a+b \text {ArcSin}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(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} (a+b \text {ArcSin}(c x)) \log \left (1-i e^{-i \text {ArcSin}(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} (a+b \text {ArcSin}(c x)) \log \left (1-i e^{-i \text {ArcSin}(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} (a+b \text {ArcSin}(c x)) \log \left (1-i e^{i \text {ArcSin}(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} (a+b \text {ArcSin}(c x)) \log \left (1-i e^{i \text {ArcSin}(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 {PolyLog}\left (2,i e^{-i \text {ArcSin}(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 {PolyLog}\left (2,i e^{-i \text {ArcSin}(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 {PolyLog}\left (2,i e^{i \text {ArcSin}(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 {PolyLog}\left (2,i e^{i \text {ArcSin}(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} (a+b \text {ArcSin}(c x)) \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right )}{12 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 {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right )}{6 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} (a+b \text {ArcSin}(c x))^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(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} (a+b \text {ArcSin}(c x))^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(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} (a+b \text {ArcSin}(c x))^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right ) \tan \left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right )}{24 c^4 d^2 \sqrt {d-c^2 d x^2}} \]

[Out]

-1/4*I*(c*f-g)^2*(c*f+2*g)*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)+I*b^2*(c*f-2*g)
*(c*f+g)^2*polylog(2,I/(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)+1/12*I*(c*f
+g)^3*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)+1/3*I*b^2*(c*f+g)^3*polylog(2,I/(I*c
*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-1/6*b^2*(c*f-g)^3*cot(1/4*Pi+1/2*arcsi
n(c*x))*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-1/12*(c*f-g)^3*(a+b*arcsin(c*x))^2*cot(1/4*Pi+1/2*arcs
in(c*x))*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-1/4*(c*f-g)^2*(c*f+2*g)*(a+b*arcsin(c*x))^2*cot(1/4*P
i+1/2*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-1/12*b*(c*f-g)^3*(a+b*arcsin(c*x))*csc(1/4*
Pi+1/2*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-1/24*(c*f-g)^3*(a+b*arcsin(c*x))^2*cot(1
/4*Pi+1/2*arcsin(c*x))*csc(1/4*Pi+1/2*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)+b*(c*f-2*
g)*(c*f+g)^2*(a+b*arcsin(c*x))*ln(1-I/(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1
/2)+1/3*b*(c*f+g)^3*(a+b*arcsin(c*x))*ln(1-I/(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^
2+d)^(1/2)+1/3*b*(c*f-g)^3*(a+b*arcsin(c*x))*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c
^2*d*x^2+d)^(1/2)+b*(c*f-g)^2*(c*f+2*g)*(a+b*arcsin(c*x))*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2
)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-I*b^2*(c*f-g)^2*(c*f+2*g)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^
(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-1/3*I*b^2*(c*f-g)^3*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1
/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)+1/4*I*(c*f-2*g)*(c*f+g)^2*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^
2*d*x^2+d)^(1/2)-1/12*I*(c*f-g)^3*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-1/12*b*(
c*f+g)^3*(a+b*arcsin(c*x))*sec(1/4*Pi+1/2*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)+1/6*b
^2*(c*f+g)^3*(-c^2*x^2+1)^(1/2)*tan(1/4*Pi+1/2*arcsin(c*x))/c^4/d^2/(-c^2*d*x^2+d)^(1/2)+1/4*(c*f-2*g)*(c*f+g)
^2*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)*tan(1/4*Pi+1/2*arcsin(c*x))/c^4/d^2/(-c^2*d*x^2+d)^(1/2)+1/12*(c*f+g
)^3*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)*tan(1/4*Pi+1/2*arcsin(c*x))/c^4/d^2/(-c^2*d*x^2+d)^(1/2)+1/24*(c*f+
g)^3*(a+b*arcsin(c*x))^2*sec(1/4*Pi+1/2*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)*tan(1/4*Pi+1/2*arcsin(c*x))/c^4/d^2/
(-c^2*d*x^2+d)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.39, 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 = {4861, 4859, 4857, 3399, 4271, 3852, 8, 4269, 3798, 2221, 2317, 2438} \begin {gather*} -\frac {i \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^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} (a+b \text {ArcSin}(c x)) \csc ^2\left (\frac {1}{2} \text {ArcSin}(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} (a+b \text {ArcSin}(c x))^2 \cot \left (\frac {1}{2} \text {ArcSin}(c x)+\frac {\pi }{4}\right ) \csc ^2\left (\frac {1}{2} \text {ArcSin}(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} (a+b \text {ArcSin}(c x))^2 \cot \left (\frac {1}{2} \text {ArcSin}(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} \text {ArcSin}(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} (a+b \text {ArcSin}(c x)) \log \left (1-i e^{i \text {ArcSin}(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 \text {ArcSin}(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} (a+b \text {ArcSin}(c x))^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} (a+b \text {ArcSin}(c x))^2 \cot \left (\frac {1}{2} \text {ArcSin}(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} (a+b \text {ArcSin}(c x)) \log \left (1-i e^{i \text {ArcSin}(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 \text {ArcSin}(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} (a+b \text {ArcSin}(c x))^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} (a+b \text {ArcSin}(c x))^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} (a+b \text {ArcSin}(c x)) \sec ^2\left (\frac {1}{2} \text {ArcSin}(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} (a+b \text {ArcSin}(c x)) \log \left (1-i e^{-i \text {ArcSin}(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} (a+b \text {ArcSin}(c x)) \log \left (1-i e^{-i \text {ArcSin}(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 \text {ArcSin}(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 \text {ArcSin}(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} (a+b \text {ArcSin}(c x))^2 \tan \left (\frac {1}{2} \text {ArcSin}(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} (a+b \text {ArcSin}(c x))^2 \tan \left (\frac {1}{2} \text {ArcSin}(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} (a+b \text {ArcSin}(c x))^2 \sec ^2\left (\frac {1}{2} \text {ArcSin}(c x)+\frac {\pi }{4}\right ) \tan \left (\frac {1}{2} \text {ArcSin}(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} \text {ArcSin}(c x)+\frac {\pi }{4}\right )}{6 c^4 d^2 \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((f + g*x)^3*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]

[Out]

((-1/12*I)*(c*f - g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((I/4)*(c*f -
2*g)*(c*f + g)^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((I/12)*(c*f + g)^3*
Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^4*d^2*Sqrt[d - c^2*d*x^2]) - ((I/4)*(c*f - g)^2*(c*f + 2*g)*Sqrt[1
- c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^4*d^2*Sqrt[d - c^2*d*x^2]) - (b^2*(c*f - g)^3*Sqrt[1 - c^2*x^2]*Cot[Pi/4
+ ArcSin[c*x]/2])/(6*c^4*d^2*Sqrt[d - c^2*d*x^2]) - ((c*f - g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Cot[
Pi/4 + ArcSin[c*x]/2])/(12*c^4*d^2*Sqrt[d - c^2*d*x^2]) - ((c*f - g)^2*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*Ar
cSin[c*x])^2*Cot[Pi/4 + ArcSin[c*x]/2])/(4*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (b*(c*f - g)^3*Sqrt[1 - c^2*x^2]*(a
+ b*ArcSin[c*x])*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(12*c^4*d^2*Sqrt[d - c^2*d*x^2]) - ((c*f - g)^3*Sqrt[1 - c^2*x^2
]*(a + b*ArcSin[c*x])^2*Cot[Pi/4 + ArcSin[c*x]/2]*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(24*c^4*d^2*Sqrt[d - c^2*d*x^2]
) + (b*(c*f - 2*g)*(c*f + g)^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I/E^(I*ArcSin[c*x])])/(c^4*d^2*Sq
rt[d - c^2*d*x^2]) + (b*(c*f + g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I/E^(I*ArcSin[c*x])])/(3*c^4
*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])])
/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^2*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I*
E^(I*ArcSin[c*x])])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + (I*b^2*(c*f - 2*g)*(c*f + g)^2*Sqrt[1 - c^2*x^2]*PolyLog[2
, I/E^(I*ArcSin[c*x])])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((I/3)*b^2*(c*f + g)^3*Sqrt[1 - c^2*x^2]*PolyLog[2, I/
E^(I*ArcSin[c*x])])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) - ((I/3)*b^2*(c*f - g)^3*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I
*ArcSin[c*x])])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) - (I*b^2*(c*f - g)^2*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*PolyLog[2, I*
E^(I*ArcSin[c*x])])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) - (b*(c*f + g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Sec[P
i/4 + ArcSin[c*x]/2]^2)/(12*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*(c*f + g)^3*Sqrt[1 - c^2*x^2]*Tan[Pi/4 + ArcSi
n[c*x]/2])/(6*c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((c*f - 2*g)*(c*f + g)^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*
Tan[Pi/4 + ArcSin[c*x]/2])/(4*c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((c*f + g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]
)^2*Tan[Pi/4 + ArcSin[c*x]/2])/(12*c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((c*f + g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin
[c*x])^2*Sec[Pi/4 + ArcSin[c*x]/2]^2*Tan[Pi/4 + ArcSin[c*x]/2])/(24*c^4*d^2*Sqrt[d - c^2*d*x^2])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3399

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])

Rule 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4271

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-b^2)*(c + d*x)^m*Cot[e
+ f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (Dist[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))), Int[(c +
d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[b^2*((n - 2)/(n - 1)), Int[(c + d*x)^m*(b*Csc[e + f*x])^
(n - 2), x], x] - Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 4857

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol]
:> Dist[1/(c^(m + 1)*Sqrt[d]), Subst[Int[(a + b*x)^n*(c*f + g*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a,
b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])

Rule 4859

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n/Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; Free
Q[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]

Rule 4861

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Dist[Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] /; F
reeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p - 1/2] &&  !GtQ[d, 0]

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 6.16, size = 715, normalized size = 0.45 \begin {gather*} \frac {\sqrt {1-c^2 x^2} \left (\frac {(c f-g)^2 (c f+2 g) \left (i b \left (\frac {(a+b \text {ArcSin}(c x))^2}{b}-4 \left (i (a+b \text {ArcSin}(c x)) \log \left (1+e^{\frac {1}{2} i (\pi -2 \text {ArcSin}(c x))}\right )-b \text {PolyLog}\left (2,-e^{\frac {1}{2} i (\pi -2 \text {ArcSin}(c x))}\right )\right )\right )-(a+b \text {ArcSin}(c x))^2 \tan \left (\frac {\pi }{4}-\frac {1}{2} \text {ArcSin}(c x)\right )\right )}{4 c^4}-\frac {(c f-g)^3 \left (2 b (a+b \text {ArcSin}(c x)) \sec ^2\left (\frac {\pi }{4}-\frac {1}{2} \text {ArcSin}(c x)\right )+4 b^2 \tan \left (\frac {\pi }{4}-\frac {1}{2} \text {ArcSin}(c x)\right )+(a+b \text {ArcSin}(c x))^2 \sec ^2\left (\frac {\pi }{4}-\frac {1}{2} \text {ArcSin}(c x)\right ) \tan \left (\frac {\pi }{4}-\frac {1}{2} \text {ArcSin}(c x)\right )-2 \left (i b \left (\frac {(a+b \text {ArcSin}(c x))^2}{b}-4 \left (i (a+b \text {ArcSin}(c x)) \log \left (1+e^{\frac {1}{2} i (\pi -2 \text {ArcSin}(c x))}\right )-b \text {PolyLog}\left (2,-e^{\frac {1}{2} i (\pi -2 \text {ArcSin}(c x))}\right )\right )\right )-(a+b \text {ArcSin}(c x))^2 \tan \left (\frac {\pi }{4}-\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right )}{24 c^4}-\frac {(c f-2 g) (c f+g)^2 \left (i b \left (\frac {(a+b \text {ArcSin}(c x))^2}{b}+4 \left (i (a+b \text {ArcSin}(c x)) \log \left (1+e^{\frac {1}{2} i (\pi +2 \text {ArcSin}(c x))}\right )+b \text {PolyLog}\left (2,-e^{\frac {1}{2} i (\pi +2 \text {ArcSin}(c x))}\right )\right )\right )-(a+b \text {ArcSin}(c x))^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right )\right )}{4 c^4}-\frac {(c f+g)^3 \left (2 b (a+b \text {ArcSin}(c x)) \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right )-4 b^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right )-(a+b \text {ArcSin}(c x))^2 \sec ^2\left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right ) \tan \left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right )+2 \left (i b \left (\frac {(a+b \text {ArcSin}(c x))^2}{b}+4 \left (i (a+b \text {ArcSin}(c x)) \log \left (1+e^{\frac {1}{2} i (\pi +2 \text {ArcSin}(c x))}\right )+b \text {PolyLog}\left (2,-e^{\frac {1}{2} i (\pi +2 \text {ArcSin}(c x))}\right )\right )\right )-(a+b \text {ArcSin}(c x))^2 \tan \left (\frac {\pi }{4}+\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right )}{24 c^4}\right )}{d^2 \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)^3*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]

[Out]

(Sqrt[1 - c^2*x^2]*(((c*f - g)^2*(c*f + 2*g)*(I*b*((a + b*ArcSin[c*x])^2/b - 4*(I*(a + b*ArcSin[c*x])*Log[1 +
E^((I/2)*(Pi - 2*ArcSin[c*x]))] - b*PolyLog[2, -E^((I/2)*(Pi - 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[
Pi/4 - ArcSin[c*x]/2]))/(4*c^4) - ((c*f - g)^3*(2*b*(a + b*ArcSin[c*x])*Sec[Pi/4 - ArcSin[c*x]/2]^2 + 4*b^2*Ta
n[Pi/4 - ArcSin[c*x]/2] + (a + b*ArcSin[c*x])^2*Sec[Pi/4 - ArcSin[c*x]/2]^2*Tan[Pi/4 - ArcSin[c*x]/2] - 2*(I*b
*((a + b*ArcSin[c*x])^2/b - 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi - 2*ArcSin[c*x]))] - b*PolyLog[2, -E
^((I/2)*(Pi - 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 - ArcSin[c*x]/2])))/(24*c^4) - ((c*f - 2*g)*
(c*f + g)^2*(I*b*((a + b*ArcSin[c*x])^2/b + 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi + 2*ArcSin[c*x]))] +
b*PolyLog[2, -E^((I/2)*(Pi + 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2]))/(4*c^4) -
((c*f + g)^3*(2*b*(a + b*ArcSin[c*x])*Sec[Pi/4 + ArcSin[c*x]/2]^2 - 4*b^2*Tan[Pi/4 + ArcSin[c*x]/2] - (a + b*
ArcSin[c*x])^2*Sec[Pi/4 + ArcSin[c*x]/2]^2*Tan[Pi/4 + ArcSin[c*x]/2] + 2*(I*b*((a + b*ArcSin[c*x])^2/b + 4*(I*
(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi + 2*ArcSin[c*x]))] + b*PolyLog[2, -E^((I/2)*(Pi + 2*ArcSin[c*x]))]))
- (a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2])))/(24*c^4)))/(d^2*Sqrt[d - c^2*d*x^2])

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 13139 vs. \(2 (1473 ) = 2946\).
time = 1.00, size = 13140, normalized size = 8.27

method	result	size
default	\(\text {Expression too large to display}\)	\(13140\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

1/3*a*b*c*f^3*(1/(c^4*d^(5/2)*x^2 - c^2*d^(5/2)) + 2*log(c*x + 1)/(c^2*d^(5/2)) + 2*log(c*x - 1)/(c^2*d^(5/2))
) + 2/3*a*b*f^3*(2*x/(sqrt(-c^2*d*x^2 + d)*d^2) + x/((-c^2*d*x^2 + d)^(3/2)*d))*arcsin(c*x) + 1/3*a^2*f^3*(2*x
/(sqrt(-c^2*d*x^2 + d)*d^2) + x/((-c^2*d*x^2 + d)^(3/2)*d)) + 1/3*a^2*g^3*(3*x^2/((-c^2*d*x^2 + d)^(3/2)*c^2*d
) - 2/((-c^2*d*x^2 + d)^(3/2)*c^4*d)) - a^2*f*g^2*(x/(sqrt(-c^2*d*x^2 + d)*c^2*d^2) - x/((-c^2*d*x^2 + d)^(3/2
)*c^2*d)) + sqrt(d)*integrate(((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)*arctan2(c*x, sqrt(c*x
+ 1)*sqrt(-c*x + 1))^2 + 2*(a*b*g^3*x^3 + 3*a*b*f*g^2*x^2 + 3*a*b*f^2*g*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c
*x + 1)))/((c^4*d^3*x^4 - 2*c^2*d^3*x^2 + d^3)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x) + a^2*f^2*g/((-c^2*d*x^2 + d)
^(3/2)*c^2*d)

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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.

[In]

integrate((g*x+f)^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(-(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 + (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)*arcsin(c*x)^2 + 2*(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)*arcsin(c*x))*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)

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Sympy [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.

[In]

integrate((g*x+f)**3*(a+b*asin(c*x))**2/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

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Giac [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.

[In]

integrate((g*x+f)^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((g*x + f)^3*(b*arcsin(c*x) + a)^2/(-c^2*d*x^2 + d)^(5/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)^3*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^(5/2),x)

[Out]

int(((f + g*x)^3*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^(5/2), x)

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