\(\int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx\) [22]

Optimal result

Integrand size = 18, antiderivative size = 362 \[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=-\frac {d^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {2 d e x \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {e^2 x^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}+\frac {2 d e \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b^2 c}+\frac {e^2 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{4 b^2 c^3}-\frac {3 e^2 \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b^2 c^3}-\frac {d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2 c^3}+\frac {2 d e \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {3 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2 c^3} \] Output:

-d^2*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arcsin(c*x))-2*d*e*x*(-c^2*x^2+1)^(1/2)/b 
/c/(a+b*arcsin(c*x))-e^2*x^2*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arcsin(c*x))+2*d* 
e*cos(2*a/b)*Ci(2*(a+b*arcsin(c*x))/b)/b^2/c^2+d^2*Ci((a+b*arcsin(c*x))/b) 
*sin(a/b)/b^2/c+1/4*e^2*Ci((a+b*arcsin(c*x))/b)*sin(a/b)/b^2/c^3-3/4*e^2*C 
i(3*(a+b*arcsin(c*x))/b)*sin(3*a/b)/b^2/c^3-d^2*cos(a/b)*Si((a+b*arcsin(c* 
x))/b)/b^2/c-1/4*e^2*cos(a/b)*Si((a+b*arcsin(c*x))/b)/b^2/c^3+2*d*e*sin(2* 
a/b)*Si(2*(a+b*arcsin(c*x))/b)/b^2/c^2+3/4*e^2*cos(3*a/b)*Si(3*(a+b*arcsin 
(c*x))/b)/b^2/c^3
 

Mathematica [A] (verified)

Time = 1.30 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.80 \[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=-\frac {\frac {4 b c^2 d^2 \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}+\frac {8 b c^2 d e x \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}+\frac {4 b c^2 e^2 x^2 \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}-8 c d e \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-\left (4 c^2 d^2+e^2\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right ) \sin \left (\frac {a}{b}\right )+3 e^2 \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )+4 c^2 d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )-8 c d e \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-3 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{4 b^2 c^3} \] Input:

Integrate[(d + e*x)^2/(a + b*ArcSin[c*x])^2,x]
 

Output:

-1/4*((4*b*c^2*d^2*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) + (8*b*c^2*d*e*x 
*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) + (4*b*c^2*e^2*x^2*Sqrt[1 - c^2*x^ 
2])/(a + b*ArcSin[c*x]) - 8*c*d*e*Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin 
[c*x])] - (4*c^2*d^2 + e^2)*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] + 3*e^ 
2*CosIntegral[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] + 4*c^2*d^2*Cos[a/b]*Sin 
Integral[a/b + ArcSin[c*x]] + e^2*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] 
- 8*c*d*e*Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])] - 3*e^2*Cos[(3*a 
)/b]*SinIntegral[3*(a/b + ArcSin[c*x])])/(b^2*c^3)
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5244, 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.

\(\displaystyle \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx\)

\(\Big \downarrow \) 5244

\(\displaystyle \int \left (\frac {d^2}{(a+b \arcsin (c x))^2}+\frac {2 d e x}{(a+b \arcsin (c x))^2}+\frac {e^2 x^2}{(a+b \arcsin (c x))^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^2 \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2 c^3}-\frac {3 e^2 \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2 c^3}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2 c^3}+\frac {3 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2 c^3}+\frac {2 d e \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {2 d e \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {d^2 \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {d^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {2 d e x \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {e^2 x^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}\)

Input:

Int[(d + e*x)^2/(a + b*ArcSin[c*x])^2,x]
 

Output:

-((d^2*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x]))) - (2*d*e*x*Sqrt[1 - c 
^2*x^2])/(b*c*(a + b*ArcSin[c*x])) - (e^2*x^2*Sqrt[1 - c^2*x^2])/(b*c*(a + 
 b*ArcSin[c*x])) + (2*d*e*Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c*x])) 
/b])/(b^2*c^2) + (d^2*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(b^2*c) 
 + (e^2*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(4*b^2*c^3) - (3*e^2* 
CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(4*b^2*c^3) - (d^2*Co 
s[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b^2*c) - (e^2*Cos[a/b]*SinInte 
gral[(a + b*ArcSin[c*x])/b])/(4*b^2*c^3) + (2*d*e*Sin[(2*a)/b]*SinIntegral 
[(2*(a + b*ArcSin[c*x]))/b])/(b^2*c^2) + (3*e^2*Cos[(3*a)/b]*SinIntegral[( 
3*(a + b*ArcSin[c*x]))/b])/(4*b^2*c^3)
 

Defintions of rubi rules used

rule 2009

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

rule 5244

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_.), x_Sy 
mbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcSin[c*x])^n, x], x] /; F 
reeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]
 

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.45

method	result	size
derivativedivides	\(\frac {4 \arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b \,c^{2} d^{2}-4 \arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b \,c^{2} d^{2}+8 \arcsin \left (c x \right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b c d e +8 \arcsin \left (c x \right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b c d e +4 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a \,c^{2} d^{2}-4 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a \,c^{2} d^{2}+\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b \,e^{2}+3 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b \,e^{2}-3 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b \,e^{2}-\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b \,e^{2}-4 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d^{2}+8 \,\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a c d e +8 \,\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a c d e +\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a \,e^{2}+3 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a \,e^{2}-3 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a \,e^{2}-\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a \,e^{2}-4 \sin \left (2 \arcsin \left (c x \right )\right ) d c e b -\sqrt {-c^{2} x^{2}+1}\, b \,e^{2}+\cos \left (3 \arcsin \left (c x \right )\right ) b \,e^{2}}{4 c^{3} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\)	\(526\)
default	\(\frac {4 \arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b \,c^{2} d^{2}-4 \arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b \,c^{2} d^{2}+8 \arcsin \left (c x \right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b c d e +8 \arcsin \left (c x \right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b c d e +4 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a \,c^{2} d^{2}-4 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a \,c^{2} d^{2}+\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b \,e^{2}+3 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b \,e^{2}-3 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b \,e^{2}-\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b \,e^{2}-4 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d^{2}+8 \,\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a c d e +8 \,\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a c d e +\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a \,e^{2}+3 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a \,e^{2}-3 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a \,e^{2}-\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a \,e^{2}-4 \sin \left (2 \arcsin \left (c x \right )\right ) d c e b -\sqrt {-c^{2} x^{2}+1}\, b \,e^{2}+\cos \left (3 \arcsin \left (c x \right )\right ) b \,e^{2}}{4 c^{3} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\)	\(526\)

Input:

int((e*x+d)^2/(a+b*arcsin(c*x))^2,x,method=_RETURNVERBOSE)
 

Output:

1/4/c^3*(4*arcsin(c*x)*Ci(arcsin(c*x)+a/b)*sin(a/b)*b*c^2*d^2-4*arcsin(c*x 
)*Si(arcsin(c*x)+a/b)*cos(a/b)*b*c^2*d^2+8*arcsin(c*x)*Si(2*arcsin(c*x)+2* 
a/b)*sin(2*a/b)*b*c*d*e+8*arcsin(c*x)*Ci(2*arcsin(c*x)+2*a/b)*cos(2*a/b)*b 
*c*d*e+4*Ci(arcsin(c*x)+a/b)*sin(a/b)*a*c^2*d^2-4*Si(arcsin(c*x)+a/b)*cos( 
a/b)*a*c^2*d^2+arcsin(c*x)*Ci(arcsin(c*x)+a/b)*sin(a/b)*b*e^2+3*arcsin(c*x 
)*Si(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*b*e^2-3*arcsin(c*x)*Ci(3*arcsin(c*x)+ 
3*a/b)*sin(3*a/b)*b*e^2-arcsin(c*x)*Si(arcsin(c*x)+a/b)*cos(a/b)*b*e^2-4*( 
-c^2*x^2+1)^(1/2)*b*c^2*d^2+8*Si(2*arcsin(c*x)+2*a/b)*sin(2*a/b)*a*c*d*e+8 
*Ci(2*arcsin(c*x)+2*a/b)*cos(2*a/b)*a*c*d*e+Ci(arcsin(c*x)+a/b)*sin(a/b)*a 
*e^2+3*Si(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*a*e^2-3*Ci(3*arcsin(c*x)+3*a/b)* 
sin(3*a/b)*a*e^2-Si(arcsin(c*x)+a/b)*cos(a/b)*a*e^2-4*sin(2*arcsin(c*x))*d 
*c*e*b-(-c^2*x^2+1)^(1/2)*b*e^2+cos(3*arcsin(c*x))*b*e^2)/(a+b*arcsin(c*x) 
)/b^2
 

Fricas [F]

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

integrate((e*x+d)^2/(a+b*arcsin(c*x))^2,x, algorithm="fricas")
 

Output:

integral((e^2*x^2 + 2*d*e*x + d^2)/(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) 
+ a^2), x)
 

Sympy [F]

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

integrate((e*x+d)**2/(a+b*asin(c*x))**2,x)
 

Output:

Integral((d + e*x)**2/(a + b*asin(c*x))**2, x)
 

Maxima [F]

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

integrate((e*x+d)^2/(a+b*arcsin(c*x))^2,x, algorithm="maxima")
 

Output:

-((e^2*x^2 + 2*d*e*x + d^2)*sqrt(c*x + 1)*sqrt(-c*x + 1) - (b^2*c*arctan2( 
c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)*integrate((3*c^2*e^2*x^3 + 4*c 
^2*d*e*x^2 - 2*d*e + (c^2*d^2 - 2*e^2)*x)*sqrt(c*x + 1)*sqrt(-c*x + 1)/(a* 
b*c^3*x^2 - a*b*c + (b^2*c^3*x^2 - b^2*c)*arctan2(c*x, sqrt(c*x + 1)*sqrt( 
-c*x + 1))), x))/(b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c 
)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1276 vs. \(2 (348) = 696\).

Time = 0.25 (sec) , antiderivative size = 1276, normalized size of antiderivative = 3.52 \[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^2/(a+b*arcsin(c*x))^2,x, algorithm="giac")
 

Output:

4*b*c*d*e*arcsin(c*x)*cos(a/b)^2*cos_integral(2*a/b + 2*arcsin(c*x))/(b^3* 
c^3*arcsin(c*x) + a*b^2*c^3) - 3*b*e^2*arcsin(c*x)*cos(a/b)^2*cos_integral 
(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + b*c^2 
*d^2*arcsin(c*x)*cos_integral(a/b + arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin( 
c*x) + a*b^2*c^3) + 3*b*e^2*arcsin(c*x)*cos(a/b)^3*sin_integral(3*a/b + 3* 
arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 4*b*c*d*e*arcsin(c*x)*cos 
(a/b)*sin(a/b)*sin_integral(2*a/b + 2*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + 
a*b^2*c^3) - b*c^2*d^2*arcsin(c*x)*cos(a/b)*sin_integral(a/b + arcsin(c*x) 
)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 4*a*c*d*e*cos(a/b)^2*cos_integral(2* 
a/b + 2*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 3*a*e^2*cos(a/b)^ 
2*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^ 
2*c^3) + a*c^2*d^2*cos_integral(a/b + arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsi 
n(c*x) + a*b^2*c^3) + 3*a*e^2*cos(a/b)^3*sin_integral(3*a/b + 3*arcsin(c*x 
))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 4*a*c*d*e*cos(a/b)*sin(a/b)*sin_int 
egral(2*a/b + 2*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - a*c^2*d^2 
*cos(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3 
) - 2*sqrt(-c^2*x^2 + 1)*b*c^2*d*e*x/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 2 
*b*c*d*e*arcsin(c*x)*cos_integral(2*a/b + 2*arcsin(c*x))/(b^3*c^3*arcsin(c 
*x) + a*b^2*c^3) + 3/4*b*e^2*arcsin(c*x)*cos_integral(3*a/b + 3*arcsin(c*x 
))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 1/4*b*e^2*arcsin(c*x)*c...
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int((d + e*x)^2/(a + b*asin(c*x))^2,x)
 

Output:

int((d + e*x)^2/(a + b*asin(c*x))^2, x)
 

Reduce [F]

\[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=\left (\int \frac {x^{2}}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) e^{2}+2 \left (\int \frac {x}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) d e +\left (\int \frac {1}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) d^{2} \] Input:

int((e*x+d)^2/(a+b*asin(c*x))^2,x)
 

Output:

int(x**2/(asin(c*x)**2*b**2 + 2*asin(c*x)*a*b + a**2),x)*e**2 + 2*int(x/(a 
sin(c*x)**2*b**2 + 2*asin(c*x)*a*b + a**2),x)*d*e + int(1/(asin(c*x)**2*b* 
*2 + 2*asin(c*x)*a*b + a**2),x)*d**2