3.233 \(\int \frac{(c e+d e x)^4}{(a+b \sin ^{-1}(c+d x))^4} \, dx\)

Optimal. Leaf size=416 \[ -\frac{e^4 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c+d x)}{b}\right )}{48 b^4 d}+\frac{27 e^4 \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{32 b^4 d}-\frac{125 e^4 \sin \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (\frac{5 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{96 b^4 d}+\frac{e^4 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c+d x)}{b}\right )}{48 b^4 d}-\frac{27 e^4 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{32 b^4 d}+\frac{125 e^4 \cos \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{96 b^4 d}+\frac{5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{25 e^4 \sqrt{1-(c+d x)^2} (c+d x)^4}{6 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{2 e^4 \sqrt{1-(c+d x)^2} (c+d x)^2}{b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{e^4 \sqrt{1-(c+d x)^2} (c+d x)^4}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3} \]

[Out]

-(e^4*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(3*b*d*(a + b*ArcSin[c + d*x])^3) - (2*e^4*(c + d*x)^3)/(3*b^2*d*(a +
b*ArcSin[c + d*x])^2) + (5*e^4*(c + d*x)^5)/(6*b^2*d*(a + b*ArcSin[c + d*x])^2) - (2*e^4*(c + d*x)^2*Sqrt[1 -
(c + d*x)^2])/(b^3*d*(a + b*ArcSin[c + d*x])) + (25*e^4*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(6*b^3*d*(a + b*Ar
cSin[c + d*x])) - (e^4*CosIntegral[(a + b*ArcSin[c + d*x])/b]*Sin[a/b])/(48*b^4*d) + (27*e^4*CosIntegral[(3*(a
+ b*ArcSin[c + d*x]))/b]*Sin[(3*a)/b])/(32*b^4*d) - (125*e^4*CosIntegral[(5*(a + b*ArcSin[c + d*x]))/b]*Sin[(
5*a)/b])/(96*b^4*d) + (e^4*Cos[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(48*b^4*d) - (27*e^4*Cos[(3*a)/b]*
SinIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/(32*b^4*d) + (125*e^4*Cos[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c
+ d*x]))/b])/(96*b^4*d)

________________________________________________________________________________________

Rubi [A]  time = 0.863495, antiderivative size = 412, normalized size of antiderivative = 0.99, number of steps used = 24, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {4805, 12, 4633, 4719, 4631, 3303, 3299, 3302} \[ -\frac{e^4 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )}{48 b^4 d}+\frac{27 e^4 \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 a}{b}+3 \sin ^{-1}(c+d x)\right )}{32 b^4 d}-\frac{125 e^4 \sin \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (\frac{5 a}{b}+5 \sin ^{-1}(c+d x)\right )}{96 b^4 d}+\frac{e^4 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )}{48 b^4 d}-\frac{27 e^4 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c+d x)\right )}{32 b^4 d}+\frac{125 e^4 \cos \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 a}{b}+5 \sin ^{-1}(c+d x)\right )}{96 b^4 d}+\frac{5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{25 e^4 \sqrt{1-(c+d x)^2} (c+d x)^4}{6 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{2 e^4 \sqrt{1-(c+d x)^2} (c+d x)^2}{b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{e^4 \sqrt{1-(c+d x)^2} (c+d x)^4}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^4/(a + b*ArcSin[c + d*x])^4,x]

[Out]

-(e^4*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(3*b*d*(a + b*ArcSin[c + d*x])^3) - (2*e^4*(c + d*x)^3)/(3*b^2*d*(a +
b*ArcSin[c + d*x])^2) + (5*e^4*(c + d*x)^5)/(6*b^2*d*(a + b*ArcSin[c + d*x])^2) - (2*e^4*(c + d*x)^2*Sqrt[1 -
(c + d*x)^2])/(b^3*d*(a + b*ArcSin[c + d*x])) + (25*e^4*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(6*b^3*d*(a + b*Ar
cSin[c + d*x])) - (e^4*CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b])/(48*b^4*d) + (27*e^4*CosIntegral[(3*a)/b +
3*ArcSin[c + d*x]]*Sin[(3*a)/b])/(32*b^4*d) - (125*e^4*CosIntegral[(5*a)/b + 5*ArcSin[c + d*x]]*Sin[(5*a)/b])
/(96*b^4*d) + (e^4*Cos[a/b]*SinIntegral[a/b + ArcSin[c + d*x]])/(48*b^4*d) - (27*e^4*Cos[(3*a)/b]*SinIntegral[
(3*a)/b + 3*ArcSin[c + d*x]])/(32*b^4*d) + (125*e^4*Cos[(5*a)/b]*SinIntegral[(5*a)/b + 5*ArcSin[c + d*x]])/(96
*b^4*d)

Rule 4805

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]

Rule 12

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

Rule 4633

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 4719

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
((f*x)^m*(a + b*ArcSin[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] - Dist[(f*m)/(b*c*Sqrt[d]*(n + 1)), 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] && GtQ[d, 0]

Rule 4631

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

Rule 3303

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 3299

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 3302

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]

Rubi steps

\begin{align*} \int \frac{(c e+d e x)^4}{\left (a+b \sin ^{-1}(c+d x)\right )^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^4 x^4}{\left (a+b \sin ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^4 \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b \sin ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=-\frac{e^4 (c+d x)^4 \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac{\left (4 e^4\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{1-x^2} \left (a+b \sin ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}-\frac{\left (5 e^4\right ) \operatorname{Subst}\left (\int \frac{x^5}{\sqrt{1-x^2} \left (a+b \sin ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}\\ &=-\frac{e^4 (c+d x)^4 \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac{2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{\left (2 e^4\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{b^2 d}-\frac{\left (25 e^4\right ) \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{6 b^2 d}\\ &=-\frac{e^4 (c+d x)^4 \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac{2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{2 e^4 (c+d x)^2 \sqrt{1-(c+d x)^2}}{b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{25 e^4 (c+d x)^4 \sqrt{1-(c+d x)^2}}{6 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{\left (2 e^4\right ) \operatorname{Subst}\left (\int \left (-\frac{\sin (x)}{4 (a+b x)}+\frac{3 \sin (3 x)}{4 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{b^3 d}-\frac{\left (25 e^4\right ) \operatorname{Subst}\left (\int \left (-\frac{\sin (x)}{8 (a+b x)}+\frac{9 \sin (3 x)}{16 (a+b x)}-\frac{5 \sin (5 x)}{16 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{6 b^3 d}\\ &=-\frac{e^4 (c+d x)^4 \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac{2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{2 e^4 (c+d x)^2 \sqrt{1-(c+d x)^2}}{b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{25 e^4 (c+d x)^4 \sqrt{1-(c+d x)^2}}{6 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{e^4 \operatorname{Subst}\left (\int \frac{\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{2 b^3 d}+\frac{\left (25 e^4\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{48 b^3 d}+\frac{\left (125 e^4\right ) \operatorname{Subst}\left (\int \frac{\sin (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{96 b^3 d}+\frac{\left (3 e^4\right ) \operatorname{Subst}\left (\int \frac{\sin (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{2 b^3 d}-\frac{\left (75 e^4\right ) \operatorname{Subst}\left (\int \frac{\sin (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{32 b^3 d}\\ &=-\frac{e^4 (c+d x)^4 \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac{2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{2 e^4 (c+d x)^2 \sqrt{1-(c+d x)^2}}{b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{25 e^4 (c+d x)^4 \sqrt{1-(c+d x)^2}}{6 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{\left (e^4 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{2 b^3 d}+\frac{\left (25 e^4 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{48 b^3 d}+\frac{\left (3 e^4 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{2 b^3 d}-\frac{\left (75 e^4 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{32 b^3 d}+\frac{\left (125 e^4 \cos \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{96 b^3 d}+\frac{\left (e^4 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{2 b^3 d}-\frac{\left (25 e^4 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{48 b^3 d}-\frac{\left (3 e^4 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{2 b^3 d}+\frac{\left (75 e^4 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{32 b^3 d}-\frac{\left (125 e^4 \sin \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{96 b^3 d}\\ &=-\frac{e^4 (c+d x)^4 \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac{2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{2 e^4 (c+d x)^2 \sqrt{1-(c+d x)^2}}{b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{25 e^4 (c+d x)^4 \sqrt{1-(c+d x)^2}}{6 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{e^4 \text{Ci}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right ) \sin \left (\frac{a}{b}\right )}{48 b^4 d}+\frac{27 e^4 \text{Ci}\left (\frac{3 a}{b}+3 \sin ^{-1}(c+d x)\right ) \sin \left (\frac{3 a}{b}\right )}{32 b^4 d}-\frac{125 e^4 \text{Ci}\left (\frac{5 a}{b}+5 \sin ^{-1}(c+d x)\right ) \sin \left (\frac{5 a}{b}\right )}{96 b^4 d}+\frac{e^4 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )}{48 b^4 d}-\frac{27 e^4 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c+d x)\right )}{32 b^4 d}+\frac{125 e^4 \cos \left (\frac{5 a}{b}\right ) \text{Si}\left (\frac{5 a}{b}+5 \sin ^{-1}(c+d x)\right )}{96 b^4 d}\\ \end{align*}

Mathematica [A]  time = 1.64222, size = 414, normalized size = 1. \[ \frac{e^4 \left (-\frac{32 b^3 \sqrt{1-(c+d x)^2} (c+d x)^4}{\left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac{16 b^2 \left (5 (c+d x)^5-4 (c+d x)^3\right )}{\left (a+b \sin ^{-1}(c+d x)\right )^2}+384 \left (\cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )-\sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )+544 \left (3 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )-\sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )-3 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )+\cos \left (\frac{3 a}{b}\right ) \text{Si}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )\right )-125 \left (10 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )-5 \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )+\sin \left (\frac{5 a}{b}\right ) \text{CosIntegral}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )-10 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )+5 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )-\cos \left (\frac{5 a}{b}\right ) \text{Si}\left (5 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )\right )+\frac{16 b \sqrt{1-(c+d x)^2} \left (25 (c+d x)^4-12 (c+d x)^2\right )}{a+b \sin ^{-1}(c+d x)}\right )}{96 b^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^4/(a + b*ArcSin[c + d*x])^4,x]

[Out]

(e^4*((-32*b^3*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^3 + (16*b^2*(-4*(c + d*x)^3 + 5*(c +
d*x)^5))/(a + b*ArcSin[c + d*x])^2 + (16*b*Sqrt[1 - (c + d*x)^2]*(-12*(c + d*x)^2 + 25*(c + d*x)^4))/(a + b*A
rcSin[c + d*x]) + 384*(-(CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b]) + Cos[a/b]*SinIntegral[a/b + ArcSin[c +
d*x]]) + 544*(3*CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b] - CosIntegral[3*(a/b + ArcSin[c + d*x])]*Sin[(3*a)
/b] - 3*Cos[a/b]*SinIntegral[a/b + ArcSin[c + d*x]] + Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])]) - 1
25*(10*CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b] - 5*CosIntegral[3*(a/b + ArcSin[c + d*x])]*Sin[(3*a)/b] + C
osIntegral[5*(a/b + ArcSin[c + d*x])]*Sin[(5*a)/b] - 10*Cos[a/b]*SinIntegral[a/b + ArcSin[c + d*x]] + 5*Cos[(3
*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])] - Cos[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c + d*x])])))/(96*b^4
*d)

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Maple [B]  time = 0.099, size = 1138, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^4/(a+b*arcsin(d*x+c))^4,x)

[Out]

-1/96/d*e^4*(-4*arcsin(d*x+c)*(1-(d*x+c)^2)^(1/2)*a*b^2+81*arcsin(d*x+c)^3*Si(3*arcsin(d*x+c)+3*a/b)*cos(3*a/b
)*b^3-81*arcsin(d*x+c)^3*Ci(3*arcsin(d*x+c)+3*a/b)*sin(3*a/b)*b^3+54*arcsin(d*x+c)*cos(3*arcsin(d*x+c))*a*b^2-
125*arcsin(d*x+c)^3*Si(5*arcsin(d*x+c)+5*a/b)*cos(5*a/b)*b^3+125*arcsin(d*x+c)^3*Ci(5*arcsin(d*x+c)+5*a/b)*sin
(5*a/b)*b^3-50*arcsin(d*x+c)*cos(5*arcsin(d*x+c))*a*b^2-2*arcsin(d*x+c)^3*Si(arcsin(d*x+c)+a/b)*cos(a/b)*b^3+2
*arcsin(d*x+c)^3*Ci(arcsin(d*x+c)+a/b)*sin(a/b)*b^3-243*arcsin(d*x+c)^2*Ci(3*arcsin(d*x+c)+3*a/b)*sin(3*a/b)*a
*b^2-375*arcsin(d*x+c)*Si(5*arcsin(d*x+c)+5*a/b)*cos(5*a/b)*a^2*b+6*arcsin(d*x+c)*Ci(arcsin(d*x+c)+a/b)*sin(a/
b)*a^2*b-6*arcsin(d*x+c)*Si(arcsin(d*x+c)+a/b)*cos(a/b)*a^2*b-2*arcsin(d*x+c)*(d*x+c)*b^3-2*Si(arcsin(d*x+c)+a
/b)*cos(a/b)*a^3+2*Ci(arcsin(d*x+c)+a/b)*sin(a/b)*a^3-2*arcsin(d*x+c)^2*(1-(d*x+c)^2)^(1/2)*b^3-2*(1-(d*x+c)^2
)^(1/2)*a^2*b-2*(d*x+c)*a*b^2+9*arcsin(d*x+c)*sin(3*arcsin(d*x+c))*b^3+81*Si(3*arcsin(d*x+c)+3*a/b)*cos(3*a/b)
*a^3-81*Ci(3*arcsin(d*x+c)+3*a/b)*sin(3*a/b)*a^3+27*cos(3*arcsin(d*x+c))*a^2*b+9*sin(3*arcsin(d*x+c))*a*b^2-25
*arcsin(d*x+c)^2*cos(5*arcsin(d*x+c))*b^3-5*arcsin(d*x+c)*sin(5*arcsin(d*x+c))*b^3-125*Si(5*arcsin(d*x+c)+5*a/
b)*cos(5*a/b)*a^3+125*Ci(5*arcsin(d*x+c)+5*a/b)*sin(5*a/b)*a^3-25*cos(5*arcsin(d*x+c))*a^2*b+27*arcsin(d*x+c)^
2*cos(3*arcsin(d*x+c))*b^3+375*arcsin(d*x+c)^2*Ci(5*arcsin(d*x+c)+5*a/b)*sin(5*a/b)*a*b^2+243*arcsin(d*x+c)*Si
(3*arcsin(d*x+c)+3*a/b)*cos(3*a/b)*a^2*b-5*sin(5*arcsin(d*x+c))*a*b^2+243*arcsin(d*x+c)^2*Si(3*arcsin(d*x+c)+3
*a/b)*cos(3*a/b)*a*b^2+4*(1-(d*x+c)^2)^(1/2)*b^3-6*cos(3*arcsin(d*x+c))*b^3+2*cos(5*arcsin(d*x+c))*b^3-243*arc
sin(d*x+c)*Ci(3*arcsin(d*x+c)+3*a/b)*sin(3*a/b)*a^2*b+375*arcsin(d*x+c)*Ci(5*arcsin(d*x+c)+5*a/b)*sin(5*a/b)*a
^2*b-6*arcsin(d*x+c)^2*Si(arcsin(d*x+c)+a/b)*cos(a/b)*a*b^2+6*arcsin(d*x+c)^2*Ci(arcsin(d*x+c)+a/b)*sin(a/b)*a
*b^2-375*arcsin(d*x+c)^2*Si(5*arcsin(d*x+c)+5*a/b)*cos(5*a/b)*a*b^2)/(a+b*arcsin(d*x+c))^3/b^4

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4/(a+b*arcsin(d*x+c))^4,x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{d^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}}{b^{4} \arcsin \left (d x + c\right )^{4} + 4 \, a b^{3} \arcsin \left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \arcsin \left (d x + c\right )^{2} + 4 \, a^{3} b \arcsin \left (d x + c\right ) + a^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4/(a+b*arcsin(d*x+c))^4,x, algorithm="fricas")

[Out]

integral((d^4*e^4*x^4 + 4*c*d^3*e^4*x^3 + 6*c^2*d^2*e^4*x^2 + 4*c^3*d*e^4*x + c^4*e^4)/(b^4*arcsin(d*x + c)^4
+ 4*a*b^3*arcsin(d*x + c)^3 + 6*a^2*b^2*arcsin(d*x + c)^2 + 4*a^3*b*arcsin(d*x + c) + a^4), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**4/(a+b*asin(d*x+c))**4,x)

[Out]

Timed out

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Giac [B]  time = 2.73301, size = 7835, normalized size = 18.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4/(a+b*arcsin(d*x+c))^4,x, algorithm="giac")

[Out]

-125/6*b^3*arcsin(d*x + c)^3*cos(a/b)^4*cos_integral(5*a/b + 5*arcsin(d*x + c))*e^4*sin(a/b)/(b^7*d*arcsin(d*x
+ c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 125/6*b^3*arcsin(d*x + c)^3
*cos(a/b)^5*e^4*sin_integral(5*a/b + 5*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2
+ 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 125/2*a*b^2*arcsin(d*x + c)^2*cos(a/b)^4*cos_integral(5*a/b + 5*
arcsin(d*x + c))*e^4*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x
+ c) + a^3*b^4*d) + 125/2*a*b^2*arcsin(d*x + c)^2*cos(a/b)^5*e^4*sin_integral(5*a/b + 5*arcsin(d*x + c))/(b^7*
d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 125/8*b^3*arcsi
n(d*x + c)^3*cos(a/b)^2*cos_integral(5*a/b + 5*arcsin(d*x + c))*e^4*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^
6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 125/2*a^2*b*arcsin(d*x + c)*cos(a/b)^4*cos_
integral(5*a/b + 5*arcsin(d*x + c))*e^4*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^
2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 27/8*b^3*arcsin(d*x + c)^3*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(d*x
+ c))*e^4*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3
*b^4*d) - 625/24*b^3*arcsin(d*x + c)^3*cos(a/b)^3*e^4*sin_integral(5*a/b + 5*arcsin(d*x + c))/(b^7*d*arcsin(d*
x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 125/2*a^2*b*arcsin(d*x + c
)*cos(a/b)^5*e^4*sin_integral(5*a/b + 5*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^
2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 27/8*b^3*arcsin(d*x + c)^3*cos(a/b)^3*e^4*sin_integral(3*a/b +
3*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*
b^4*d) + 375/8*a*b^2*arcsin(d*x + c)^2*cos(a/b)^2*cos_integral(5*a/b + 5*arcsin(d*x + c))*e^4*sin(a/b)/(b^7*d*
arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 125/6*a^3*cos(a/b
)^4*cos_integral(5*a/b + 5*arcsin(d*x + c))*e^4*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^
2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 81/8*a*b^2*arcsin(d*x + c)^2*cos(a/b)^2*cos_integral(3*a/b + 3*
arcsin(d*x + c))*e^4*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x
+ c) + a^3*b^4*d) - 625/8*a*b^2*arcsin(d*x + c)^2*cos(a/b)^3*e^4*sin_integral(5*a/b + 5*arcsin(d*x + c))/(b^7*
d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 125/6*a^3*cos(a
/b)^5*e^4*sin_integral(5*a/b + 5*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a
^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 81/8*a*b^2*arcsin(d*x + c)^2*cos(a/b)^3*e^4*sin_integral(3*a/b + 3*arc
sin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d
) - 125/96*b^3*arcsin(d*x + c)^3*cos_integral(5*a/b + 5*arcsin(d*x + c))*e^4*sin(a/b)/(b^7*d*arcsin(d*x + c)^3
+ 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 375/8*a^2*b*arcsin(d*x + c)*cos(a/
b)^2*cos_integral(5*a/b + 5*arcsin(d*x + c))*e^4*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)
^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 27/32*b^3*arcsin(d*x + c)^3*cos_integral(3*a/b + 3*arcsin(d*x
+ c))*e^4*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*
b^4*d) + 81/8*a^2*b*arcsin(d*x + c)*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(d*x + c))*e^4*sin(a/b)/(b^7*d*arc
sin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 1/48*b^3*arcsin(d*x
+ c)^3*cos_integral(a/b + arcsin(d*x + c))*e^4*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2
+ 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 625/96*b^3*arcsin(d*x + c)^3*cos(a/b)*e^4*sin_integral(5*a/b + 5
*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b
^4*d) - 625/8*a^2*b*arcsin(d*x + c)*cos(a/b)^3*e^4*sin_integral(5*a/b + 5*arcsin(d*x + c))/(b^7*d*arcsin(d*x +
c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 81/32*b^3*arcsin(d*x + c)^3*c
os(a/b)*e^4*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3
*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 81/8*a^2*b*arcsin(d*x + c)*cos(a/b)^3*e^4*sin_integral(3*a/b + 3*arc
sin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d
) + 1/48*b^3*arcsin(d*x + c)^3*cos(a/b)*e^4*sin_integral(a/b + arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a
*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 25/6*((d*x + c)^2 - 1)^2*sqrt(-(d*x + c)
^2 + 1)*b^3*arcsin(d*x + c)^2*e^4/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(
d*x + c) + a^3*b^4*d) + 5/6*((d*x + c)^2 - 1)^2*(d*x + c)*b^3*arcsin(d*x + c)*e^4/(b^7*d*arcsin(d*x + c)^3 + 3
*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 125/32*a*b^2*arcsin(d*x + c)^2*cos_int
egral(5*a/b + 5*arcsin(d*x + c))*e^4*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b
^5*d*arcsin(d*x + c) + a^3*b^4*d) + 125/8*a^3*cos(a/b)^2*cos_integral(5*a/b + 5*arcsin(d*x + c))*e^4*sin(a/b)/
(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 81/32*a*b^
2*arcsin(d*x + c)^2*cos_integral(3*a/b + 3*arcsin(d*x + c))*e^4*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*
arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 27/8*a^3*cos(a/b)^2*cos_integral(3*a/b + 3*arcs
in(d*x + c))*e^4*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c)
+ a^3*b^4*d) - 1/16*a*b^2*arcsin(d*x + c)^2*cos_integral(a/b + arcsin(d*x + c))*e^4*sin(a/b)/(b^7*d*arcsin(d*
x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 625/32*a*b^2*arcsin(d*x +
c)^2*cos(a/b)*e^4*sin_integral(5*a/b + 5*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)
^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 625/24*a^3*cos(a/b)^3*e^4*sin_integral(5*a/b + 5*arcsin(d*x +
c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 243/32
*a*b^2*arcsin(d*x + c)^2*cos(a/b)*e^4*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b
^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 27/8*a^3*cos(a/b)^3*e^4*sin_integral(3*a/b
+ 3*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a
^3*b^4*d) + 1/16*a*b^2*arcsin(d*x + c)^2*cos(a/b)*e^4*sin_integral(a/b + arcsin(d*x + c))/(b^7*d*arcsin(d*x +
c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 25/3*((d*x + c)^2 - 1)^2*sqrt(
-(d*x + c)^2 + 1)*a*b^2*arcsin(d*x + c)*e^4/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5
*d*arcsin(d*x + c) + a^3*b^4*d) - 19/3*(-(d*x + c)^2 + 1)^(3/2)*b^3*arcsin(d*x + c)^2*e^4/(b^7*d*arcsin(d*x +
c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 5/6*((d*x + c)^2 - 1)^2*(d*x +
c)*a*b^2*e^4/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d
) + ((d*x + c)^2 - 1)*(d*x + c)*b^3*arcsin(d*x + c)*e^4/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2
+ 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 125/32*a^2*b*arcsin(d*x + c)*cos_integral(5*a/b + 5*arcsin(d*x +
c))*e^4*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b
^4*d) - 81/32*a^2*b*arcsin(d*x + c)*cos_integral(3*a/b + 3*arcsin(d*x + c))*e^4*sin(a/b)/(b^7*d*arcsin(d*x + c
)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 1/16*a^2*b*arcsin(d*x + c)*cos_
integral(a/b + arcsin(d*x + c))*e^4*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^
5*d*arcsin(d*x + c) + a^3*b^4*d) + 625/32*a^2*b*arcsin(d*x + c)*cos(a/b)*e^4*sin_integral(5*a/b + 5*arcsin(d*x
+ c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 243
/32*a^2*b*arcsin(d*x + c)*cos(a/b)*e^4*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*
b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 1/16*a^2*b*arcsin(d*x + c)*cos(a/b)*e^4*s
in_integral(a/b + arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin
(d*x + c) + a^3*b^4*d) + 25/6*((d*x + c)^2 - 1)^2*sqrt(-(d*x + c)^2 + 1)*a^2*b*e^4/(b^7*d*arcsin(d*x + c)^3 +
3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 1/3*((d*x + c)^2 - 1)^2*sqrt(-(d*x +
c)^2 + 1)*b^3*e^4/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b
^4*d) - 38/3*(-(d*x + c)^2 + 1)^(3/2)*a*b^2*arcsin(d*x + c)*e^4/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*
x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 13/6*sqrt(-(d*x + c)^2 + 1)*b^3*arcsin(d*x + c)^2*e^4/(b
^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + ((d*x + c)^2
- 1)*(d*x + c)*a*b^2*e^4/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c)
+ a^3*b^4*d) + 1/6*(d*x + c)*b^3*arcsin(d*x + c)*e^4/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 +
3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 125/96*a^3*cos_integral(5*a/b + 5*arcsin(d*x + c))*e^4*sin(a/b)/(b
^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 27/32*a^3*co
s_integral(3*a/b + 3*arcsin(d*x + c))*e^4*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*
a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 1/48*a^3*cos_integral(a/b + arcsin(d*x + c))*e^4*sin(a/b)/(b^7*d*arcs
in(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 625/96*a^3*cos(a/b)*e
^4*sin_integral(5*a/b + 5*arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*
d*arcsin(d*x + c) + a^3*b^4*d) + 81/32*a^3*cos(a/b)*e^4*sin_integral(3*a/b + 3*arcsin(d*x + c))/(b^7*d*arcsin(
d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 1/48*a^3*cos(a/b)*e^4*si
n_integral(a/b + arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(
d*x + c) + a^3*b^4*d) - 19/3*(-(d*x + c)^2 + 1)^(3/2)*a^2*b*e^4/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*
x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 2/3*(-(d*x + c)^2 + 1)^(3/2)*b^3*e^4/(b^7*d*arcsin(d*x +
c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 13/3*sqrt(-(d*x + c)^2 + 1)*a
*b^2*arcsin(d*x + c)*e^4/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c)
+ a^3*b^4*d) + 1/6*(d*x + c)*a*b^2*e^4/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*ar
csin(d*x + c) + a^3*b^4*d) + 13/6*sqrt(-(d*x + c)^2 + 1)*a^2*b*e^4/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin
(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 1/3*sqrt(-(d*x + c)^2 + 1)*b^3*e^4/(b^7*d*arcsin(d*x
+ c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d)