\(\int \frac {(d+e x)^3}{a+b \arcsin (c x)} \, dx\) [16]

Optimal result

Integrand size = 18, antiderivative size = 393 \[ \int \frac {(d+e x)^3}{a+b \arcsin (c x)} \, dx=\frac {d^3 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {3 d e^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{4 b c^3}-\frac {3 d e^2 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 a}{b}+3 \arcsin (c x)\right )}{4 b c^3}-\frac {3 d^2 e \operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \arcsin (c x)\right ) \sin \left (\frac {2 a}{b}\right )}{2 b c^2}-\frac {e^3 \operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \arcsin (c x)\right ) \sin \left (\frac {2 a}{b}\right )}{4 b c^4}+\frac {e^3 \operatorname {CosIntegral}\left (\frac {4 a}{b}+4 \arcsin (c x)\right ) \sin \left (\frac {4 a}{b}\right )}{8 b c^4}+\frac {d^3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {3 d e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{4 b c^3}+\frac {3 d^2 e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{2 b c^2}+\frac {e^3 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{4 b c^4}-\frac {3 d e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \arcsin (c x)\right )}{4 b c^3}-\frac {e^3 \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 a}{b}+4 \arcsin (c x)\right )}{8 b c^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {4831, 6874, 3384, 3380, 3383, 4491, 12} \[ \int \frac {(d+e x)^3}{a+b \arcsin (c x)} \, dx=-\frac {e^3 \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{4 b c^4}+\frac {e^3 \sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 a}{b}+4 \arcsin (c x)\right )}{8 b c^4}+\frac {e^3 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{4 b c^4}-\frac {e^3 \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 a}{b}+4 \arcsin (c x)\right )}{8 b c^4}+\frac {3 d e^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{4 b c^3}-\frac {3 d e^2 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 a}{b}+3 \arcsin (c x)\right )}{4 b c^3}+\frac {3 d e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{4 b c^3}-\frac {3 d e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \arcsin (c x)\right )}{4 b c^3}-\frac {3 d^2 e \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{2 b c^2}+\frac {3 d^2 e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{2 b c^2}+\frac {d^3 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {d^3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c} \]

[In]

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

[Out]

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

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 4831

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

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cos (x) (c d+e \sin (x))^3}{a+b x} \, dx,x,\arcsin (c x)\right )}{c^4} \\ & = \frac {\text {Subst}\left (\int \left (\frac {c^3 d^3 \cos (x)}{a+b x}+\frac {3 c^2 d^2 e \cos (x) \sin (x)}{a+b x}+\frac {3 c d e^2 \cos (x) \sin ^2(x)}{a+b x}+\frac {e^3 \cos (x) \sin ^3(x)}{a+b x}\right ) \, dx,x,\arcsin (c x)\right )}{c^4} \\ & = \frac {d^3 \text {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\arcsin (c x)\right )}{c}+\frac {\left (3 d^2 e\right ) \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{a+b x} \, dx,x,\arcsin (c x)\right )}{c^2}+\frac {\left (3 d e^2\right ) \text {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{a+b x} \, dx,x,\arcsin (c x)\right )}{c^3}+\frac {e^3 \text {Subst}\left (\int \frac {\cos (x) \sin ^3(x)}{a+b x} \, dx,x,\arcsin (c x)\right )}{c^4} \\ & = \frac {\left (3 d^2 e\right ) \text {Subst}\left (\int \frac {\sin (2 x)}{2 (a+b x)} \, dx,x,\arcsin (c x)\right )}{c^2}+\frac {\left (3 d e^2\right ) \text {Subst}\left (\int \left (\frac {\cos (x)}{4 (a+b x)}-\frac {\cos (3 x)}{4 (a+b x)}\right ) \, dx,x,\arcsin (c x)\right )}{c^3}+\frac {e^3 \text {Subst}\left (\int \left (\frac {\sin (2 x)}{4 (a+b x)}-\frac {\sin (4 x)}{8 (a+b x)}\right ) \, dx,x,\arcsin (c x)\right )}{c^4}+\frac {\left (d^3 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\arcsin (c x)\right )}{c}+\frac {\left (d^3 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\arcsin (c x)\right )}{c} \\ & = \frac {d^3 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {d^3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {\left (3 d^2 e\right ) \text {Subst}\left (\int \frac {\sin (2 x)}{a+b x} \, dx,x,\arcsin (c x)\right )}{2 c^2}+\frac {\left (3 d e^2\right ) \text {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\arcsin (c x)\right )}{4 c^3}-\frac {\left (3 d e^2\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{a+b x} \, dx,x,\arcsin (c x)\right )}{4 c^3}-\frac {e^3 \text {Subst}\left (\int \frac {\sin (4 x)}{a+b x} \, dx,x,\arcsin (c x)\right )}{8 c^4}+\frac {e^3 \text {Subst}\left (\int \frac {\sin (2 x)}{a+b x} \, dx,x,\arcsin (c x)\right )}{4 c^4} \\ & = \frac {d^3 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {d^3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {\left (3 d e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\arcsin (c x)\right )}{4 c^3}+\frac {\left (3 d^2 e \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\arcsin (c x)\right )}{2 c^2}+\frac {\left (e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\arcsin (c x)\right )}{4 c^4}-\frac {\left (3 d e^2 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\arcsin (c x)\right )}{4 c^3}-\frac {\left (e^3 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\arcsin (c x)\right )}{8 c^4}+\frac {\left (3 d e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\arcsin (c x)\right )}{4 c^3}-\frac {\left (3 d^2 e \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\arcsin (c x)\right )}{2 c^2}-\frac {\left (e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\arcsin (c x)\right )}{4 c^4}-\frac {\left (3 d e^2 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\arcsin (c x)\right )}{4 c^3}+\frac {\left (e^3 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\arcsin (c x)\right )}{8 c^4} \\ & = \frac {d^3 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {3 d e^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{4 b c^3}-\frac {3 d e^2 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 a}{b}+3 \arcsin (c x)\right )}{4 b c^3}-\frac {3 d^2 e \operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \arcsin (c x)\right ) \sin \left (\frac {2 a}{b}\right )}{2 b c^2}-\frac {e^3 \operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \arcsin (c x)\right ) \sin \left (\frac {2 a}{b}\right )}{4 b c^4}+\frac {e^3 \operatorname {CosIntegral}\left (\frac {4 a}{b}+4 \arcsin (c x)\right ) \sin \left (\frac {4 a}{b}\right )}{8 b c^4}+\frac {d^3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {3 d e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{4 b c^3}+\frac {3 d^2 e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{2 b c^2}+\frac {e^3 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{4 b c^4}-\frac {3 d e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \arcsin (c x)\right )}{4 b c^3}-\frac {e^3 \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 a}{b}+4 \arcsin (c x)\right )}{8 b c^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.77 \[ \int \frac {(d+e x)^3}{a+b \arcsin (c x)} \, dx=\frac {d^3 \left (\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )\right )}{b c}+\frac {3 d e^2 \left (\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )-\cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )-\sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )\right )}{4 b c^3}+\frac {e^3 \left (-2 \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {2 a}{b}\right )+\operatorname {CosIntegral}\left (4 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {4 a}{b}\right )+2 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-\cos \left (\frac {4 a}{b}\right ) \text {Si}\left (4 \left (\frac {a}{b}+\arcsin (c x)\right )\right )\right )}{8 b c^4}+\frac {3 d^2 e \left (-\operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \arcsin (c x)\right ) \sin \left (\frac {2 a}{b}\right )+\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )\right )}{2 b c^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {8 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) c^{3} d^{3}+8 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) c^{3} d^{3}+12 \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) c^{2} d^{2} e -12 \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) c^{2} d^{2} e +6 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) c d \,e^{2}+6 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) c d \,e^{2}-6 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) c d \,e^{2}-6 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) c d \,e^{2}+2 \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) e^{3}-2 \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) e^{3}-\cos \left (\frac {4 a}{b}\right ) \operatorname {Si}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) e^{3}+\sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) e^{3}}{8 c^{4} b}\)	\(327\)
default	\(\frac {8 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) c^{3} d^{3}+8 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) c^{3} d^{3}+12 \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) c^{2} d^{2} e -12 \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) c^{2} d^{2} e +6 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) c d \,e^{2}+6 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) c d \,e^{2}-6 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) c d \,e^{2}-6 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) c d \,e^{2}+2 \cos \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) e^{3}-2 \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) e^{3}-\cos \left (\frac {4 a}{b}\right ) \operatorname {Si}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) e^{3}+\sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) e^{3}}{8 c^{4} b}\)	\(327\)

[In]

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

[Out]

1/8/c^4*(8*Si(arcsin(c*x)+a/b)*sin(a/b)*c^3*d^3+8*Ci(arcsin(c*x)+a/b)*cos(a/b)*c^3*d^3+12*cos(2*a/b)*Si(2*arcs
in(c*x)+2*a/b)*c^2*d^2*e-12*sin(2*a/b)*Ci(2*arcsin(c*x)+2*a/b)*c^2*d^2*e+6*Si(arcsin(c*x)+a/b)*sin(a/b)*c*d*e^
2+6*Ci(arcsin(c*x)+a/b)*cos(a/b)*c*d*e^2-6*Si(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*c*d*e^2-6*Ci(3*arcsin(c*x)+3*a/b
)*cos(3*a/b)*c*d*e^2+2*cos(2*a/b)*Si(2*arcsin(c*x)+2*a/b)*e^3-2*sin(2*a/b)*Ci(2*arcsin(c*x)+2*a/b)*e^3-cos(4*a
/b)*Si(4*arcsin(c*x)+4*a/b)*e^3+sin(4*a/b)*Ci(4*arcsin(c*x)+4*a/b)*e^3)/b

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 609, normalized size of antiderivative = 1.55 \[ \int \frac {(d+e x)^3}{a+b \arcsin (c x)} \, dx=-\frac {3 \, d e^{2} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{b c^{3}} + \frac {d^{3} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b c} + \frac {e^{3} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b c^{4}} - \frac {3 \, d^{2} e \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b c^{2}} - \frac {e^{3} \cos \left (\frac {a}{b}\right )^{4} \operatorname {Si}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{b c^{4}} - \frac {3 \, d e^{2} \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{b c^{3}} + \frac {3 \, d^{2} e \cos \left (\frac {a}{b}\right )^{2} \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b c^{2}} + \frac {d^{3} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b c} + \frac {9 \, d e^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, b c^{3}} + \frac {3 \, d e^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{4 \, b c^{3}} - \frac {e^{3} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{2 \, b c^{4}} - \frac {e^{3} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{2 \, b c^{4}} + \frac {e^{3} \cos \left (\frac {a}{b}\right )^{2} \operatorname {Si}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{b c^{4}} + \frac {3 \, d e^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, b c^{3}} - \frac {3 \, d^{2} e \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{2 \, b c^{2}} + \frac {e^{3} \cos \left (\frac {a}{b}\right )^{2} \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{2 \, b c^{4}} + \frac {3 \, d e^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{4 \, b c^{3}} - \frac {e^{3} \operatorname {Si}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{8 \, b c^{4}} - \frac {e^{3} \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{4 \, b c^{4}} \]

[In]

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

[Out]

-3*d*e^2*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(c*x))/(b*c^3) + d^3*cos(a/b)*cos_integral(a/b + arcsin(c*x))
/(b*c) + e^3*cos(a/b)^3*cos_integral(4*a/b + 4*arcsin(c*x))*sin(a/b)/(b*c^4) - 3*d^2*e*cos(a/b)*cos_integral(2
*a/b + 2*arcsin(c*x))*sin(a/b)/(b*c^2) - e^3*cos(a/b)^4*sin_integral(4*a/b + 4*arcsin(c*x))/(b*c^4) - 3*d*e^2*
cos(a/b)^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b*c^3) + 3*d^2*e*cos(a/b)^2*sin_integral(2*a/b + 2*ar
csin(c*x))/(b*c^2) + d^3*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b*c) + 9/4*d*e^2*cos(a/b)*cos_integral(3*a/
b + 3*arcsin(c*x))/(b*c^3) + 3/4*d*e^2*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b*c^3) - 1/2*e^3*cos(a/b)*cos
_integral(4*a/b + 4*arcsin(c*x))*sin(a/b)/(b*c^4) - 1/2*e^3*cos(a/b)*cos_integral(2*a/b + 2*arcsin(c*x))*sin(a
/b)/(b*c^4) + e^3*cos(a/b)^2*sin_integral(4*a/b + 4*arcsin(c*x))/(b*c^4) + 3/4*d*e^2*sin(a/b)*sin_integral(3*a
/b + 3*arcsin(c*x))/(b*c^3) - 3/2*d^2*e*sin_integral(2*a/b + 2*arcsin(c*x))/(b*c^2) + 1/2*e^3*cos(a/b)^2*sin_i
ntegral(2*a/b + 2*arcsin(c*x))/(b*c^4) + 3/4*d*e^2*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b*c^3) - 1/8*e^3*
sin_integral(4*a/b + 4*arcsin(c*x))/(b*c^4) - 1/4*e^3*sin_integral(2*a/b + 2*arcsin(c*x))/(b*c^4)

Mupad [F(-1)]

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

[In]

int((d + e*x)^3/(a + b*asin(c*x)),x)

[Out]

int((d + e*x)^3/(a + b*asin(c*x)), x)