\(\int \frac {(c e+d e x)^2}{(a+b \arcsin (c+d x))^3} \, dx\) [229]

Optimal result

Integrand size = 23, antiderivative size = 248 \[ \int \frac {(c e+d e x)^2}{(a+b \arcsin (c+d x))^3} \, dx=-\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}-\frac {e^2 (c+d x)}{b^2 d (a+b \arcsin (c+d x))}+\frac {3 e^2 (c+d x)^3}{2 b^2 d (a+b \arcsin (c+d x))}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b^3 d}+\frac {9 e^2 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{8 b^3 d}-\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b^3 d}+\frac {9 e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{8 b^3 d} \]

[Out]

-e^2*(d*x+c)/b^2/d/(a+b*arcsin(d*x+c))+3/2*e^2*(d*x+c)^3/b^2/d/(a+b*arcsin(d*x+c))-1/8*e^2*Ci((a+b*arcsin(d*x+
c))/b)*cos(a/b)/b^3/d+9/8*e^2*Ci(3*(a+b*arcsin(d*x+c))/b)*cos(3*a/b)/b^3/d-1/8*e^2*Si((a+b*arcsin(d*x+c))/b)*s
in(a/b)/b^3/d+9/8*e^2*Si(3*(a+b*arcsin(d*x+c))/b)*sin(3*a/b)/b^3/d-1/2*e^2*(d*x+c)^2*(1-(d*x+c)^2)^(1/2)/b/d/(
a+b*arcsin(d*x+c))^2

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {4889, 12, 4729, 4807, 4731, 4491, 3384, 3380, 3383, 4719} \[ \int \frac {(c e+d e x)^2}{(a+b \arcsin (c+d x))^3} \, dx=-\frac {e^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b^3 d}+\frac {9 e^2 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{8 b^3 d}-\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b^3 d}+\frac {9 e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{8 b^3 d}+\frac {3 e^2 (c+d x)^3}{2 b^2 d (a+b \arcsin (c+d x))}-\frac {e^2 (c+d x)}{b^2 d (a+b \arcsin (c+d x))}-\frac {e^2 \sqrt {1-(c+d x)^2} (c+d x)^2}{2 b d (a+b \arcsin (c+d x))^2} \]

[In]

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

[Out]

-1/2*(e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(b*d*(a + b*ArcSin[c + d*x])^2) - (e^2*(c + d*x))/(b^2*d*(a + b*A
rcSin[c + d*x])) + (3*e^2*(c + d*x)^3)/(2*b^2*d*(a + b*ArcSin[c + d*x])) - (e^2*Cos[a/b]*CosIntegral[(a + b*Ar
cSin[c + d*x])/b])/(8*b^3*d) + (9*e^2*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/(8*b^3*d) - (e^
2*Sin[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(8*b^3*d) + (9*e^2*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSi
n[c + d*x]))/b])/(8*b^3*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 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 4719

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cos[-a/b + x/b], x], x,
a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 4729

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[c*((m + 1)/(b*(n + 1))), Int[x^(m + 1)*((a + b*ArcSin[c*x])^(n + 1)/
Sqrt[1 - c^2*x^2]), x], x] - Dist[m/(b*c*(n + 1)), Int[x^(m - 1)*((a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2
]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 4731

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

Rule 4807

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] - Dist[f*(m/(b
*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]], Int[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n + 1), x], x] /;
FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && 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))^3} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int \frac {x^2}{(a+b \arcsin (x))^3} \, dx,x,c+d x\right )}{d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}+\frac {e^2 \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} (a+b \arcsin (x))^2} \, dx,x,c+d x\right )}{b d}-\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {1-x^2} (a+b \arcsin (x))^2} \, dx,x,c+d x\right )}{2 b d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}-\frac {e^2 (c+d x)}{b^2 d (a+b \arcsin (c+d x))}+\frac {3 e^2 (c+d x)^3}{2 b^2 d (a+b \arcsin (c+d x))}+\frac {e^2 \text {Subst}\left (\int \frac {1}{a+b \arcsin (x)} \, dx,x,c+d x\right )}{b^2 d}-\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {x^2}{a+b \arcsin (x)} \, dx,x,c+d x\right )}{2 b^2 d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}-\frac {e^2 (c+d x)}{b^2 d (a+b \arcsin (c+d x))}+\frac {3 e^2 (c+d x)^3}{2 b^2 d (a+b \arcsin (c+d x))}+\frac {e^2 \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{b^3 d}-\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{2 b^3 d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}-\frac {e^2 (c+d x)}{b^2 d (a+b \arcsin (c+d x))}+\frac {3 e^2 (c+d x)^3}{2 b^2 d (a+b \arcsin (c+d x))}-\frac {\left (9 e^2\right ) \text {Subst}\left (\int \left (-\frac {\cos \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 x}+\frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{4 x}\right ) \, dx,x,a+b \arcsin (c+d x)\right )}{2 b^3 d}+\frac {\left (e^2 \cos \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 )}{b^3 d}+\frac {\left (e^2 \sin \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 )}{b^3 d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}-\frac {e^2 (c+d x)}{b^2 d (a+b \arcsin (c+d x))}+\frac {3 e^2 (c+d x)^3}{2 b^2 d (a+b \arcsin (c+d x))}+\frac {e^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{b^3 d}+\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{b^3 d}+\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{8 b^3 d}-\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{8 b^3 d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}-\frac {e^2 (c+d x)}{b^2 d (a+b \arcsin (c+d x))}+\frac {3 e^2 (c+d x)^3}{2 b^2 d (a+b \arcsin (c+d x))}+\frac {e^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{b^3 d}+\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{b^3 d}-\frac {\left (9 e^2 \cos \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 )}{8 b^3 d}+\frac {\left (9 e^2 \cos \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 )}{8 b^3 d}-\frac {\left (9 e^2 \sin \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 )}{8 b^3 d}+\frac {\left (9 e^2 \sin \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 )}{8 b^3 d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}-\frac {e^2 (c+d x)}{b^2 d (a+b \arcsin (c+d x))}+\frac {3 e^2 (c+d x)^3}{2 b^2 d (a+b \arcsin (c+d x))}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b^3 d}+\frac {9 e^2 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{8 b^3 d}-\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b^3 d}+\frac {9 e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{8 b^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.88 \[ \int \frac {(c e+d e x)^2}{(a+b \arcsin (c+d x))^3} \, dx=\frac {e^2 \left (-\frac {4 b^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{(a+b \arcsin (c+d x))^2}+\frac {4 b \left (-2 (c+d x)+3 (c+d x)^3\right )}{a+b \arcsin (c+d x)}+8 \left (\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c+d x)\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c+d x)\right )\right )+9 \left (-\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c+d x)\right )+\cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )-\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c+d x)\right )+\sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )\right )\right )}{8 b^3 d} \]

[In]

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

[Out]

(e^2*((-4*b^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^2 + (4*b*(-2*(c + d*x) + 3*(c + d*x)^
3))/(a + b*ArcSin[c + d*x]) + 8*(Cos[a/b]*CosIntegral[a/b + ArcSin[c + d*x]] + Sin[a/b]*SinIntegral[a/b + ArcS
in[c + d*x]]) + 9*(-(Cos[a/b]*CosIntegral[a/b + ArcSin[c + d*x]]) + Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c
+ d*x])] - Sin[a/b]*SinIntegral[a/b + ArcSin[c + d*x]] + Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])])
))/(8*b^3*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(475\) vs. \(2(234)=468\).

Time = 0.75 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.92

method	result	size
derivativedivides	\(\frac {e^{2} \left (9 \arcsin \left (d x +c \right )^{2} \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) b^{2}+9 \arcsin \left (d x +c \right )^{2} \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) b^{2}-\arcsin \left (d x +c \right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b^{2}-\arcsin \left (d x +c \right )^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b^{2}+18 \arcsin \left (d x +c \right ) \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a b +18 \arcsin \left (d x +c \right ) \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a b -2 \arcsin \left (d x +c \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a b -2 \arcsin \left (d x +c \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a b -3 \arcsin \left (d x +c \right ) \sin \left (3 \arcsin \left (d x +c \right )\right ) b^{2}+\arcsin \left (d x +c \right ) b^{2} \left (d x +c \right )+9 \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a^{2}+9 \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a^{2}-\sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a^{2}-\cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a^{2}+\cos \left (3 \arcsin \left (d x +c \right )\right ) b^{2}-3 \sin \left (3 \arcsin \left (d x +c \right )\right ) a b -\sqrt {1-\left (d x +c \right )^{2}}\, b^{2}+a b \left (d x +c \right )\right )}{8 d \left (a +b \arcsin \left (d x +c \right )\right )^{2} b^{3}}\)	\(476\)
default	\(\frac {e^{2} \left (9 \arcsin \left (d x +c \right )^{2} \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) b^{2}+9 \arcsin \left (d x +c \right )^{2} \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) b^{2}-\arcsin \left (d x +c \right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b^{2}-\arcsin \left (d x +c \right )^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b^{2}+18 \arcsin \left (d x +c \right ) \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a b +18 \arcsin \left (d x +c \right ) \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a b -2 \arcsin \left (d x +c \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a b -2 \arcsin \left (d x +c \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a b -3 \arcsin \left (d x +c \right ) \sin \left (3 \arcsin \left (d x +c \right )\right ) b^{2}+\arcsin \left (d x +c \right ) b^{2} \left (d x +c \right )+9 \cos \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a^{2}+9 \sin \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a^{2}-\sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a^{2}-\cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a^{2}+\cos \left (3 \arcsin \left (d x +c \right )\right ) b^{2}-3 \sin \left (3 \arcsin \left (d x +c \right )\right ) a b -\sqrt {1-\left (d x +c \right )^{2}}\, b^{2}+a b \left (d x +c \right )\right )}{8 d \left (a +b \arcsin \left (d x +c \right )\right )^{2} b^{3}}\)	\(476\)

[In]

int((d*e*x+c*e)^2/(a+b*arcsin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/8/d*e^2*(9*arcsin(d*x+c)^2*cos(3*a/b)*Ci(3*arcsin(d*x+c)+3*a/b)*b^2+9*arcsin(d*x+c)^2*sin(3*a/b)*Si(3*arcsin
(d*x+c)+3*a/b)*b^2-arcsin(d*x+c)^2*sin(a/b)*Si(arcsin(d*x+c)+a/b)*b^2-arcsin(d*x+c)^2*cos(a/b)*Ci(arcsin(d*x+c
)+a/b)*b^2+18*arcsin(d*x+c)*cos(3*a/b)*Ci(3*arcsin(d*x+c)+3*a/b)*a*b+18*arcsin(d*x+c)*sin(3*a/b)*Si(3*arcsin(d
*x+c)+3*a/b)*a*b-2*arcsin(d*x+c)*sin(a/b)*Si(arcsin(d*x+c)+a/b)*a*b-2*arcsin(d*x+c)*cos(a/b)*Ci(arcsin(d*x+c)+
a/b)*a*b-3*arcsin(d*x+c)*sin(3*arcsin(d*x+c))*b^2+arcsin(d*x+c)*b^2*(d*x+c)+9*cos(3*a/b)*Ci(3*arcsin(d*x+c)+3*
a/b)*a^2+9*sin(3*a/b)*Si(3*arcsin(d*x+c)+3*a/b)*a^2-sin(a/b)*Si(arcsin(d*x+c)+a/b)*a^2-cos(a/b)*Ci(arcsin(d*x+
c)+a/b)*a^2+cos(3*arcsin(d*x+c))*b^2-3*sin(3*arcsin(d*x+c))*a*b-(1-(d*x+c)^2)^(1/2)*b^2+a*b*(d*x+c))/(a+b*arcs
in(d*x+c))^2/b^3

Fricas [F]

\[ \int \frac {(c e+d e x)^2}{(a+b \arcsin (c+d x))^3} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

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

[Out]

integral((d^2*e^2*x^2 + 2*c*d*e^2*x + c^2*e^2)/(b^3*arcsin(d*x + c)^3 + 3*a*b^2*arcsin(d*x + c)^2 + 3*a^2*b*ar
csin(d*x + c) + a^3), x)

Sympy [F]

\[ \int \frac {(c e+d e x)^2}{(a+b \arcsin (c+d x))^3} \, dx=e^{2} \left (\int \frac {c^{2}}{a^{3} + 3 a^{2} b \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{3} + 3 a^{2} b \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{3} + 3 a^{2} b \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx\right ) \]

[In]

integrate((d*e*x+c*e)**2/(a+b*asin(d*x+c))**3,x)

[Out]

e**2*(Integral(c**2/(a**3 + 3*a**2*b*asin(c + d*x) + 3*a*b**2*asin(c + d*x)**2 + b**3*asin(c + d*x)**3), x) +
Integral(d**2*x**2/(a**3 + 3*a**2*b*asin(c + d*x) + 3*a*b**2*asin(c + d*x)**2 + b**3*asin(c + d*x)**3), x) + I
ntegral(2*c*d*x/(a**3 + 3*a**2*b*asin(c + d*x) + 3*a*b**2*asin(c + d*x)**2 + b**3*asin(c + d*x)**3), x))

Maxima [F]

\[ \int \frac {(c e+d e x)^2}{(a+b \arcsin (c+d x))^3} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

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

[Out]

1/2*(3*a*d^3*e^2*x^3 + 9*a*c*d^2*e^2*x^2 + (9*a*c^2 - 2*a)*d*e^2*x + (3*a*c^3 - 2*a*c)*e^2 - (b*d^2*e^2*x^2 +
2*b*c*d*e^2*x + b*c^2*e^2)*sqrt(d*x + c + 1)*sqrt(-d*x - c + 1) + (3*b*d^3*e^2*x^3 + 9*b*c*d^2*e^2*x^2 + (9*b*
c^2 - 2*b)*d*e^2*x + (3*b*c^3 - 2*b*c)*e^2)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) - 2*(b^4*d*
arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2 + 2*a*b^3*d*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-
d*x - c + 1)) + a^2*b^2*d)*integrate(1/2*(9*d^2*e^2*x^2 + 18*c*d*e^2*x + (9*c^2 - 2)*e^2)/(b^3*arctan2(d*x + c
, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + a*b^2), x))/(b^4*d*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c
+ 1))^2 + 2*a*b^3*d*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + a^2*b^2*d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1641 vs. \(2 (234) = 468\).

Time = 0.58 (sec) , antiderivative size = 1641, normalized size of antiderivative = 6.62 \[ \int \frac {(c e+d e x)^2}{(a+b \arcsin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

9/2*b^2*e^2*arcsin(d*x + c)^2*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*
a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) + 9/2*b^2*e^2*arcsin(d*x + c)^2*cos(a/b)^2*sin(a/b)*sin_integral(3*a/b +
3*arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) + 9*a*b*e^2*arcsin(d*x +
c)*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a
^2*b^3*d) + 9*a*b*e^2*arcsin(d*x + c)*cos(a/b)^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b^5*d*arcsi
n(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - 27/8*b^2*e^2*arcsin(d*x + c)^2*cos(a/b)*cos_integral(3
*a/b + 3*arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) + 9/2*a^2*e^2*cos(
a/b)^3*cos_integral(3*a/b + 3*arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*
d) - 1/8*b^2*e^2*arcsin(d*x + c)^2*cos(a/b)*cos_integral(a/b + arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a
*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - 9/8*b^2*e^2*arcsin(d*x + c)^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(d*x
+ c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) + 9/2*a^2*e^2*cos(a/b)^2*sin(a/b)*sin
_integral(3*a/b + 3*arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - 1/8*b
^2*e^2*arcsin(d*x + c)^2*sin(a/b)*sin_integral(a/b + arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arc
sin(d*x + c) + a^2*b^3*d) + 3/2*((d*x + c)^2 - 1)*(d*x + c)*b^2*e^2*arcsin(d*x + c)/(b^5*d*arcsin(d*x + c)^2 +
2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - 27/4*a*b*e^2*arcsin(d*x + c)*cos(a/b)*cos_integral(3*a/b + 3*arcsin(
d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - 1/4*a*b*e^2*arcsin(d*x + c)*cos(
a/b)*cos_integral(a/b + arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - 9
/4*a*b*e^2*arcsin(d*x + c)*sin(a/b)*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4
*d*arcsin(d*x + c) + a^2*b^3*d) - 1/4*a*b*e^2*arcsin(d*x + c)*sin(a/b)*sin_integral(a/b + arcsin(d*x + c))/(b^
5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) + 3/2*((d*x + c)^2 - 1)*(d*x + c)*a*b*e^2/(b^5*
d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) + 1/2*(d*x + c)*b^2*e^2*arcsin(d*x + c)/(b^5*d*ar
csin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - 27/8*a^2*e^2*cos(a/b)*cos_integral(3*a/b + 3*arcsin
(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - 1/8*a^2*e^2*cos(a/b)*cos_integr
al(a/b + arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - 9/8*a^2*e^2*sin(
a/b)*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d)
- 1/8*a^2*e^2*sin(a/b)*sin_integral(a/b + arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x +
c) + a^2*b^3*d) + 1/2*(-(d*x + c)^2 + 1)^(3/2)*b^2*e^2/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) +
a^2*b^3*d) + 1/2*(d*x + c)*a*b*e^2/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - 1/2*sqr
t(-(d*x + c)^2 + 1)*b^2*e^2/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c e+d e x)^2}{(a+b \arcsin (c+d x))^3} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3} \,d x \]

[In]

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

[Out]

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