\(\int \frac {(a+b \arcsin (c x))^2}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx\) [147]

Optimal result

Integrand size = 33, antiderivative size = 1106 \[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\frac {i c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {g \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{d \left (c^2 f^2-g^2\right ) (f+g x)}-\frac {2 b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {i c^2 f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {2 b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {i c^2 f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 b c^2 f \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 b c^2 f \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 c^2 f \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c^2 f \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}} \] Output:

I*c*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^2/(c^2*f^2-g^2)/(-c^2*d*x^2+d)^(1 
/2)+g*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/d/(c^2*f^2-g^2)/(g*x+f)-2*b 
*c*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))* 
g/(c*f-(c^2*f^2-g^2)^(1/2)))/(c^2*f^2-g^2)/(-c^2*d*x^2+d)^(1/2)-I*c^2*f*(- 
c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/( 
c*f-(c^2*f^2-g^2)^(1/2)))/(c^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^(1/2)-2*b*c*( 
-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c 
*f+(c^2*f^2-g^2)^(1/2)))/(c^2*f^2-g^2)/(-c^2*d*x^2+d)^(1/2)+I*c^2*f*(-c^2* 
x^2+1)^(1/2)*(a+b*arcsin(c*x))^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f+ 
(c^2*f^2-g^2)^(1/2)))/(c^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^(1/2)+2*I*b^2*c*( 
-c^2*x^2+1)^(1/2)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g 
^2)^(1/2)))/(c^2*f^2-g^2)/(-c^2*d*x^2+d)^(1/2)-2*b*c^2*f*(-c^2*x^2+1)^(1/2 
)*(a+b*arcsin(c*x))*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2 
-g^2)^(1/2)))/(c^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^(1/2)+2*I*b^2*c*(-c^2*x^2 
+1)^(1/2)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2 
)))/(c^2*f^2-g^2)/(-c^2*d*x^2+d)^(1/2)+2*b*c^2*f*(-c^2*x^2+1)^(1/2)*(a+b*a 
rcsin(c*x))*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1 
/2)))/(c^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^(1/2)-2*I*b^2*c^2*f*(-c^2*x^2+1)^ 
(1/2)*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))/ 
(c^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^(1/2)+2*I*b^2*c^2*f*(-c^2*x^2+1)^(1/...
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 651, normalized size of antiderivative = 0.59 \[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\frac {c \sqrt {1-c^2 x^2} \left (i (a+b \arcsin (c x))^2+\frac {g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c f+c g x}-2 b (a+b \arcsin (c x)) \log \left (1+\frac {i e^{i \arcsin (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-2 b (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )+2 i b^2 \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+2 i b^2 \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-\frac {i c f \left ((a+b \arcsin (c x))^2 \log \left (1+\frac {i e^{i \arcsin (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-2 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+2 b^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )\right )}{\sqrt {c^2 f^2-g^2}}+\frac {c f \left (2 b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )+i \left ((a+b \arcsin (c x))^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )+2 b^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )\right )}{\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

(c*Sqrt[1 - c^2*x^2]*(I*(a + b*ArcSin[c*x])^2 + (g*Sqrt[1 - c^2*x^2]*(a + 
b*ArcSin[c*x])^2)/(c*f + c*g*x) - 2*b*(a + b*ArcSin[c*x])*Log[1 + (I*E^(I* 
ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - 2*b*(a + b*ArcSin[c*x])* 
Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] + (2*I)*b^2*P 
olyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + (2*I)*b^2 
*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] - (I*c*f* 
((a + b*ArcSin[c*x])^2*Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2* 
f^2 - g^2])] - (2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x]) 
*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + 2*b^2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g 
)/(c*f - Sqrt[c^2*f^2 - g^2])]))/Sqrt[c^2*f^2 - g^2] + (c*f*(2*b*(a + b*Ar 
cSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] 
 + I*((a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^ 
2*f^2 - g^2])] + 2*b^2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2* 
f^2 - g^2])])))/Sqrt[c^2*f^2 - g^2]))/((c^2*f^2 - g^2)*Sqrt[d - c^2*d*x^2] 
)
 

Rubi [A] (verified)

Time = 3.25 (sec) , antiderivative size = 721, normalized size of antiderivative = 0.65, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.485, Rules used = {5276, 5272, 3042, 3805, 3042, 3804, 2694, 27, 2620, 3011, 2720, 5030, 2620, 2715, 2838, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2} (f+g x)^2} \, dx\)

\(\Big \downarrow \) 5276

\(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {(a+b \arcsin (c x))^2}{(f+g x)^2 \sqrt {1-c^2 x^2}}dx}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 5272

\(\displaystyle \frac {c \sqrt {1-c^2 x^2} \int \frac {(a+b \arcsin (c x))^2}{(c f+c g x)^2}d\arcsin (c x)}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {c \sqrt {1-c^2 x^2} \int \frac {(a+b \arcsin (c x))^2}{(c f+g \sin (\arcsin (c x)))^2}d\arcsin (c x)}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3805

\(\displaystyle \frac {c \sqrt {1-c^2 x^2} \left (-\frac {2 b g \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c f+c g x}d\arcsin (c x)}{c^2 f^2-g^2}+\frac {c f \int \frac {(a+b \arcsin (c x))^2}{c f+c g x}d\arcsin (c x)}{c^2 f^2-g^2}+\frac {g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {c \sqrt {1-c^2 x^2} \left (-\frac {2 b g \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c f+c g x}d\arcsin (c x)}{c^2 f^2-g^2}+\frac {c f \int \frac {(a+b \arcsin (c x))^2}{c f+g \sin (\arcsin (c x))}d\arcsin (c x)}{c^2 f^2-g^2}+\frac {g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3804

\(\displaystyle \frac {c \sqrt {1-c^2 x^2} \left (-\frac {2 b g \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c f+c g x}d\arcsin (c x)}{c^2 f^2-g^2}+\frac {2 c f \int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))^2}{2 c e^{i \arcsin (c x)} f-i e^{2 i \arcsin (c x)} g+i g}d\arcsin (c x)}{c^2 f^2-g^2}+\frac {g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2694

\(\displaystyle \frac {c \sqrt {1-c^2 x^2} \left (-\frac {2 b g \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c f+c g x}d\arcsin (c x)}{c^2 f^2-g^2}+\frac {2 c f \left (\frac {i g \int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))^2}{2 \left (c f-i e^{i \arcsin (c x)} g+\sqrt {c^2 f^2-g^2}\right )}d\arcsin (c x)}{\sqrt {c^2 f^2-g^2}}-\frac {i g \int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))^2}{2 \left (c f-i e^{i \arcsin (c x)} g-\sqrt {c^2 f^2-g^2}\right )}d\arcsin (c x)}{\sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}+\frac {g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {c \sqrt {1-c^2 x^2} \left (-\frac {2 b g \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c f+c g x}d\arcsin (c x)}{c^2 f^2-g^2}+\frac {2 c f \left (\frac {i g \int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))^2}{c f-i e^{i \arcsin (c x)} g+\sqrt {c^2 f^2-g^2}}d\arcsin (c x)}{2 \sqrt {c^2 f^2-g^2}}-\frac {i g \int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))^2}{c f-i e^{i \arcsin (c x)} g-\sqrt {c^2 f^2-g^2}}d\arcsin (c x)}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}+\frac {g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {c \sqrt {1-c^2 x^2} \left (-\frac {2 b g \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c f+c g x}d\arcsin (c x)}{c^2 f^2-g^2}+\frac {2 c f \left (\frac {i g \left (\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {2 b \int (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )d\arcsin (c x)}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {i g \left (\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}-\frac {2 b \int (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )d\arcsin (c x)}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}+\frac {g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {c \sqrt {1-c^2 x^2} \left (\frac {2 c f \left (\frac {i g \left (\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-i b \int \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )d\arcsin (c x)\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {i g \left (\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-i b \int \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )d\arcsin (c x)\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {2 b g \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c f+c g x}d\arcsin (c x)}{c^2 f^2-g^2}+\frac {g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {c \sqrt {1-c^2 x^2} \left (\frac {2 c f \left (\frac {i g \left (\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )de^{i \arcsin (c x)}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {i g \left (\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )de^{i \arcsin (c x)}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {2 b g \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c f+c g x}d\arcsin (c x)}{c^2 f^2-g^2}+\frac {g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 5030

\(\displaystyle \frac {c \sqrt {1-c^2 x^2} \left (\frac {2 c f \left (\frac {i g \left (\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )de^{i \arcsin (c x)}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {i g \left (\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )de^{i \arcsin (c x)}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {2 b g \left (\int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))}{c f-i e^{i \arcsin (c x)} g-\sqrt {c^2 f^2-g^2}}d\arcsin (c x)+\int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))}{c f-i e^{i \arcsin (c x)} g+\sqrt {c^2 f^2-g^2}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^2}{2 b g}\right )}{c^2 f^2-g^2}+\frac {g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {c \sqrt {1-c^2 x^2} \left (\frac {2 c f \left (\frac {i g \left (\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )de^{i \arcsin (c x)}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {i g \left (\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )de^{i \arcsin (c x)}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {2 b g \left (-\frac {b \int \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )d\arcsin (c x)}{g}-\frac {b \int \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )d\arcsin (c x)}{g}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {i (a+b \arcsin (c x))^2}{2 b g}\right )}{c^2 f^2-g^2}+\frac {g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {c \sqrt {1-c^2 x^2} \left (\frac {g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{\left (c^2 f^2-g^2\right ) (c f+c g x)}-\frac {2 b g \left (-\frac {i (a+b \arcsin (c x))^2}{2 b g}+\frac {\log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) (a+b \arcsin (c x))}{g}+\frac {\log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) (a+b \arcsin (c x))}{g}+\frac {i b \int e^{-i \arcsin (c x)} \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )de^{i \arcsin (c x)}}{g}+\frac {i b \int e^{-i \arcsin (c x)} \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )de^{i \arcsin (c x)}}{g}\right )}{c^2 f^2-g^2}+\frac {2 c f \left (\frac {i g \left (\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )de^{i \arcsin (c x)}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {i g \left (\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )de^{i \arcsin (c x)}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {c \sqrt {1-c^2 x^2} \left (\frac {2 c f \left (\frac {i g \left (\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )de^{i \arcsin (c x)}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {i g \left (\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )de^{i \arcsin (c x)}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {2 b g \left (\frac {(a+b \arcsin (c x)) \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {i (a+b \arcsin (c x))^2}{2 b g}-\frac {i b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}-\frac {i b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g}\right )}{c^2 f^2-g^2}+\frac {g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {c \sqrt {1-c^2 x^2} \left (-\frac {2 b g \left (\frac {(a+b \arcsin (c x)) \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {i (a+b \arcsin (c x))^2}{2 b g}-\frac {i b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}-\frac {i b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g}\right )}{c^2 f^2-g^2}+\frac {2 c f \left (\frac {i g \left (\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-b \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {i g \left (\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-b \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}+\frac {g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

Input:

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

Output:

(c*Sqrt[1 - c^2*x^2]*((g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/((c^2*f^ 
2 - g^2)*(c*f + c*g*x)) - (2*b*g*(((-1/2*I)*(a + b*ArcSin[c*x])^2)/(b*g) + 
 ((a + b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 
- g^2])])/g + ((a + b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + 
Sqrt[c^2*f^2 - g^2])])/g - (I*b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - 
Sqrt[c^2*f^2 - g^2])])/g - (I*b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + 
Sqrt[c^2*f^2 - g^2])])/g))/(c^2*f^2 - g^2) + (2*c*f*(((-1/2*I)*g*(((a + b* 
ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]) 
])/g - (2*b*(I*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f 
 - Sqrt[c^2*f^2 - g^2])] - b*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqr 
t[c^2*f^2 - g^2])]))/g))/Sqrt[c^2*f^2 - g^2] + ((I/2)*g*(((a + b*ArcSin[c* 
x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g - (2 
*b*(I*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c 
^2*f^2 - g^2])] - b*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 
 - g^2])]))/g))/Sqrt[c^2*f^2 - g^2]))/(c^2*f^2 - g^2)))/Sqrt[d - c^2*d*x^2 
]
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2620

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] - Si 
mp[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 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 2838

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 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3804

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Sy 
mbol] :> Simp[2   Int[(c + d*x)^m*(E^(I*(e + f*x))/(I*b + 2*a*E^(I*(e + f*x 
)) - I*b*E^(2*I*(e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ 
[a^2 - b^2, 0] && IGtQ[m, 0]
 

rule 3805

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_ 
Symbol] :> Simp[b*(c + d*x)^m*(Cos[e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f 
*x]))), x] + (Simp[a/(a^2 - b^2)   Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x] 
, x] - Simp[b*d*(m/(f*(a^2 - b^2)))   Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/( 
a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - 
b^2, 0] && IGtQ[m, 0]
 

rule 5030

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[ 
(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1 
))), x] + (Int[(e + f*x)^m*(E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2] - I*b*E^( 
I*(c + d*x)))), x] + Int[(e + f*x)^m*(E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] 
 - I*b*E^(I*(c + d*x)))), x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] 
 && PosQ[a^2 - b^2]
 

rule 5272

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sq 
rt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[1/(c^(m + 1)*Sqrt[d])   Subst[In 
t[(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] && (G 
tQ[m, 0] || IGtQ[n, 0])
 

rule 5276

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

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

\[\int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (g x +f \right )^{2} \sqrt {-c^{2} d \,x^{2}+d}}d x\]

Input:

int((a+b*arcsin(c*x))^2/(g*x+f)^2/(-c^2*d*x^2+d)^(1/2),x)
 

Output:

int((a+b*arcsin(c*x))^2/(g*x+f)^2/(-c^2*d*x^2+d)^(1/2),x)
 

Fricas [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2}} \,d x } \] Input:

integrate((a+b*arcsin(c*x))^2/(g*x+f)^2/(-c^2*d*x^2+d)^(1/2),x, algorithm= 
"fricas")
 

Output:

integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^ 
2)/(c^2*d*g^2*x^4 + 2*c^2*d*f*g*x^3 - 2*d*f*g*x - d*f^2 + (c^2*d*f^2 - d*g 
^2)*x^2), x)
 

Sympy [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (f + g x\right )^{2}}\, dx \] Input:

integrate((a+b*asin(c*x))**2/(g*x+f)**2/(-c**2*d*x**2+d)**(1/2),x)
 

Output:

Integral((a + b*asin(c*x))**2/(sqrt(-d*(c*x - 1)*(c*x + 1))*(f + g*x)**2), 
 x)
 

Maxima [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2}} \,d x } \] Input:

integrate((a+b*arcsin(c*x))^2/(g*x+f)^2/(-c^2*d*x^2+d)^(1/2),x, algorithm= 
"maxima")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*arcsin(c*x))^2/(g*x+f)^2/(-c^2*d*x^2+d)^(1/2),x, algorithm= 
"giac")
                                                                                    
                                                                                    
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (f+g\,x\right )}^2\,\sqrt {d-c^2\,d\,x^2}} \,d x \] Input:

int((a + b*asin(c*x))^2/((f + g*x)^2*(d - c^2*d*x^2)^(1/2)),x)
 

Output:

int((a + b*asin(c*x))^2/((f + g*x)^2*(d - c^2*d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx =\text {Too large to display} \] Input:

int((a+b*asin(c*x))^2/(g*x+f)^2/(-c^2*d*x^2+d)^(1/2),x)
 

Output:

(2*sqrt(c**2*f**2 - g**2)*atan((tan(asin(c*x)/2)*c*f + g)/sqrt(c**2*f**2 - 
 g**2))*a**2*c**2*f**2 + 2*sqrt(c**2*f**2 - g**2)*atan((tan(asin(c*x)/2)*c 
*f + g)/sqrt(c**2*f**2 - g**2))*a**2*c**2*f*g*x + sqrt( - c**2*x**2 + 1)*a 
**2*c**2*f**2*g - sqrt( - c**2*x**2 + 1)*a**2*g**3 + 2*int(asin(c*x)/(sqrt 
( - c**2*x**2 + 1)*f**2 + 2*sqrt( - c**2*x**2 + 1)*f*g*x + sqrt( - c**2*x* 
*2 + 1)*g**2*x**2),x)*a*b*c**4*f**5 + 2*int(asin(c*x)/(sqrt( - c**2*x**2 + 
 1)*f**2 + 2*sqrt( - c**2*x**2 + 1)*f*g*x + sqrt( - c**2*x**2 + 1)*g**2*x* 
*2),x)*a*b*c**4*f**4*g*x - 4*int(asin(c*x)/(sqrt( - c**2*x**2 + 1)*f**2 + 
2*sqrt( - c**2*x**2 + 1)*f*g*x + sqrt( - c**2*x**2 + 1)*g**2*x**2),x)*a*b* 
c**2*f**3*g**2 - 4*int(asin(c*x)/(sqrt( - c**2*x**2 + 1)*f**2 + 2*sqrt( - 
c**2*x**2 + 1)*f*g*x + sqrt( - c**2*x**2 + 1)*g**2*x**2),x)*a*b*c**2*f**2* 
g**3*x + 2*int(asin(c*x)/(sqrt( - c**2*x**2 + 1)*f**2 + 2*sqrt( - c**2*x** 
2 + 1)*f*g*x + sqrt( - c**2*x**2 + 1)*g**2*x**2),x)*a*b*f*g**4 + 2*int(asi 
n(c*x)/(sqrt( - c**2*x**2 + 1)*f**2 + 2*sqrt( - c**2*x**2 + 1)*f*g*x + sqr 
t( - c**2*x**2 + 1)*g**2*x**2),x)*a*b*g**5*x + int(asin(c*x)**2/(sqrt( - c 
**2*x**2 + 1)*f**2 + 2*sqrt( - c**2*x**2 + 1)*f*g*x + sqrt( - c**2*x**2 + 
1)*g**2*x**2),x)*b**2*c**4*f**5 + int(asin(c*x)**2/(sqrt( - c**2*x**2 + 1) 
*f**2 + 2*sqrt( - c**2*x**2 + 1)*f*g*x + sqrt( - c**2*x**2 + 1)*g**2*x**2) 
,x)*b**2*c**4*f**4*g*x - 2*int(asin(c*x)**2/(sqrt( - c**2*x**2 + 1)*f**2 + 
 2*sqrt( - c**2*x**2 + 1)*f*g*x + sqrt( - c**2*x**2 + 1)*g**2*x**2),x)*...