3.257 \(\int (a+b \sin ^{-1}(c+d x))^{7/2} \, dx\)
Optimal. Leaf size=243 \[ \frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d}+\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d}-\frac{105 b^3 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{8 d}-\frac{35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{7 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d} \]
[Out]
(-105*b^3*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(8*d) - (35*b^2*(c + d*x)*(a + b*ArcSin[c + d*x])
^(3/2))/(4*d) + (7*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(5/2))/(2*d) + ((c + d*x)*(a + b*ArcSin[c +
d*x])^(7/2))/d + (105*b^(7/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])
/(8*d) + (105*b^(7/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(8*d)
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Rubi [A] time = 0.410371, antiderivative size = 243, normalized size of antiderivative = 1.,
number of steps used = 11, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used
= {4803, 4619, 4677, 4623, 3306, 3305, 3351, 3304, 3352} \[ \frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d}+\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d}-\frac{105 b^3 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{8 d}-\frac{35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{7 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*ArcSin[c + d*x])^(7/2),x]
[Out]
(-105*b^3*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(8*d) - (35*b^2*(c + d*x)*(a + b*ArcSin[c + d*x])
^(3/2))/(4*d) + (7*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(5/2))/(2*d) + ((c + d*x)*(a + b*ArcSin[c +
d*x])^(7/2))/d + (105*b^(7/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])
/(8*d) + (105*b^(7/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(8*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 4619
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
(x*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Rule 4677
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcSin[c*x])^n)/(2*e*(p + 1)), x] + Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
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 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 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}-\frac{(7 b) \operatorname{Subst}\left (\int \frac{x \left (a+b \sin ^{-1}(x)\right )^{5/2}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{2 d}\\ &=\frac{7 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}-\frac{\left (35 b^2\right ) \operatorname{Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{4 d}\\ &=-\frac{35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{7 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}+\frac{\left (105 b^3\right ) \operatorname{Subst}\left (\int \frac{x \sqrt{a+b \sin ^{-1}(x)}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{8 d}\\ &=-\frac{105 b^3 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{8 d}-\frac{35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{7 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}+\frac{\left (105 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{16 d}\\ &=-\frac{105 b^3 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{8 d}-\frac{35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{7 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}+\frac{\left (105 b^3\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{16 d}\\ &=-\frac{105 b^3 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{8 d}-\frac{35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{7 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}+\frac{\left (105 b^3 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{16 d}+\frac{\left (105 b^3 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{16 d}\\ &=-\frac{105 b^3 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{8 d}-\frac{35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{7 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}+\frac{\left (105 b^3 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{8 d}+\frac{\left (105 b^3 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{8 d}\\ &=-\frac{105 b^3 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{8 d}-\frac{35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{7 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}+\frac{105 b^{7/2} \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d}+\frac{105 b^{7/2} \sqrt{\frac{\pi }{2}} S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{8 d}\\ \end{align*}
Mathematica [C] time = 2.00505, size = 551, normalized size = 2.27 \[ \frac{e^{-\frac{i a}{b}} \left (\frac{4 \left (4 a^3 \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+4 a^3 e^{\frac{2 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right ) \left (7 \left (4 a^2 \sqrt{-c^2-2 c d x-d^2 x^2+1}-10 a b (c+d x)-15 b^2 \sqrt{-c^2-2 c d x-d^2 x^2+1}\right )+\sin ^{-1}(c+d x) \left (24 a^2 (c+d x)+56 a b \sqrt{-c^2-2 c d x-d^2 x^2+1}-70 b^2 (c+d x)\right )+4 b \sin ^{-1}(c+d x)^2 \left (6 a (c+d x)+7 b \sqrt{-c^2-2 c d x-d^2 x^2+1}\right )+8 b^2 (c+d x) \sin ^{-1}(c+d x)^3\right )\right )}{\sqrt{\frac{1}{b}}}+\sqrt{2 \pi } \left (8 i a^3 \left (-1+e^{\frac{2 i a}{b}}\right )+105 b^3 \left (1+e^{\frac{2 i a}{b}}\right )\right ) \sqrt{a+b \sin ^{-1}(c+d x)} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}\right )-i \sqrt{2 \pi } \left (8 i a^3 \left (1+e^{\frac{2 i a}{b}}\right )+105 b^3 \left (-1+e^{\frac{2 i a}{b}}\right )\right ) \sqrt{a+b \sin ^{-1}(c+d x)} S\left (\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right )\right )}{32 \sqrt{\frac{1}{b}} d \sqrt{a+b \sin ^{-1}(c+d x)}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(a + b*ArcSin[c + d*x])^(7/2),x]
[Out]
(((8*I)*a^3*(-1 + E^(((2*I)*a)/b)) + 105*b^3*(1 + E^(((2*I)*a)/b)))*Sqrt[2*Pi]*Sqrt[a + b*ArcSin[c + d*x]]*Fre
snelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]] - I*(105*b^3*(-1 + E^(((2*I)*a)/b)) + (8*I)*a^3*(1
+ E^(((2*I)*a)/b)))*Sqrt[2*Pi]*Sqrt[a + b*ArcSin[c + d*x]]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[
c + d*x]]] + (4*(E^((I*a)/b)*(a + b*ArcSin[c + d*x])*(7*(-10*a*b*(c + d*x) + 4*a^2*Sqrt[1 - c^2 - 2*c*d*x - d^
2*x^2] - 15*b^2*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]) + (24*a^2*(c + d*x) - 70*b^2*(c + d*x) + 56*a*b*Sqrt[1 - c^
2 - 2*c*d*x - d^2*x^2])*ArcSin[c + d*x] + 4*b*(6*a*(c + d*x) + 7*b*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])*ArcSin[c
+ d*x]^2 + 8*b^2*(c + d*x)*ArcSin[c + d*x]^3) + 4*a^3*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)
*(a + b*ArcSin[c + d*x]))/b] + 4*a^3*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, (I*(a + b*
ArcSin[c + d*x]))/b]))/Sqrt[b^(-1)])/(32*Sqrt[b^(-1)]*d*E^((I*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])
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Maple [B] time = 0., size = 608, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*arcsin(d*x+c))^(7/2),x)
[Out]
1/16/d/(a+b*arcsin(d*x+c))^(1/2)*(105*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(a/b)*FresnelC
(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^4+105*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*
x+c))^(1/2)*sin(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^4+16*arcsin(d*x+c)^4
*sin((a+b*arcsin(d*x+c))/b-a/b)*b^4+64*arcsin(d*x+c)^3*sin((a+b*arcsin(d*x+c))/b-a/b)*a*b^3+56*arcsin(d*x+c)^3
*cos((a+b*arcsin(d*x+c))/b-a/b)*b^4+96*arcsin(d*x+c)^2*sin((a+b*arcsin(d*x+c))/b-a/b)*a^2*b^2-140*arcsin(d*x+c
)^2*sin((a+b*arcsin(d*x+c))/b-a/b)*b^4+168*arcsin(d*x+c)^2*cos((a+b*arcsin(d*x+c))/b-a/b)*a*b^3+64*arcsin(d*x+
c)*sin((a+b*arcsin(d*x+c))/b-a/b)*a^3*b-280*arcsin(d*x+c)*sin((a+b*arcsin(d*x+c))/b-a/b)*a*b^3+168*arcsin(d*x+
c)*cos((a+b*arcsin(d*x+c))/b-a/b)*a^2*b^2-210*arcsin(d*x+c)*cos((a+b*arcsin(d*x+c))/b-a/b)*b^4+16*sin((a+b*arc
sin(d*x+c))/b-a/b)*a^4-140*sin((a+b*arcsin(d*x+c))/b-a/b)*a^2*b^2+56*cos((a+b*arcsin(d*x+c))/b-a/b)*a^3*b-210*
cos((a+b*arcsin(d*x+c))/b-a/b)*a*b^3)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arcsin(d*x+c))^(7/2),x, algorithm="maxima")
[Out]
integrate((b*arcsin(d*x + c) + a)^(7/2), 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((a+b*arcsin(d*x+c))^(7/2),x, algorithm="fricas")
[Out]
Exception raised: UnboundLocalError
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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((a+b*asin(d*x+c))**(7/2),x)
[Out]
Timed out
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Giac [B] time = 3.28124, size = 2175, normalized size = 8.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arcsin(d*x+c))^(7/2),x, algorithm="giac")
[Out]
-sqrt(2)*sqrt(pi)*a^3*b^3*i*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*a
rcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^3*i/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d) - sqrt(2)*sqrt(pi)*a
^3*b^3*i*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*
sqrt(abs(b))/b)*e^(-a*i/b)/((b^3*i/sqrt(abs(b)) - b^2*sqrt(abs(b)))*d) - 1/2*sqrt(b*arcsin(d*x + c) + a)*b^3*i
*arcsin(d*x + c)^3*e^(i*arcsin(d*x + c))/d + 1/2*sqrt(b*arcsin(d*x + c) + a)*b^3*i*arcsin(d*x + c)^3*e^(-i*arc
sin(d*x + c))/d + 9/8*sqrt(2)*sqrt(pi)*a^2*b^4*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1
/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^3*i/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d) +
sqrt(2)*sqrt(pi)*a^3*b^2*i*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*a
rcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^2*i/sqrt(abs(b)) + b*sqrt(abs(b)))*d) - 9/8*sqrt(2)*sqrt(pi)
*a^2*b^4*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*
sqrt(abs(b))/b)*e^(-a*i/b)/((b^3*i/sqrt(abs(b)) - b^2*sqrt(abs(b)))*d) + sqrt(2)*sqrt(pi)*a^3*b^2*i*erf(1/2*sq
rt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(
-a*i/b)/((b^2*i/sqrt(abs(b)) - b*sqrt(abs(b)))*d) - 3/2*sqrt(b*arcsin(d*x + c) + a)*a*b^2*i*arcsin(d*x + c)^2*
e^(i*arcsin(d*x + c))/d + 3/2*sqrt(b*arcsin(d*x + c) + a)*a*b^2*i*arcsin(d*x + c)^2*e^(-i*arcsin(d*x + c))/d -
9/8*sqrt(2)*sqrt(pi)*a^2*b^3*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b
*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^2*i/sqrt(abs(b)) + b*sqrt(abs(b)))*d) - 105/32*sqrt(2)*sqr
t(pi)*b^5*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a
)*sqrt(abs(b))/b)*e^(a*i/b)/((b^2*i/sqrt(abs(b)) + b*sqrt(abs(b)))*d) + 9/8*sqrt(2)*sqrt(pi)*a^2*b^3*erf(1/2*s
qrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^
(-a*i/b)/((b^2*i/sqrt(abs(b)) - b*sqrt(abs(b)))*d) + 105/32*sqrt(2)*sqrt(pi)*b^5*erf(1/2*sqrt(2)*sqrt(b*arcsin
(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-a*i/b)/((b^2*i/sqr
t(abs(b)) - b*sqrt(abs(b)))*d) - 3/2*sqrt(b*arcsin(d*x + c) + a)*a^2*b*i*arcsin(d*x + c)*e^(i*arcsin(d*x + c))
/d + 35/8*sqrt(b*arcsin(d*x + c) + a)*b^3*i*arcsin(d*x + c)*e^(i*arcsin(d*x + c))/d + 7/4*sqrt(b*arcsin(d*x +
c) + a)*b^3*arcsin(d*x + c)^2*e^(i*arcsin(d*x + c))/d + 3/2*sqrt(b*arcsin(d*x + c) + a)*a^2*b*i*arcsin(d*x + c
)*e^(-i*arcsin(d*x + c))/d - 35/8*sqrt(b*arcsin(d*x + c) + a)*b^3*i*arcsin(d*x + c)*e^(-i*arcsin(d*x + c))/d +
7/4*sqrt(b*arcsin(d*x + c) + a)*b^3*arcsin(d*x + c)^2*e^(-i*arcsin(d*x + c))/d - 1/2*sqrt(b*arcsin(d*x + c) +
a)*a^3*i*e^(i*arcsin(d*x + c))/d + 35/8*sqrt(b*arcsin(d*x + c) + a)*a*b^2*i*e^(i*arcsin(d*x + c))/d + 7/2*sqr
t(b*arcsin(d*x + c) + a)*a*b^2*arcsin(d*x + c)*e^(i*arcsin(d*x + c))/d + 1/2*sqrt(b*arcsin(d*x + c) + a)*a^3*i
*e^(-i*arcsin(d*x + c))/d - 35/8*sqrt(b*arcsin(d*x + c) + a)*a*b^2*i*e^(-i*arcsin(d*x + c))/d + 7/2*sqrt(b*arc
sin(d*x + c) + a)*a*b^2*arcsin(d*x + c)*e^(-i*arcsin(d*x + c))/d + 7/4*sqrt(b*arcsin(d*x + c) + a)*a^2*b*e^(i*
arcsin(d*x + c))/d - 105/16*sqrt(b*arcsin(d*x + c) + a)*b^3*e^(i*arcsin(d*x + c))/d + 7/4*sqrt(b*arcsin(d*x +
c) + a)*a^2*b*e^(-i*arcsin(d*x + c))/d - 105/16*sqrt(b*arcsin(d*x + c) + a)*b^3*e^(-i*arcsin(d*x + c))/d