\(\int \frac {(c e+d e x)^2}{(a+b \arcsin (c+d x))^2} \, dx\) [223]

Optimal result

Integrand size = 23, antiderivative size = 186 \[ \int \frac {(c e+d e x)^2}{(a+b \arcsin (c+d x))^2} \, dx=-\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))}+\frac {e^2 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{4 b^2 d}-\frac {3 e^2 \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b^2 d}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{4 b^2 d}+\frac {3 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{4 b^2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4889, 12, 4727, 3384, 3380, 3383} \[ \int \frac {(c e+d e x)^2}{(a+b \arcsin (c+d x))^2} \, dx=\frac {e^2 \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{4 b^2 d}-\frac {3 e^2 \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{4 b^2 d}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{4 b^2 d}+\frac {3 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{4 b^2 d}-\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 4727

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 - c^2*x^2]*((a + b*ArcSin
[c*x])^(n + 1)/(b*c*(n + 1))), x] - Dist[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), Sin[
-a/b + x/b]^(m - 1)*(m - (m + 1)*Sin[-a/b + x/b]^2), x], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c}, x]
&& IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 4889

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e^2 x^2}{(a+b \arcsin (x))^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int \frac {x^2}{(a+b \arcsin (x))^2} \, dx,x,c+d x\right )}{d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))}+\frac {e^2 \text {Subst}\left (\int \left (-\frac {3 \sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 x}+\frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{4 x}\right ) \, dx,x,a+b \arcsin (c+d x)\right )}{b^2 d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))}+\frac {e^2 \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{4 b^2 d}-\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{4 b^2 d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))}-\frac {\left (e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{4 b^2 d}+\frac {\left (3 e^2 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{4 b^2 d}+\frac {\left (e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{4 b^2 d}-\frac {\left (3 e^2 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{4 b^2 d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))}+\frac {e^2 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{4 b^2 d}-\frac {3 e^2 \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b^2 d}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{4 b^2 d}+\frac {3 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{4 b^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.23 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.75 \[ \int \frac {(c e+d e x)^2}{(a+b \arcsin (c+d x))^2} \, dx=\frac {e^2 \left (-\frac {4 b (c+d x)^2 \sqrt {1-(c+d x)^2}}{a+b \arcsin (c+d x)}+\operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c+d x)\right ) \sin \left (\frac {a}{b}\right )-3 \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right ) \sin \left (\frac {3 a}{b}\right )-\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c+d x)\right )+3 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )\right )}{4 b^2 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.43

method	result	size
derivativedivides	\(\frac {e^{2} \left (\arcsin \left (d x +c \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b -\arcsin \left (d x +c \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b +3 \arcsin \left (d x +c \right ) \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) b -3 \arcsin \left (d x +c \right ) \operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b +\sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a -\cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a +3 \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a -3 \,\operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +\cos \left (3 \arcsin \left (d x +c \right )\right ) b -\sqrt {1-\left (d x +c \right )^{2}}\, b \right )}{4 d \left (a +b \arcsin \left (d x +c \right )\right ) b^{2}}\)	\(266\)
default	\(\frac {e^{2} \left (\arcsin \left (d x +c \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b -\arcsin \left (d x +c \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b +3 \arcsin \left (d x +c \right ) \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) b -3 \arcsin \left (d x +c \right ) \operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b +\sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a -\cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a +3 \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a -3 \,\operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +\cos \left (3 \arcsin \left (d x +c \right )\right ) b -\sqrt {1-\left (d x +c \right )^{2}}\, b \right )}{4 d \left (a +b \arcsin \left (d x +c \right )\right ) b^{2}}\)	\(266\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-((d^2*e^2*x^2 + 2*c*d*e^2*x + c^2*e^2)*sqrt(d*x + c + 1)*sqrt(-d*x - c + 1) - (b^2*d*arctan2(d*x + c, sqrt(d*
x + c + 1)*sqrt(-d*x - c + 1)) + a*b*d)*integrate((3*d^3*e^2*x^3 + 9*c*d^2*e^2*x^2 + (9*c^2 - 2)*d*e^2*x + (3*
c^3 - 2*c)*e^2)*sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)/(a*b*d^2*x^2 + 2*a*b*c*d*x + a*b*c^2 - a*b + (b^2*d^2*x^2
+ 2*b^2*c*d*x + b^2*c^2 - b^2)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))), x))/(b^2*d*arctan2(d*
x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + a*b*d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 698 vs. \(2 (176) = 352\).

Time = 0.41 (sec) , antiderivative size = 698, normalized size of antiderivative = 3.75 \[ \int \frac {(c e+d e x)^2}{(a+b \arcsin (c+d x))^2} \, dx=-\frac {3 \, b e^{2} \arcsin \left (d x + c\right ) \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d} + \frac {3 \, b e^{2} \arcsin \left (d x + c\right ) \cos \left (\frac {a}{b}\right )^{3} \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right )}{b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d} - \frac {3 \, a e^{2} \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d} + \frac {3 \, a e^{2} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right )}{b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d} + \frac {3 \, b e^{2} \arcsin \left (d x + c\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, {\left (b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d\right )}} + \frac {b e^{2} \arcsin \left (d x + c\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (d x + c\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, {\left (b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d\right )}} - \frac {9 \, b e^{2} \arcsin \left (d x + c\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right )}{4 \, {\left (b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d\right )}} - \frac {b e^{2} \arcsin \left (d x + c\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (d x + c\right )\right )}{4 \, {\left (b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d\right )}} + \frac {3 \, a e^{2} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, {\left (b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d\right )}} + \frac {a e^{2} \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (d x + c\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, {\left (b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d\right )}} - \frac {9 \, a e^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right )}{4 \, {\left (b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d\right )}} - \frac {a e^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (d x + c\right )\right )}{4 \, {\left (b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d\right )}} + \frac {{\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} b e^{2}}{b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d} - \frac {\sqrt {-{\left (d x + c\right )}^{2} + 1} b e^{2}}{b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d} \]

[In]

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

[Out]

-3*b*e^2*arcsin(d*x + c)*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(d*x + c))*sin(a/b)/(b^3*d*arcsin(d*x + c) +
a*b^2*d) + 3*b*e^2*arcsin(d*x + c)*cos(a/b)^3*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b^3*d*arcsin(d*x + c) +
a*b^2*d) - 3*a*e^2*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(d*x + c))*sin(a/b)/(b^3*d*arcsin(d*x + c) + a*b^2
*d) + 3*a*e^2*cos(a/b)^3*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b^3*d*arcsin(d*x + c) + a*b^2*d) + 3/4*b*e^2
*arcsin(d*x + c)*cos_integral(3*a/b + 3*arcsin(d*x + c))*sin(a/b)/(b^3*d*arcsin(d*x + c) + a*b^2*d) + 1/4*b*e^
2*arcsin(d*x + c)*cos_integral(a/b + arcsin(d*x + c))*sin(a/b)/(b^3*d*arcsin(d*x + c) + a*b^2*d) - 9/4*b*e^2*a
rcsin(d*x + c)*cos(a/b)*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b^3*d*arcsin(d*x + c) + a*b^2*d) - 1/4*b*e^2*
arcsin(d*x + c)*cos(a/b)*sin_integral(a/b + arcsin(d*x + c))/(b^3*d*arcsin(d*x + c) + a*b^2*d) + 3/4*a*e^2*cos
_integral(3*a/b + 3*arcsin(d*x + c))*sin(a/b)/(b^3*d*arcsin(d*x + c) + a*b^2*d) + 1/4*a*e^2*cos_integral(a/b +
arcsin(d*x + c))*sin(a/b)/(b^3*d*arcsin(d*x + c) + a*b^2*d) - 9/4*a*e^2*cos(a/b)*sin_integral(3*a/b + 3*arcsi
n(d*x + c))/(b^3*d*arcsin(d*x + c) + a*b^2*d) - 1/4*a*e^2*cos(a/b)*sin_integral(a/b + arcsin(d*x + c))/(b^3*d*
arcsin(d*x + c) + a*b^2*d) + (-(d*x + c)^2 + 1)^(3/2)*b*e^2/(b^3*d*arcsin(d*x + c) + a*b^2*d) - sqrt(-(d*x + c
)^2 + 1)*b*e^2/(b^3*d*arcsin(d*x + c) + a*b^2*d)

Mupad [F(-1)]

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

[In]

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

[Out]

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