3.239 \(\int \frac{1}{(a+b \sin ^{-1}(c+d x))^5} \, dx\)

Optimal. Leaf size=191 \[ \frac{\cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c+d x)}{b}\right )}{24 b^5 d}+\frac{\sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c+d x)}{b}\right )}{24 b^5 d}-\frac{c+d x}{24 b^4 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{c+d x}{12 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac{\sqrt{1-(c+d x)^2}}{24 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{\sqrt{1-(c+d x)^2}}{4 b d \left (a+b \sin ^{-1}(c+d x)\right )^4} \]

[Out]

-Sqrt[1 - (c + d*x)^2]/(4*b*d*(a + b*ArcSin[c + d*x])^4) + (c + d*x)/(12*b^2*d*(a + b*ArcSin[c + d*x])^3) + Sq
rt[1 - (c + d*x)^2]/(24*b^3*d*(a + b*ArcSin[c + d*x])^2) - (c + d*x)/(24*b^4*d*(a + b*ArcSin[c + d*x])) + (Cos
[a/b]*CosIntegral[(a + b*ArcSin[c + d*x])/b])/(24*b^5*d) + (Sin[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(
24*b^5*d)

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Rubi [A]  time = 0.281997, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {4803, 4621, 4719, 4623, 3303, 3299, 3302} \[ \frac{\cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c+d x)}{b}\right )}{24 b^5 d}+\frac{\sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c+d x)}{b}\right )}{24 b^5 d}-\frac{c+d x}{24 b^4 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{c+d x}{12 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac{\sqrt{1-(c+d x)^2}}{24 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{\sqrt{1-(c+d x)^2}}{4 b d \left (a+b \sin ^{-1}(c+d x)\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c + d*x])^(-5),x]

[Out]

-Sqrt[1 - (c + d*x)^2]/(4*b*d*(a + b*ArcSin[c + d*x])^4) + (c + d*x)/(12*b^2*d*(a + b*ArcSin[c + d*x])^3) + Sq
rt[1 - (c + d*x)^2]/(24*b^3*d*(a + b*ArcSin[c + d*x])^2) - (c + d*x)/(24*b^4*d*(a + b*ArcSin[c + d*x])) + (Cos
[a/b]*CosIntegral[(a + b*ArcSin[c + d*x])/b])/(24*b^5*d) + (Sin[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(
24*b^5*d)

Rule 4803

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

Rule 4621

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(n + 1))
/(b*c*(n + 1)), x] + Dist[c/(b*(n + 1)), Int[(x*(a + b*ArcSin[c*x])^(n + 1))/Sqrt[1 - c^2*x^2], x], x] /; Free
Q[{a, b, c}, x] && LtQ[n, -1]

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 4623

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

Rule 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{1}{\left (a+b \sin ^{-1}(c+d x)\right )^5} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b \sin ^{-1}(x)\right )^5} \, dx,x,c+d x\right )}{d}\\ &=-\frac{\sqrt{1-(c+d x)^2}}{4 b d \left (a+b \sin ^{-1}(c+d x)\right )^4}-\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^2} \left (a+b \sin ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{4 b d}\\ &=-\frac{\sqrt{1-(c+d x)^2}}{4 b d \left (a+b \sin ^{-1}(c+d x)\right )^4}+\frac{c+d x}{12 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b \sin ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{12 b^2 d}\\ &=-\frac{\sqrt{1-(c+d x)^2}}{4 b d \left (a+b \sin ^{-1}(c+d x)\right )^4}+\frac{c+d x}{12 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac{\sqrt{1-(c+d x)^2}}{24 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^2} \left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{24 b^3 d}\\ &=-\frac{\sqrt{1-(c+d x)^2}}{4 b d \left (a+b \sin ^{-1}(c+d x)\right )^4}+\frac{c+d x}{12 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac{\sqrt{1-(c+d x)^2}}{24 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{c+d x}{24 b^4 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{24 b^4 d}\\ &=-\frac{\sqrt{1-(c+d x)^2}}{4 b d \left (a+b \sin ^{-1}(c+d x)\right )^4}+\frac{c+d x}{12 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac{\sqrt{1-(c+d x)^2}}{24 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{c+d x}{24 b^4 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}-\frac{x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{24 b^5 d}\\ &=-\frac{\sqrt{1-(c+d x)^2}}{4 b d \left (a+b \sin ^{-1}(c+d x)\right )^4}+\frac{c+d x}{12 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac{\sqrt{1-(c+d x)^2}}{24 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{c+d x}{24 b^4 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{\cos \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{24 b^5 d}+\frac{\sin \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{24 b^5 d}\\ &=-\frac{\sqrt{1-(c+d x)^2}}{4 b d \left (a+b \sin ^{-1}(c+d x)\right )^4}+\frac{c+d x}{12 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac{\sqrt{1-(c+d x)^2}}{24 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{c+d x}{24 b^4 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{\cos \left (\frac{a}{b}\right ) \text{Ci}\left (\frac{a+b \sin ^{-1}(c+d x)}{b}\right )}{24 b^5 d}+\frac{\sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c+d x)}{b}\right )}{24 b^5 d}\\ \end{align*}

Mathematica [A]  time = 0.413517, size = 156, normalized size = 0.82 \[ \frac{-\frac{6 b^4 \sqrt{1-(c+d x)^2}}{\left (a+b \sin ^{-1}(c+d x)\right )^4}+\frac{2 b^3 (c+d x)}{\left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac{b^2 \sqrt{1-(c+d x)^2}}{\left (a+b \sin ^{-1}(c+d x)\right )^2}+\cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )+\sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )-\frac{b (c+d x)}{a+b \sin ^{-1}(c+d x)}}{24 b^5 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSin[c + d*x])^(-5),x]

[Out]

((-6*b^4*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^4 + (2*b^3*(c + d*x))/(a + b*ArcSin[c + d*x])^3 + (b^2
*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^2 - (b*(c + d*x))/(a + b*ArcSin[c + d*x]) + Cos[a/b]*CosIntegr
al[a/b + ArcSin[c + d*x]] + Sin[a/b]*SinIntegral[a/b + ArcSin[c + d*x]])/(24*b^5*d)

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Maple [B]  time = 0.075, size = 387, normalized size = 2. \begin{align*}{\frac{1}{d} \left ( -{\frac{1}{4\, \left ( a+b\arcsin \left ( dx+c \right ) \right ) ^{4}b}\sqrt{1- \left ( dx+c \right ) ^{2}}}+{\frac{1}{24\, \left ( a+b\arcsin \left ( dx+c \right ) \right ) ^{3}{b}^{5}} \left ( \left ( \arcsin \left ( dx+c \right ) \right ) ^{3}{\it Si} \left ( \arcsin \left ( dx+c \right ) +{\frac{a}{b}} \right ) \sin \left ({\frac{a}{b}} \right ){b}^{3}+ \left ( \arcsin \left ( dx+c \right ) \right ) ^{3}{\it Ci} \left ( \arcsin \left ( dx+c \right ) +{\frac{a}{b}} \right ) \cos \left ({\frac{a}{b}} \right ){b}^{3}+3\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2}{\it Si} \left ( \arcsin \left ( dx+c \right ) +{\frac{a}{b}} \right ) \sin \left ({\frac{a}{b}} \right ) a{b}^{2}+3\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2}{\it Ci} \left ( \arcsin \left ( dx+c \right ) +{\frac{a}{b}} \right ) \cos \left ({\frac{a}{b}} \right ) a{b}^{2}- \left ( \arcsin \left ( dx+c \right ) \right ) ^{2} \left ( dx+c \right ){b}^{3}+3\,\arcsin \left ( dx+c \right ){\it Si} \left ( \arcsin \left ( dx+c \right ) +{\frac{a}{b}} \right ) \sin \left ({\frac{a}{b}} \right ){a}^{2}b+3\,\arcsin \left ( dx+c \right ){\it Ci} \left ( \arcsin \left ( dx+c \right ) +{\frac{a}{b}} \right ) \cos \left ({\frac{a}{b}} \right ){a}^{2}b+\sqrt{1- \left ( dx+c \right ) ^{2}}\arcsin \left ( dx+c \right ){b}^{3}-2\,\arcsin \left ( dx+c \right ) \left ( dx+c \right ) a{b}^{2}+{\it Si} \left ( \arcsin \left ( dx+c \right ) +{\frac{a}{b}} \right ) \sin \left ({\frac{a}{b}} \right ){a}^{3}+{\it Ci} \left ( \arcsin \left ( dx+c \right ) +{\frac{a}{b}} \right ) \cos \left ({\frac{a}{b}} \right ){a}^{3}+\sqrt{1- \left ( dx+c \right ) ^{2}}a{b}^{2}- \left ( dx+c \right ){a}^{2}b+2\, \left ( dx+c \right ){b}^{3} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*arcsin(d*x+c))^5,x)

[Out]

1/d*(-1/4*(1-(d*x+c)^2)^(1/2)/(a+b*arcsin(d*x+c))^4/b+1/24*(arcsin(d*x+c)^3*Si(arcsin(d*x+c)+a/b)*sin(a/b)*b^3
+arcsin(d*x+c)^3*Ci(arcsin(d*x+c)+a/b)*cos(a/b)*b^3+3*arcsin(d*x+c)^2*Si(arcsin(d*x+c)+a/b)*sin(a/b)*a*b^2+3*a
rcsin(d*x+c)^2*Ci(arcsin(d*x+c)+a/b)*cos(a/b)*a*b^2-arcsin(d*x+c)^2*(d*x+c)*b^3+3*arcsin(d*x+c)*Si(arcsin(d*x+
c)+a/b)*sin(a/b)*a^2*b+3*arcsin(d*x+c)*Ci(arcsin(d*x+c)+a/b)*cos(a/b)*a^2*b+(1-(d*x+c)^2)^(1/2)*arcsin(d*x+c)*
b^3-2*arcsin(d*x+c)*(d*x+c)*a*b^2+Si(arcsin(d*x+c)+a/b)*sin(a/b)*a^3+Ci(arcsin(d*x+c)+a/b)*cos(a/b)*a^3+(1-(d*
x+c)^2)^(1/2)*a*b^2-(d*x+c)*a^2*b+2*(d*x+c)*b^3)/(a+b*arcsin(d*x+c))^3/b^5)

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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(1/(a+b*arcsin(d*x+c))^5,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{1}{b^{5} \arcsin \left (d x + c\right )^{5} + 5 \, a b^{4} \arcsin \left (d x + c\right )^{4} + 10 \, a^{2} b^{3} \arcsin \left (d x + c\right )^{3} + 10 \, a^{3} b^{2} \arcsin \left (d x + c\right )^{2} + 5 \, a^{4} b \arcsin \left (d x + c\right ) + a^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*arcsin(d*x+c))^5,x, algorithm="fricas")

[Out]

integral(1/(b^5*arcsin(d*x + c)^5 + 5*a*b^4*arcsin(d*x + c)^4 + 10*a^2*b^3*arcsin(d*x + c)^3 + 10*a^3*b^2*arcs
in(d*x + c)^2 + 5*a^4*b*arcsin(d*x + c) + a^5), 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(1/(a+b*asin(d*x+c))**5,x)

[Out]

Timed out

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Giac [B]  time = 1.26283, size = 2585, normalized size = 13.53 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*arcsin(d*x+c))^5,x, algorithm="giac")

[Out]

1/24*b^4*arcsin(d*x + c)^4*cos(a/b)*cos_integral(a/b + arcsin(d*x + c))/(b^9*d*arcsin(d*x + c)^4 + 4*a*b^8*d*a
rcsin(d*x + c)^3 + 6*a^2*b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) + a^4*b^5*d) + 1/24*b^4*arcsin(
d*x + c)^4*sin(a/b)*sin_integral(a/b + arcsin(d*x + c))/(b^9*d*arcsin(d*x + c)^4 + 4*a*b^8*d*arcsin(d*x + c)^3
+ 6*a^2*b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) + a^4*b^5*d) + 1/6*a*b^3*arcsin(d*x + c)^3*cos(
a/b)*cos_integral(a/b + arcsin(d*x + c))/(b^9*d*arcsin(d*x + c)^4 + 4*a*b^8*d*arcsin(d*x + c)^3 + 6*a^2*b^7*d*
arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) + a^4*b^5*d) + 1/6*a*b^3*arcsin(d*x + c)^3*sin(a/b)*sin_integr
al(a/b + arcsin(d*x + c))/(b^9*d*arcsin(d*x + c)^4 + 4*a*b^8*d*arcsin(d*x + c)^3 + 6*a^2*b^7*d*arcsin(d*x + c)
^2 + 4*a^3*b^6*d*arcsin(d*x + c) + a^4*b^5*d) - 1/24*(d*x + c)*b^4*arcsin(d*x + c)^3/(b^9*d*arcsin(d*x + c)^4
+ 4*a*b^8*d*arcsin(d*x + c)^3 + 6*a^2*b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) + a^4*b^5*d) + 1/4
*a^2*b^2*arcsin(d*x + c)^2*cos(a/b)*cos_integral(a/b + arcsin(d*x + c))/(b^9*d*arcsin(d*x + c)^4 + 4*a*b^8*d*a
rcsin(d*x + c)^3 + 6*a^2*b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) + a^4*b^5*d) + 1/4*a^2*b^2*arcs
in(d*x + c)^2*sin(a/b)*sin_integral(a/b + arcsin(d*x + c))/(b^9*d*arcsin(d*x + c)^4 + 4*a*b^8*d*arcsin(d*x + c
)^3 + 6*a^2*b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) + a^4*b^5*d) - 1/8*(d*x + c)*a*b^3*arcsin(d*
x + c)^2/(b^9*d*arcsin(d*x + c)^4 + 4*a*b^8*d*arcsin(d*x + c)^3 + 6*a^2*b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*
arcsin(d*x + c) + a^4*b^5*d) + 1/6*a^3*b*arcsin(d*x + c)*cos(a/b)*cos_integral(a/b + arcsin(d*x + c))/(b^9*d*a
rcsin(d*x + c)^4 + 4*a*b^8*d*arcsin(d*x + c)^3 + 6*a^2*b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) +
a^4*b^5*d) + 1/6*a^3*b*arcsin(d*x + c)*sin(a/b)*sin_integral(a/b + arcsin(d*x + c))/(b^9*d*arcsin(d*x + c)^4
+ 4*a*b^8*d*arcsin(d*x + c)^3 + 6*a^2*b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) + a^4*b^5*d) + 1/2
4*sqrt(-(d*x + c)^2 + 1)*b^4*arcsin(d*x + c)^2/(b^9*d*arcsin(d*x + c)^4 + 4*a*b^8*d*arcsin(d*x + c)^3 + 6*a^2*
b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) + a^4*b^5*d) - 1/8*(d*x + c)*a^2*b^2*arcsin(d*x + c)/(b^
9*d*arcsin(d*x + c)^4 + 4*a*b^8*d*arcsin(d*x + c)^3 + 6*a^2*b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x +
c) + a^4*b^5*d) + 1/12*(d*x + c)*b^4*arcsin(d*x + c)/(b^9*d*arcsin(d*x + c)^4 + 4*a*b^8*d*arcsin(d*x + c)^3 +
6*a^2*b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) + a^4*b^5*d) + 1/24*a^4*cos(a/b)*cos_integral(a/b
+ arcsin(d*x + c))/(b^9*d*arcsin(d*x + c)^4 + 4*a*b^8*d*arcsin(d*x + c)^3 + 6*a^2*b^7*d*arcsin(d*x + c)^2 + 4
*a^3*b^6*d*arcsin(d*x + c) + a^4*b^5*d) + 1/24*a^4*sin(a/b)*sin_integral(a/b + arcsin(d*x + c))/(b^9*d*arcsin(
d*x + c)^4 + 4*a*b^8*d*arcsin(d*x + c)^3 + 6*a^2*b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) + a^4*b
^5*d) + 1/12*sqrt(-(d*x + c)^2 + 1)*a*b^3*arcsin(d*x + c)/(b^9*d*arcsin(d*x + c)^4 + 4*a*b^8*d*arcsin(d*x + c)
^3 + 6*a^2*b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) + a^4*b^5*d) - 1/24*(d*x + c)*a^3*b/(b^9*d*ar
csin(d*x + c)^4 + 4*a*b^8*d*arcsin(d*x + c)^3 + 6*a^2*b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) +
a^4*b^5*d) + 1/12*(d*x + c)*a*b^3/(b^9*d*arcsin(d*x + c)^4 + 4*a*b^8*d*arcsin(d*x + c)^3 + 6*a^2*b^7*d*arcsin(
d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) + a^4*b^5*d) + 1/24*sqrt(-(d*x + c)^2 + 1)*a^2*b^2/(b^9*d*arcsin(d*x
+ c)^4 + 4*a*b^8*d*arcsin(d*x + c)^3 + 6*a^2*b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) + a^4*b^5*d
) - 1/4*sqrt(-(d*x + c)^2 + 1)*b^4/(b^9*d*arcsin(d*x + c)^4 + 4*a*b^8*d*arcsin(d*x + c)^3 + 6*a^2*b^7*d*arcsin
(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) + a^4*b^5*d)