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

Optimal. Leaf size=208 \[ -\frac{2 e \cos \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}-\frac{2 e \sin \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}+\frac{e (c+d x)^2}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{2 e \sqrt{1-(c+d x)^2} (c+d x)}{3 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{e}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{e \sqrt{1-(c+d x)^2} (c+d x)}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3} \]

[Out]

-(e*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(3*b*d*(a + b*ArcSin[c + d*x])^3) - e/(6*b^2*d*(a + b*ArcSin[c + d*x])^2)
+ (e*(c + d*x)^2)/(3*b^2*d*(a + b*ArcSin[c + d*x])^2) + (2*e*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(3*b^3*d*(a + b
*ArcSin[c + d*x])) - (2*e*Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/(3*b^4*d) - (2*e*Sin[(2*a)/
b]*SinIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/(3*b^4*d)

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Rubi [A]  time = 0.330869, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4805, 12, 4633, 4719, 4631, 3303, 3299, 3302, 4641} \[ -\frac{2 e \cos \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 a}{b}+2 \sin ^{-1}(c+d x)\right )}{3 b^4 d}-\frac{2 e \sin \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sin ^{-1}(c+d x)\right )}{3 b^4 d}+\frac{e (c+d x)^2}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{2 e \sqrt{1-(c+d x)^2} (c+d x)}{3 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{e}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{e \sqrt{1-(c+d x)^2} (c+d x)}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(3*b*d*(a + b*ArcSin[c + d*x])^3) - e/(6*b^2*d*(a + b*ArcSin[c + d*x])^2)
+ (e*(c + d*x)^2)/(3*b^2*d*(a + b*ArcSin[c + d*x])^2) + (2*e*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(3*b^3*d*(a + b
*ArcSin[c + d*x])) - (2*e*Cos[(2*a)/b]*CosIntegral[(2*a)/b + 2*ArcSin[c + d*x]])/(3*b^4*d) - (2*e*Sin[(2*a)/b]
*SinIntegral[(2*a)/b + 2*ArcSin[c + d*x]])/(3*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]

Rule 4641

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSin[c*x])^
(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n,
-1]

Rubi steps

\begin{align*} \int \frac{c e+d e x}{\left (a+b \sin ^{-1}(c+d x)\right )^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e x}{\left (a+b \sin ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=\frac{e \operatorname{Subst}\left (\int \frac{x}{\left (a+b \sin ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=-\frac{e (c+d x) \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac{e \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \left (a+b \sin ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}-\frac{(2 e) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x^2} \left (a+b \sin ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}\\ &=-\frac{e (c+d x) \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac{e}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{e (c+d x)^2}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{(2 e) \operatorname{Subst}\left (\int \frac{x}{\left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{3 b^2 d}\\ &=-\frac{e (c+d x) \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac{e}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{e (c+d x)^2}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{2 e (c+d x) \sqrt{1-(c+d x)^2}}{3 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{(2 e) \operatorname{Subst}\left (\int \frac{\cos (2 x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{3 b^3 d}\\ &=-\frac{e (c+d x) \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac{e}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{e (c+d x)^2}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{2 e (c+d x) \sqrt{1-(c+d x)^2}}{3 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{\left (2 e \cos \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{3 b^3 d}-\frac{\left (2 e \sin \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{3 b^3 d}\\ &=-\frac{e (c+d x) \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac{e}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{e (c+d x)^2}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{2 e (c+d x) \sqrt{1-(c+d x)^2}}{3 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{2 e \cos \left (\frac{2 a}{b}\right ) \text{Ci}\left (\frac{2 a}{b}+2 \sin ^{-1}(c+d x)\right )}{3 b^4 d}-\frac{2 e \sin \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sin ^{-1}(c+d x)\right )}{3 b^4 d}\\ \end{align*}

Mathematica [A]  time = 0.728307, size = 186, normalized size = 0.89 \[ \frac{e \left (-\frac{2 b^3 (c+d x) \sqrt{1-(c+d x)^2}}{\left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac{b^2 \left (2 (c+d x)^2-1\right )}{\left (a+b \sin ^{-1}(c+d x)\right )^2}-4 \left (\cos \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )+\sin \left (\frac{2 a}{b}\right ) \text{Si}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )-\log \left (a+b \sin ^{-1}(c+d x)\right )\right )+\frac{4 b (c+d x) \sqrt{1-(c+d x)^2}}{a+b \sin ^{-1}(c+d x)}-4 \log \left (a+b \sin ^{-1}(c+d x)\right )\right )}{6 b^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*((-2*b^3*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^3 + (b^2*(-1 + 2*(c + d*x)^2))/(a + b*Arc
Sin[c + d*x])^2 + (4*b*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x]) - 4*Log[a + b*ArcSin[c + d*x]]
- 4*(Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c + d*x])] - Log[a + b*ArcSin[c + d*x]] + Sin[(2*a)/b]*SinInteg
ral[2*(a/b + ArcSin[c + d*x])])))/(6*b^4*d)

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Maple [B]  time = 0.037, size = 399, normalized size = 1.9 \begin{align*} -{\frac{e}{6\,d \left ( a+b\arcsin \left ( dx+c \right ) \right ) ^{3}{b}^{4}} \left ( 4\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{3}{\it Si} \left ( 2\,\arcsin \left ( dx+c \right ) +2\,{\frac{a}{b}} \right ) \sin \left ( 2\,{\frac{a}{b}} \right ){b}^{3}+4\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{3}{\it Ci} \left ( 2\,\arcsin \left ( dx+c \right ) +2\,{\frac{a}{b}} \right ) \cos \left ( 2\,{\frac{a}{b}} \right ){b}^{3}+12\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2}{\it Si} \left ( 2\,\arcsin \left ( dx+c \right ) +2\,{\frac{a}{b}} \right ) \sin \left ( 2\,{\frac{a}{b}} \right ) a{b}^{2}+12\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2}{\it Ci} \left ( 2\,\arcsin \left ( dx+c \right ) +2\,{\frac{a}{b}} \right ) \cos \left ( 2\,{\frac{a}{b}} \right ) a{b}^{2}-2\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2}\sin \left ( 2\,\arcsin \left ( dx+c \right ) \right ){b}^{3}+12\,\arcsin \left ( dx+c \right ){\it Si} \left ( 2\,\arcsin \left ( dx+c \right ) +2\,{\frac{a}{b}} \right ) \sin \left ( 2\,{\frac{a}{b}} \right ){a}^{2}b+12\,\arcsin \left ( dx+c \right ){\it Ci} \left ( 2\,\arcsin \left ( dx+c \right ) +2\,{\frac{a}{b}} \right ) \cos \left ( 2\,{\frac{a}{b}} \right ){a}^{2}b-4\,\arcsin \left ( dx+c \right ) \sin \left ( 2\,\arcsin \left ( dx+c \right ) \right ) a{b}^{2}+\arcsin \left ( dx+c \right ) \cos \left ( 2\,\arcsin \left ( dx+c \right ) \right ){b}^{3}+4\,{\it Si} \left ( 2\,\arcsin \left ( dx+c \right ) +2\,{\frac{a}{b}} \right ) \sin \left ( 2\,{\frac{a}{b}} \right ){a}^{3}+4\,{\it Ci} \left ( 2\,\arcsin \left ( dx+c \right ) +2\,{\frac{a}{b}} \right ) \cos \left ( 2\,{\frac{a}{b}} \right ){a}^{3}-2\,\sin \left ( 2\,\arcsin \left ( dx+c \right ) \right ){a}^{2}b+\sin \left ( 2\,\arcsin \left ( dx+c \right ) \right ){b}^{3}+\cos \left ( 2\,\arcsin \left ( dx+c \right ) \right ) a{b}^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/d*e*(4*arcsin(d*x+c)^3*Si(2*arcsin(d*x+c)+2*a/b)*sin(2*a/b)*b^3+4*arcsin(d*x+c)^3*Ci(2*arcsin(d*x+c)+2*a/
b)*cos(2*a/b)*b^3+12*arcsin(d*x+c)^2*Si(2*arcsin(d*x+c)+2*a/b)*sin(2*a/b)*a*b^2+12*arcsin(d*x+c)^2*Ci(2*arcsin
(d*x+c)+2*a/b)*cos(2*a/b)*a*b^2-2*arcsin(d*x+c)^2*sin(2*arcsin(d*x+c))*b^3+12*arcsin(d*x+c)*Si(2*arcsin(d*x+c)
+2*a/b)*sin(2*a/b)*a^2*b+12*arcsin(d*x+c)*Ci(2*arcsin(d*x+c)+2*a/b)*cos(2*a/b)*a^2*b-4*arcsin(d*x+c)*sin(2*arc
sin(d*x+c))*a*b^2+arcsin(d*x+c)*cos(2*arcsin(d*x+c))*b^3+4*Si(2*arcsin(d*x+c)+2*a/b)*sin(2*a/b)*a^3+4*Ci(2*arc
sin(d*x+c)+2*a/b)*cos(2*a/b)*a^3-2*sin(2*arcsin(d*x+c))*a^2*b+sin(2*arcsin(d*x+c))*b^3+cos(2*arcsin(d*x+c))*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)/(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 e x + c e}{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)/(a+b*arcsin(d*x+c))^4,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [B]  time = 2.29532, size = 2275, normalized size = 10.94 \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)/(a+b*arcsin(d*x+c))^4,x, algorithm="giac")

[Out]

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