\(\int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^3} \, dx\) [228]

Optimal result

Integrand size = 23, antiderivative size = 249 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^3} \, dx=-\frac {e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}-\frac {3 e^3 (c+d x)^2}{2 b^2 d (a+b \arcsin (c+d x))}+\frac {2 e^3 (c+d x)^4}{b^2 d (a+b \arcsin (c+d x))}+\frac {e^3 \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{2 b^3 d}-\frac {e^3 \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {4 a}{b}\right )}{b^3 d}-\frac {e^3 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 b^3 d}+\frac {e^3 \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{b^3 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {4889, 12, 4729, 4807, 4731, 4491, 3384, 3380, 3383} \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^3} \, dx=\frac {e^3 \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 b^3 d}-\frac {e^3 \sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{b^3 d}-\frac {e^3 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 b^3 d}+\frac {e^3 \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{b^3 d}+\frac {2 e^3 (c+d x)^4}{b^2 d (a+b \arcsin (c+d x))}-\frac {3 e^3 (c+d x)^2}{2 b^2 d (a+b \arcsin (c+d x))}-\frac {e^3 \sqrt {1-(c+d x)^2} (c+d x)^3}{2 b d (a+b \arcsin (c+d x))^2} \]

[In]

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

[Out]

-1/2*(e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(b*d*(a + b*ArcSin[c + d*x])^2) - (3*e^3*(c + d*x)^2)/(2*b^2*d*(a
+ b*ArcSin[c + d*x])) + (2*e^3*(c + d*x)^4)/(b^2*d*(a + b*ArcSin[c + d*x])) + (e^3*CosIntegral[(2*(a + b*ArcS
in[c + d*x]))/b]*Sin[(2*a)/b])/(2*b^3*d) - (e^3*CosIntegral[(4*(a + b*ArcSin[c + d*x]))/b]*Sin[(4*a)/b])/(b^3*
d) - (e^3*Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/(2*b^3*d) + (e^3*Cos[(4*a)/b]*SinIntegral[(
4*(a + b*ArcSin[c + d*x]))/b])/(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 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^3 x^3}{(a+b \arcsin (x))^3} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 \text {Subst}\left (\int \frac {x^3}{(a+b \arcsin (x))^3} \, dx,x,c+d x\right )}{d} \\ & = -\frac {e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}+\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2} (a+b \arcsin (x))^2} \, dx,x,c+d x\right )}{2 b d}-\frac {\left (2 e^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1-x^2} (a+b \arcsin (x))^2} \, dx,x,c+d x\right )}{b d} \\ & = -\frac {e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}-\frac {3 e^3 (c+d x)^2}{2 b^2 d (a+b \arcsin (c+d x))}+\frac {2 e^3 (c+d x)^4}{b^2 d (a+b \arcsin (c+d x))}+\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {x}{a+b \arcsin (x)} \, dx,x,c+d x\right )}{b^2 d}-\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {x^3}{a+b \arcsin (x)} \, dx,x,c+d x\right )}{b^2 d} \\ & = -\frac {e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}-\frac {3 e^3 (c+d x)^2}{2 b^2 d (a+b \arcsin (c+d x))}+\frac {2 e^3 (c+d x)^4}{b^2 d (a+b \arcsin (c+d x))}-\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{b^3 d}+\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{b^3 d} \\ & = -\frac {e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}-\frac {3 e^3 (c+d x)^2}{2 b^2 d (a+b \arcsin (c+d x))}+\frac {2 e^3 (c+d x)^4}{b^2 d (a+b \arcsin (c+d x))}-\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 x} \, dx,x,a+b \arcsin (c+d x)\right )}{b^3 d}+\frac {\left (8 e^3\right ) \text {Subst}\left (\int \left (-\frac {\sin \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 x}+\frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{4 x}\right ) \, dx,x,a+b \arcsin (c+d x)\right )}{b^3 d} \\ & = -\frac {e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}-\frac {3 e^3 (c+d x)^2}{2 b^2 d (a+b \arcsin (c+d x))}+\frac {2 e^3 (c+d x)^4}{b^2 d (a+b \arcsin (c+d x))}-\frac {e^3 \text {Subst}\left (\int \frac {\sin \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{b^3 d}-\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{2 b^3 d}+\frac {\left (2 e^3\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{b^3 d} \\ & = -\frac {e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}-\frac {3 e^3 (c+d x)^2}{2 b^2 d (a+b \arcsin (c+d x))}+\frac {2 e^3 (c+d x)^4}{b^2 d (a+b \arcsin (c+d x))}+\frac {\left (3 e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{2 b^3 d}-\frac {\left (2 e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{b^3 d}+\frac {\left (e^3 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{b^3 d}-\frac {\left (3 e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{2 b^3 d}+\frac {\left (2 e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{b^3 d}-\frac {\left (e^3 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{b^3 d} \\ & = -\frac {e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}-\frac {3 e^3 (c+d x)^2}{2 b^2 d (a+b \arcsin (c+d x))}+\frac {2 e^3 (c+d x)^4}{b^2 d (a+b \arcsin (c+d x))}+\frac {e^3 \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{2 b^3 d}-\frac {e^3 \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {4 a}{b}\right )}{b^3 d}-\frac {e^3 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 b^3 d}+\frac {e^3 \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{b^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.73 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^3} \, dx=\frac {e^3 \left (-\frac {b^2 (c+d x)^3 \sqrt {1-(c+d x)^2}}{(a+b \arcsin (c+d x))^2}+\frac {b \left (-3 (c+d x)^2+4 (c+d x)^4\right )}{a+b \arcsin (c+d x)}+\operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right ) \sin \left (\frac {2 a}{b}\right )-2 \operatorname {CosIntegral}\left (4 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right ) \sin \left (\frac {4 a}{b}\right )-\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )+2 \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (4 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )\right )}{2 b^3 d} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(506\) vs. \(2(239)=478\).

Time = 0.34 (sec) , antiderivative size = 507, normalized size of antiderivative = 2.04

method	result	size
derivativedivides	\(-\frac {e^{3} \left (16 \arcsin \left (d x +c \right )^{2} \sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) b^{2}-16 \arcsin \left (d x +c \right )^{2} \operatorname {Si}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) b^{2}+8 \arcsin \left (d x +c \right )^{2} \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) b^{2}-8 \arcsin \left (d x +c \right )^{2} \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) b^{2}+32 \arcsin \left (d x +c \right ) \sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) a b -32 \arcsin \left (d x +c \right ) \operatorname {Si}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) a b +16 \arcsin \left (d x +c \right ) \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) a b -16 \arcsin \left (d x +c \right ) \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) a b -4 \arcsin \left (d x +c \right ) \cos \left (4 \arcsin \left (d x +c \right )\right ) b^{2}+4 \arcsin \left (d x +c \right ) \cos \left (2 \arcsin \left (d x +c \right )\right ) b^{2}+16 \sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) a^{2}-16 \,\operatorname {Si}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) a^{2}+8 \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) a^{2}-8 \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) a^{2}-4 \cos \left (4 \arcsin \left (d x +c \right )\right ) a b +2 \sin \left (2 \arcsin \left (d x +c \right )\right ) b^{2}+4 \cos \left (2 \arcsin \left (d x +c \right )\right ) a b -\sin \left (4 \arcsin \left (d x +c \right )\right ) b^{2}\right )}{16 d \left (a +b \arcsin \left (d x +c \right )\right )^{2} b^{3}}\)	\(507\)
default	\(-\frac {e^{3} \left (16 \arcsin \left (d x +c \right )^{2} \sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) b^{2}-16 \arcsin \left (d x +c \right )^{2} \operatorname {Si}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) b^{2}+8 \arcsin \left (d x +c \right )^{2} \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) b^{2}-8 \arcsin \left (d x +c \right )^{2} \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) b^{2}+32 \arcsin \left (d x +c \right ) \sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) a b -32 \arcsin \left (d x +c \right ) \operatorname {Si}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) a b +16 \arcsin \left (d x +c \right ) \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) a b -16 \arcsin \left (d x +c \right ) \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) a b -4 \arcsin \left (d x +c \right ) \cos \left (4 \arcsin \left (d x +c \right )\right ) b^{2}+4 \arcsin \left (d x +c \right ) \cos \left (2 \arcsin \left (d x +c \right )\right ) b^{2}+16 \sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) a^{2}-16 \,\operatorname {Si}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) a^{2}+8 \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) a^{2}-8 \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) a^{2}-4 \cos \left (4 \arcsin \left (d x +c \right )\right ) a b +2 \sin \left (2 \arcsin \left (d x +c \right )\right ) b^{2}+4 \cos \left (2 \arcsin \left (d x +c \right )\right ) a b -\sin \left (4 \arcsin \left (d x +c \right )\right ) b^{2}\right )}{16 d \left (a +b \arcsin \left (d x +c \right )\right )^{2} b^{3}}\)	\(507\)

[In]

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

[Out]

-1/16/d*e^3*(16*arcsin(d*x+c)^2*sin(4*a/b)*Ci(4*arcsin(d*x+c)+4*a/b)*b^2-16*arcsin(d*x+c)^2*Si(4*arcsin(d*x+c)
+4*a/b)*cos(4*a/b)*b^2+8*arcsin(d*x+c)^2*cos(2*a/b)*Si(2*arcsin(d*x+c)+2*a/b)*b^2-8*arcsin(d*x+c)^2*sin(2*a/b)
*Ci(2*arcsin(d*x+c)+2*a/b)*b^2+32*arcsin(d*x+c)*sin(4*a/b)*Ci(4*arcsin(d*x+c)+4*a/b)*a*b-32*arcsin(d*x+c)*Si(4
*arcsin(d*x+c)+4*a/b)*cos(4*a/b)*a*b+16*arcsin(d*x+c)*cos(2*a/b)*Si(2*arcsin(d*x+c)+2*a/b)*a*b-16*arcsin(d*x+c
)*sin(2*a/b)*Ci(2*arcsin(d*x+c)+2*a/b)*a*b-4*arcsin(d*x+c)*cos(4*arcsin(d*x+c))*b^2+4*arcsin(d*x+c)*cos(2*arcs
in(d*x+c))*b^2+16*sin(4*a/b)*Ci(4*arcsin(d*x+c)+4*a/b)*a^2-16*Si(4*arcsin(d*x+c)+4*a/b)*cos(4*a/b)*a^2+8*cos(2
*a/b)*Si(2*arcsin(d*x+c)+2*a/b)*a^2-8*sin(2*a/b)*Ci(2*arcsin(d*x+c)+2*a/b)*a^2-4*cos(4*arcsin(d*x+c))*a*b+2*si
n(2*arcsin(d*x+c))*b^2+4*cos(2*arcsin(d*x+c))*a*b-sin(4*arcsin(d*x+c))*b^2)/(a+b*arcsin(d*x+c))^2/b^3

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

e**3*(Integral(c**3/(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**3*x**3/(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(3*c*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)
+ Integral(3*c**2*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]

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

[In]

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

[Out]

1/2*(4*a*d^4*e^3*x^4 + 16*a*c*d^3*e^3*x^3 + 3*(8*a*c^2 - a)*d^2*e^3*x^2 + 2*(8*a*c^3 - 3*a*c)*d*e^3*x + (4*a*c
^4 - 3*a*c^2)*e^3 - (b*d^3*e^3*x^3 + 3*b*c*d^2*e^3*x^2 + 3*b*c^2*d*e^3*x + b*c^3*e^3)*sqrt(d*x + c + 1)*sqrt(-
d*x - c + 1) + (4*b*d^4*e^3*x^4 + 16*b*c*d^3*e^3*x^3 + 3*(8*b*c^2 - b)*d^2*e^3*x^2 + 2*(8*b*c^3 - 3*b*c)*d*e^3
*x + (4*b*c^4 - 3*b*c^2)*e^3)*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((8*d^3*e^3*x^3 + 24*c*d^2*e^3*x^2 + 3*(8*c^2 - 1)*d*e^3*x + (8*c^3 - 3*c)*e^3)/(b^3*arc
tan2(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)*sq
rt(-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. 2201 vs. \(2 (239) = 478\).

Time = 0.54 (sec) , antiderivative size = 2201, normalized size of antiderivative = 8.84 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-8*b^2*e^3*arcsin(d*x + c)^2*cos(a/b)^3*cos_integral(4*a/b + 4*arcsin(d*x + c))*sin(a/b)/(b^5*d*arcsin(d*x + c
)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) + 8*b^2*e^3*arcsin(d*x + c)^2*cos(a/b)^4*sin_integral(4*a/b + 4*a
rcsin(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) - 16*a*b*e^3*arcsin(d*x + c)
*cos(a/b)^3*cos_integral(4*a/b + 4*arcsin(d*x + c))*sin(a/b)/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x +
c) + a^2*b^3*d) + 16*a*b*e^3*arcsin(d*x + c)*cos(a/b)^4*sin_integral(4*a/b + 4*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) + 4*b^2*e^3*arcsin(d*x + c)^2*cos(a/b)*cos_integral(4*a/b
+ 4*arcsin(d*x + c))*sin(a/b)/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - 8*a^2*e^3*c
os(a/b)^3*cos_integral(4*a/b + 4*arcsin(d*x + c))*sin(a/b)/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c
) + a^2*b^3*d) + b^2*e^3*arcsin(d*x + c)^2*cos(a/b)*cos_integral(2*a/b + 2*arcsin(d*x + c))*sin(a/b)/(b^5*d*ar
csin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - 8*b^2*e^3*arcsin(d*x + c)^2*cos(a/b)^2*sin_integral
(4*a/b + 4*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) + 8*a^2*e^3*cos(
a/b)^4*sin_integral(4*a/b + 4*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) - b^2*e^3*arcsin(d*x + c)^2*cos(a/b)^2*sin_integral(2*a/b + 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) + 8*a*b*e^3*arcsin(d*x + c)*cos(a/b)*cos_integral(4*a/b + 4*arcsin(d*x +
c))*sin(a/b)/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) + 2*a*b*e^3*arcsin(d*x + c)*co
s(a/b)*cos_integral(2*a/b + 2*arcsin(d*x + c))*sin(a/b)/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) +
a^2*b^3*d) - 16*a*b*e^3*arcsin(d*x + c)*cos(a/b)^2*sin_integral(4*a/b + 4*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) - 2*a*b*e^3*arcsin(d*x + c)*cos(a/b)^2*sin_integral(2*a/b + 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) + 2*((d*x + c)^2 - 1)^2*b^2
*e^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) + 4*a^2*e^3*cos(a/b)*co
s_integral(4*a/b + 4*arcsin(d*x + c))*sin(a/b)/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*
d) + a^2*e^3*cos(a/b)*cos_integral(2*a/b + 2*arcsin(d*x + c))*sin(a/b)/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*ar
csin(d*x + c) + a^2*b^3*d) + b^2*e^3*arcsin(d*x + c)^2*sin_integral(4*a/b + 4*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) - 8*a^2*e^3*cos(a/b)^2*sin_integral(4*a/b + 4*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*b^2*e^3*arcsin(d*x + c)^2*sin_inte
gral(2*a/b + 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) - a^2*e^3*co
s(a/b)^2*sin_integral(2*a/b + 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) + 1/2*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)*b^2*e^3/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a
^2*b^3*d) + 2*((d*x + c)^2 - 1)^2*a*b*e^3/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) +
5/2*((d*x + c)^2 - 1)*b^2*e^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
) + 2*a*b*e^3*arcsin(d*x + c)*sin_integral(4*a/b + 4*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) + a*b*e^3*arcsin(d*x + c)*sin_integral(2*a/b + 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) - 1/2*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*b^2*e^3/(b^5*d*arcsin(d
*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) + 5/2*((d*x + c)^2 - 1)*a*b*e^3/(b^5*d*arcsin(d*x + c)^2 +
2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) + 1/2*b^2*e^3*arcsin(d*x + c)/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcs
in(d*x + c) + a^2*b^3*d) + a^2*e^3*sin_integral(4*a/b + 4*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*a^2*e^3*sin_integral(2*a/b + 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) + 1/2*a*b*e^3/(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. \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^3} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3} \,d x \]

[In]

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

[Out]

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