3.247 \(\int (c e+d e x) (a+b \sin ^{-1}(c+d x))^{3/2} \, dx\)

Optimal. Leaf size=199 \[ \frac{3 \sqrt{\pi } b^{3/2} e \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{32 d}-\frac{3 \sqrt{\pi } b^{3/2} e \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{32 d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac{3 b e \sqrt{1-(c+d x)^2} (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{8 d}-\frac{e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d} \]

[Out]

(3*b*e*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(8*d) - (e*(a + b*ArcSin[c + d*x])^(3/2))/
(4*d) + (e*(c + d*x)^2*(a + b*ArcSin[c + d*x])^(3/2))/(2*d) - (3*b^(3/2)*e*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*S
qrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(32*d) + (3*b^(3/2)*e*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[
c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(32*d)

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Rubi [A]  time = 0.49308, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {4805, 12, 4629, 4707, 4641, 4635, 4406, 3306, 3305, 3351, 3304, 3352} \[ \frac{3 \sqrt{\pi } b^{3/2} e \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{32 d}-\frac{3 \sqrt{\pi } b^{3/2} e \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{32 d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac{3 b e \sqrt{1-(c+d x)^2} (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{8 d}-\frac{e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)*(a + b*ArcSin[c + d*x])^(3/2),x]

[Out]

(3*b*e*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(8*d) - (e*(a + b*ArcSin[c + d*x])^(3/2))/
(4*d) + (e*(c + d*x)^2*(a + b*ArcSin[c + d*x])^(3/2))/(2*d) - (3*b^(3/2)*e*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*S
qrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(32*d) + (3*b^(3/2)*e*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[
c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(32*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 4629

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*(a + b*ArcSin[c*x])^n)/(m
+ 1), x] - Dist[(b*c*n)/(m + 1), Int[(x^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; Fre
eQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 4707

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

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]

Rule 4635

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

Rule 4406

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 3306

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]

Rule 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(f*x^2)/d], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(f*x^2)/d],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rubi steps

\begin{align*} \int (c e+d e x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int e x \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e \operatorname{Subst}\left (\int x \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{(3 b e) \operatorname{Subst}\left (\int \frac{x^2 \sqrt{a+b \sin ^{-1}(x)}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{4 d}\\ &=\frac{3 b e (c+d x) \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{8 d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{(3 b e) \operatorname{Subst}\left (\int \frac{\sqrt{a+b \sin ^{-1}(x)}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{8 d}-\frac{\left (3 b^2 e\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{16 d}\\ &=\frac{3 b e (c+d x) \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{8 d}-\frac{e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{\left (3 b^2 e\right ) \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 d}\\ &=\frac{3 b e (c+d x) \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{8 d}-\frac{e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{\left (3 b^2 e\right ) \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 d}\\ &=\frac{3 b e (c+d x) \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{8 d}-\frac{e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{\left (3 b^2 e\right ) \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{32 d}\\ &=\frac{3 b e (c+d x) \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{8 d}-\frac{e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{\left (3 b^2 e \cos \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{2 a}{b}+2 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{32 d}+\frac{\left (3 b^2 e \sin \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{2 a}{b}+2 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{32 d}\\ &=\frac{3 b e (c+d x) \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{8 d}-\frac{e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{\left (3 b e \cos \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{16 d}+\frac{\left (3 b e \sin \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{16 d}\\ &=\frac{3 b e (c+d x) \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{8 d}-\frac{e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{3 b^{3/2} e \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{32 d}+\frac{3 b^{3/2} e \sqrt{\pi } C\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right ) \sin \left (\frac{2 a}{b}\right )}{32 d}\\ \end{align*}

Mathematica [C]  time = 0.0621734, size = 137, normalized size = 0.69 \[ \frac{b^2 e e^{-\frac{2 i a}{b}} \left (\sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{5}{2},-\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac{4 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{5}{2},\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{16 \sqrt{2} d \sqrt{a+b \sin ^{-1}(c+d x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c*e + d*e*x)*(a + b*ArcSin[c + d*x])^(3/2),x]

[Out]

(b^2*e*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((4*I)*a)/b
)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b]))/(16*Sqrt[2]*d*E^(((2*I)*
a)/b)*Sqrt[a + b*ArcSin[c + d*x]])

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Maple [A]  time = 0.088, size = 294, normalized size = 1.5 \begin{align*} -{\frac{e}{32\,d} \left ( 3\,\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{a+b\arcsin \left ( dx+c \right ) }\cos \left ( 2\,{\frac{a}{b}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{{b}^{-1}}\sqrt{\pi }b}} \right ){b}^{2}-3\,\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{a+b\arcsin \left ( dx+c \right ) }\sin \left ( 2\,{\frac{a}{b}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{{b}^{-1}}\sqrt{\pi }b}} \right ){b}^{2}+8\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2}\cos \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ){b}^{2}+16\,\arcsin \left ( dx+c \right ) \cos \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ) ab-6\,\arcsin \left ( dx+c \right ) \sin \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ){b}^{2}+8\,\cos \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ){a}^{2}-6\,\sin \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ) ab \right ){\frac{1}{\sqrt{a+b\arcsin \left ( dx+c \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))^(3/2),x)

[Out]

-1/32*e/d*(3*(1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(2*a/b)*FresnelS(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*ar
csin(d*x+c))^(1/2)/b)*b^2-3*(1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(2*a/b)*FresnelC(2/Pi^(1/2)/(1/b
)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^2+8*arcsin(d*x+c)^2*cos(2*(a+b*arcsin(d*x+c))/b-2*a/b)*b^2+16*arcsin(d*
x+c)*cos(2*(a+b*arcsin(d*x+c))/b-2*a/b)*a*b-6*arcsin(d*x+c)*sin(2*(a+b*arcsin(d*x+c))/b-2*a/b)*b^2+8*cos(2*(a+
b*arcsin(d*x+c))/b-2*a/b)*a^2-6*sin(2*(a+b*arcsin(d*x+c))/b-2*a/b)*a*b)/(a+b*arcsin(d*x+c))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}\,{d x} \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))^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} e \left (\int a c \sqrt{a + b \operatorname{asin}{\left (c + d x \right )}}\, dx + \int a d x \sqrt{a + b \operatorname{asin}{\left (c + d x \right )}}\, dx + \int b c \sqrt{a + b \operatorname{asin}{\left (c + d x \right )}} \operatorname{asin}{\left (c + d x \right )}\, dx + \int b d x \sqrt{a + b \operatorname{asin}{\left (c + d x \right )}} \operatorname{asin}{\left (c + d x \right )}\, dx\right ) \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))**(3/2),x)

[Out]

e*(Integral(a*c*sqrt(a + b*asin(c + d*x)), x) + Integral(a*d*x*sqrt(a + b*asin(c + d*x)), x) + Integral(b*c*sq
rt(a + b*asin(c + d*x))*asin(c + d*x), x) + Integral(b*d*x*sqrt(a + b*asin(c + d*x))*asin(c + d*x), x))

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Giac [B]  time = 1.76328, size = 894, normalized size = 4.49 \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))^(3/2),x, algorithm="giac")

[Out]

3/64*sqrt(pi)*b^(7/2)*i*erf(-sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b
))*e^(2*a*i/b + 1)/((b^3*i/abs(b) + b^2)*d) + 3/64*sqrt(pi)*b^(7/2)*i*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*
i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-2*a*i/b + 1)/((b^3*i/abs(b) - b^2)*d) + 1/16*sqrt(pi)*a*b^
(5/2)*erf(-sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b + 1)
/((b^3*i/abs(b) + b^2)*d) - 1/16*sqrt(pi)*a*b^(5/2)*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*
arcsin(d*x + c) + a)/sqrt(b))*e^(-2*a*i/b + 1)/((b^3*i/abs(b) - b^2)*d) - 1/16*sqrt(pi)*a*b^(3/2)*erf(-sqrt(b*
arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b + 1)/((b^2*i/abs(b) +
b)*d) + 1/16*sqrt(pi)*a*b^(3/2)*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)
/sqrt(b))*e^(-2*a*i/b + 1)/((b^2*i/abs(b) - b)*d) - 3/32*sqrt(b*arcsin(d*x + c) + a)*b*i*e^(2*i*arcsin(d*x + c
) + 1)/d - 1/8*sqrt(b*arcsin(d*x + c) + a)*b*arcsin(d*x + c)*e^(2*i*arcsin(d*x + c) + 1)/d + 3/32*sqrt(b*arcsi
n(d*x + c) + a)*b*i*e^(-2*i*arcsin(d*x + c) + 1)/d - 1/8*sqrt(b*arcsin(d*x + c) + a)*b*arcsin(d*x + c)*e^(-2*i
*arcsin(d*x + c) + 1)/d - 1/8*sqrt(b*arcsin(d*x + c) + a)*a*e^(2*i*arcsin(d*x + c) + 1)/d - 1/8*sqrt(b*arcsin(
d*x + c) + a)*a*e^(-2*i*arcsin(d*x + c) + 1)/d