\(\int \frac {(a+b \arcsin (c x))^2}{(f+g x) (d-c^2 d x^2)^{3/2}} \, dx\) [151]

Optimal result

Integrand size = 33, antiderivative size = 1137 \[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Output:

-1/2*I*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^2/d/(c*f-g)/(-c^2*d*x^2+d)^(1/ 
2)+1/2*I*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^2/d/(c*f+g)/(-c^2*d*x^2+d)^( 
1/2)-1/2*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^2*cot(1/4*Pi+1/2*arcsin(c*x) 
)/d/(c*f-g)/(-c^2*d*x^2+d)^(1/2)+2*b*(-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)))/d/(c*f+g)/(-c^2*d*x^2+d)^(1/2)+2*b*(-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)))/d/(c*f 
-g)/(-c^2*d*x^2+d)^(1/2)+I*g^2*(-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)))/d/(c^2*f^2-g^2) 
^(3/2)/(-c^2*d*x^2+d)^(1/2)-I*g^2*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^2*l 
n(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/d/(c^2*f^2-g 
^2)^(3/2)/(-c^2*d*x^2+d)^(1/2)+2*I*b^2*(-c^2*x^2+1)^(1/2)*polylog(2,I/(I*c 
*x+(-c^2*x^2+1)^(1/2)))/d/(c*f+g)/(-c^2*d*x^2+d)^(1/2)-2*I*b^2*(-c^2*x^2+1 
)^(1/2)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d/(c*f-g)/(-c^2*d*x^2+d)^( 
1/2)+2*b*g^2*(-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)))/d/(c^2*f^2-g^2)^(3/2)/(-c^2*d* 
x^2+d)^(1/2)-2*b*g^2*(-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)))/d/(c^2*f^2-g^2)^(3/2)/ 
(-c^2*d*x^2+d)^(1/2)+2*I*b^2*g^2*(-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)))/d/(c^2*f^2-g^2)^(3/2)/(-c^2* 
d*x^2+d)^(1/2)-2*I*b^2*g^2*(-c^2*x^2+1)^(1/2)*polylog(3,I*(I*c*x+(-c^2*...
 

Mathematica [A] (warning: unable to verify)

Time = 3.24 (sec) , antiderivative size = 597, normalized size of antiderivative = 0.53 \[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {1-c^2 x^2} \left (\frac {-\left ((a+b \arcsin (c x)) \left (-i a+a \cot \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )+b \arcsin (c x) \left (-i+\cot \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-4 b \log \left (1+i e^{-i \arcsin (c x)}\right )\right )\right )+4 i b^2 \operatorname {PolyLog}\left (2,-i e^{-i \arcsin (c x)}\right )}{c f-g}+\frac {2 i g^2 \left ((a+b \arcsin (c x))^2 \log \left (1+\frac {i e^{i \arcsin (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-(a+b \arcsin (c x))^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-2 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+2 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )+2 b^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-2 b^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )}{(c f-g) (c f+g) \sqrt {c^2 f^2-g^2}}+\frac {-4 i b^2 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+(a+b \arcsin (c x)) \left (-i a+4 b \log \left (1+i e^{i \arcsin (c x)}\right )+a \tan \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )+b \arcsin (c x) \left (-i+\tan \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )}{c f+g}\right )}{2 d \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 2.68 (sec) , antiderivative size = 722, normalized size of antiderivative = 0.64, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5276, 5274, 2009}

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.

\(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2} (f+g x)} \, dx\)

\(\Big \downarrow \) 5276

\(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {(a+b \arcsin (c x))^2}{(f+g x) \left (1-c^2 x^2\right )^{3/2}}dx}{d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 5274

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (\frac {2 b g^2 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2}}-\frac {2 b g^2 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2}}+\frac {i g^2 (a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2}}-\frac {i g^2 (a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{\left (c^2 f^2-g^2\right )^{3/2}}-\frac {i (a+b \arcsin (c x))^2}{2 (c f-g)}+\frac {i (a+b \arcsin (c x))^2}{2 (c f+g)}+\frac {2 b \log \left (1-i e^{-i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{c f+g}+\frac {2 b \log \left (1-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{c f-g}+\frac {\tan \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) (a+b \arcsin (c x))^2}{2 (c f+g)}-\frac {\cot \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) (a+b \arcsin (c x))^2}{2 (c f-g)}+\frac {2 i b^2 g^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2}}-\frac {2 i b^2 g^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2}}+\frac {2 i b^2 \operatorname {PolyLog}\left (2,i e^{-i \arcsin (c x)}\right )}{c f+g}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c f-g}\right )}{d \sqrt {d-c^2 d x^2}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5274

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] /; FreeQ[{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 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]
 

Maple [F]

\[\int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (g x +f \right ) \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (f + g x\right )}\, dx \] Input:

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

Output:

Integral((a + b*asin(c*x))**2/((-d*(c*x - 1)*(c*x + 1))**(3/2)*(f + g*x)), 
 x)
 

Maxima [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}} \,d x } \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

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. \[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\left (f+g\,x\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-2 \sqrt {c^{2} f^{2}-g^{2}}\, \sqrt {-c^{2} x^{2}+1}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a^{2} g^{2}-2 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{2} f \,x^{2}+\sqrt {-c^{2} x^{2}+1}\, c^{2} g \,x^{3}-\sqrt {-c^{2} x^{2}+1}\, f -\sqrt {-c^{2} x^{2}+1}\, g x}d x \right ) a b \,c^{4} f^{4}+4 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{2} f \,x^{2}+\sqrt {-c^{2} x^{2}+1}\, c^{2} g \,x^{3}-\sqrt {-c^{2} x^{2}+1}\, f -\sqrt {-c^{2} x^{2}+1}\, g x}d x \right ) a b \,c^{2} f^{2} g^{2}-2 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{2} f \,x^{2}+\sqrt {-c^{2} x^{2}+1}\, c^{2} g \,x^{3}-\sqrt {-c^{2} x^{2}+1}\, f -\sqrt {-c^{2} x^{2}+1}\, g x}d x \right ) a b \,g^{4}-\sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{2} f \,x^{2}+\sqrt {-c^{2} x^{2}+1}\, c^{2} g \,x^{3}-\sqrt {-c^{2} x^{2}+1}\, f -\sqrt {-c^{2} x^{2}+1}\, g x}d x \right ) b^{2} c^{4} f^{4}+2 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{2} f \,x^{2}+\sqrt {-c^{2} x^{2}+1}\, c^{2} g \,x^{3}-\sqrt {-c^{2} x^{2}+1}\, f -\sqrt {-c^{2} x^{2}+1}\, g x}d x \right ) b^{2} c^{2} f^{2} g^{2}-\sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{2} f \,x^{2}+\sqrt {-c^{2} x^{2}+1}\, c^{2} g \,x^{3}-\sqrt {-c^{2} x^{2}+1}\, f -\sqrt {-c^{2} x^{2}+1}\, g x}d x \right ) b^{2} g^{4}+\sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} f^{2} g -\sqrt {-c^{2} x^{2}+1}\, a^{2} g^{3}+a^{2} c^{4} f^{3} x -a^{2} c^{2} f^{2} g -a^{2} c^{2} f \,g^{2} x +a^{2} g^{3}}{\sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, d \left (c^{4} f^{4}-2 c^{2} f^{2} g^{2}+g^{4}\right )} \] Input:

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

Output:

( - 2*sqrt(c**2*f**2 - g**2)*sqrt( - c**2*x**2 + 1)*atan((tan(asin(c*x)/2) 
*c*f + g)/sqrt(c**2*f**2 - g**2))*a**2*g**2 - 2*sqrt( - c**2*x**2 + 1)*int 
(asin(c*x)/(sqrt( - c**2*x**2 + 1)*c**2*f*x**2 + sqrt( - c**2*x**2 + 1)*c* 
*2*g*x**3 - sqrt( - c**2*x**2 + 1)*f - sqrt( - c**2*x**2 + 1)*g*x),x)*a*b* 
c**4*f**4 + 4*sqrt( - c**2*x**2 + 1)*int(asin(c*x)/(sqrt( - c**2*x**2 + 1) 
*c**2*f*x**2 + sqrt( - c**2*x**2 + 1)*c**2*g*x**3 - sqrt( - c**2*x**2 + 1) 
*f - sqrt( - c**2*x**2 + 1)*g*x),x)*a*b*c**2*f**2*g**2 - 2*sqrt( - c**2*x* 
*2 + 1)*int(asin(c*x)/(sqrt( - c**2*x**2 + 1)*c**2*f*x**2 + sqrt( - c**2*x 
**2 + 1)*c**2*g*x**3 - sqrt( - c**2*x**2 + 1)*f - sqrt( - c**2*x**2 + 1)*g 
*x),x)*a*b*g**4 - sqrt( - c**2*x**2 + 1)*int(asin(c*x)**2/(sqrt( - c**2*x* 
*2 + 1)*c**2*f*x**2 + sqrt( - c**2*x**2 + 1)*c**2*g*x**3 - sqrt( - c**2*x* 
*2 + 1)*f - sqrt( - c**2*x**2 + 1)*g*x),x)*b**2*c**4*f**4 + 2*sqrt( - c**2 
*x**2 + 1)*int(asin(c*x)**2/(sqrt( - c**2*x**2 + 1)*c**2*f*x**2 + sqrt( - 
c**2*x**2 + 1)*c**2*g*x**3 - sqrt( - c**2*x**2 + 1)*f - sqrt( - c**2*x**2 
+ 1)*g*x),x)*b**2*c**2*f**2*g**2 - sqrt( - c**2*x**2 + 1)*int(asin(c*x)**2 
/(sqrt( - c**2*x**2 + 1)*c**2*f*x**2 + sqrt( - c**2*x**2 + 1)*c**2*g*x**3 
- sqrt( - c**2*x**2 + 1)*f - sqrt( - c**2*x**2 + 1)*g*x),x)*b**2*g**4 + sq 
rt( - c**2*x**2 + 1)*a**2*c**2*f**2*g - sqrt( - c**2*x**2 + 1)*a**2*g**3 + 
 a**2*c**4*f**3*x - a**2*c**2*f**2*g - a**2*c**2*f*g**2*x + a**2*g**3)/(sq 
rt(d)*sqrt( - c**2*x**2 + 1)*d*(c**4*f**4 - 2*c**2*f**2*g**2 + g**4))