3.165 \(\int (e+f x)^3 \sin (a+b (c+d x)^2) \, dx\)
Optimal. Leaf size=341 \[ \frac{3 \sqrt{\frac{\pi }{2}} f^2 \cos (a) (d e-c f) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} (c+d x)\right )}{2 b^{3/2} d^4}-\frac{3 \sqrt{\frac{\pi }{2}} f^2 \sin (a) (d e-c f) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right )}{2 b^{3/2} d^4}+\frac{f^3 \sin \left (a+b (c+d x)^2\right )}{2 b^2 d^4}-\frac{3 f^2 (c+d x) (d e-c f) \cos \left (a+b (c+d x)^2\right )}{2 b d^4}+\frac{\sqrt{\frac{\pi }{2}} \sin (a) (d e-c f)^3 \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} (c+d x)\right )}{\sqrt{b} d^4}+\frac{\sqrt{\frac{\pi }{2}} \cos (a) (d e-c f)^3 S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right )}{\sqrt{b} d^4}-\frac{3 f (d e-c f)^2 \cos \left (a+b (c+d x)^2\right )}{2 b d^4}-\frac{f^3 (c+d x)^2 \cos \left (a+b (c+d x)^2\right )}{2 b d^4} \]
[Out]
(-3*f*(d*e - c*f)^2*Cos[a + b*(c + d*x)^2])/(2*b*d^4) - (3*f^2*(d*e - c*f)*(c + d*x)*Cos[a + b*(c + d*x)^2])/(
2*b*d^4) - (f^3*(c + d*x)^2*Cos[a + b*(c + d*x)^2])/(2*b*d^4) + (3*f^2*(d*e - c*f)*Sqrt[Pi/2]*Cos[a]*FresnelC[
Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/(2*b^(3/2)*d^4) + ((d*e - c*f)^3*Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*
(c + d*x)])/(Sqrt[b]*d^4) + ((d*e - c*f)^3*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*Sin[a])/(Sqrt[b]*
d^4) - (3*f^2*(d*e - c*f)*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*Sin[a])/(2*b^(3/2)*d^4) + (f^3*Sin
[a + b*(c + d*x)^2])/(2*b^2*d^4)
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Rubi [A] time = 0.57157, antiderivative size = 341, normalized size of antiderivative
= 1., number of steps used = 14, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\)
= 0.5, Rules used = {3433, 3353, 3352, 3351, 3379, 2638, 3385, 3354, 3296, 2637}
\[ \frac{3 \sqrt{\frac{\pi }{2}} f^2 \cos (a) (d e-c f) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} (c+d x)\right )}{2 b^{3/2} d^4}-\frac{3 \sqrt{\frac{\pi }{2}} f^2 \sin (a) (d e-c f) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right )}{2 b^{3/2} d^4}+\frac{f^3 \sin \left (a+b (c+d x)^2\right )}{2 b^2 d^4}-\frac{3 f^2 (c+d x) (d e-c f) \cos \left (a+b (c+d x)^2\right )}{2 b d^4}+\frac{\sqrt{\frac{\pi }{2}} \sin (a) (d e-c f)^3 \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} (c+d x)\right )}{\sqrt{b} d^4}+\frac{\sqrt{\frac{\pi }{2}} \cos (a) (d e-c f)^3 S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right )}{\sqrt{b} d^4}-\frac{3 f (d e-c f)^2 \cos \left (a+b (c+d x)^2\right )}{2 b d^4}-\frac{f^3 (c+d x)^2 \cos \left (a+b (c+d x)^2\right )}{2 b d^4} \]
Antiderivative was successfully verified.
[In]
Int[(e + f*x)^3*Sin[a + b*(c + d*x)^2],x]
[Out]
(-3*f*(d*e - c*f)^2*Cos[a + b*(c + d*x)^2])/(2*b*d^4) - (3*f^2*(d*e - c*f)*(c + d*x)*Cos[a + b*(c + d*x)^2])/(
2*b*d^4) - (f^3*(c + d*x)^2*Cos[a + b*(c + d*x)^2])/(2*b*d^4) + (3*f^2*(d*e - c*f)*Sqrt[Pi/2]*Cos[a]*FresnelC[
Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/(2*b^(3/2)*d^4) + ((d*e - c*f)^3*Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*
(c + d*x)])/(Sqrt[b]*d^4) + ((d*e - c*f)^3*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*Sin[a])/(Sqrt[b]*
d^4) - (3*f^2*(d*e - c*f)*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*Sin[a])/(2*b^(3/2)*d^4) + (f^3*Sin
[a + b*(c + d*x)^2])/(2*b^2*d^4)
Rule 3433
Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :
> Module[{k = If[FractionQ[n], Denominator[n], 1]}, Dist[k/f^(m + 1), Subst[Int[ExpandIntegrand[(a + b*Sin[c +
d*x^(k*n)])^p, x^(k - 1)*(f*g - e*h + h*x^k)^m, x], x], x, (e + f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f,
g, h}, x] && IGtQ[p, 0] && IGtQ[m, 0]
Rule 3353
Int[Sin[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Dist[Sin[c], Int[Cos[d*(e + f*x)^2], x], x] + Dist[
Cos[c], Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]
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]
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 3379
Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))
Rule 2638
Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Rule 3385
Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> -Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Cos[c + d
*x^n])/(d*n), x] + Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e},
x] && IGtQ[n, 0] && LtQ[n, m + 1]
Rule 3354
Int[Cos[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Dist[Cos[c], Int[Cos[d*(e + f*x)^2], x], x] - Dist[
Sin[c], Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]
Rule 3296
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Rule 2637
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Rubi steps
\begin{align*} \int (e+f x)^3 \sin \left (a+b (c+d x)^2\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (d^3 e^3 \left (1-\frac{c f \left (3 d^2 e^2-3 c d e f+c^2 f^2\right )}{d^3 e^3}\right ) \sin \left (a+b x^2\right )+3 d^2 e^2 f \left (1+\frac{c f (-2 d e+c f)}{d^2 e^2}\right ) x \sin \left (a+b x^2\right )+3 d e f^2 \left (1-\frac{c f}{d e}\right ) x^2 \sin \left (a+b x^2\right )+f^3 x^3 \sin \left (a+b x^2\right )\right ) \, dx,x,c+d x\right )}{d^4}\\ &=\frac{f^3 \operatorname{Subst}\left (\int x^3 \sin \left (a+b x^2\right ) \, dx,x,c+d x\right )}{d^4}+\frac{\left (3 f^2 (d e-c f)\right ) \operatorname{Subst}\left (\int x^2 \sin \left (a+b x^2\right ) \, dx,x,c+d x\right )}{d^4}+\frac{\left (3 f (d e-c f)^2\right ) \operatorname{Subst}\left (\int x \sin \left (a+b x^2\right ) \, dx,x,c+d x\right )}{d^4}+\frac{(d e-c f)^3 \operatorname{Subst}\left (\int \sin \left (a+b x^2\right ) \, dx,x,c+d x\right )}{d^4}\\ &=-\frac{3 f^2 (d e-c f) (c+d x) \cos \left (a+b (c+d x)^2\right )}{2 b d^4}+\frac{f^3 \operatorname{Subst}\left (\int x \sin (a+b x) \, dx,x,(c+d x)^2\right )}{2 d^4}+\frac{\left (3 f^2 (d e-c f)\right ) \operatorname{Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,c+d x\right )}{2 b d^4}+\frac{\left (3 f (d e-c f)^2\right ) \operatorname{Subst}\left (\int \sin (a+b x) \, dx,x,(c+d x)^2\right )}{2 d^4}+\frac{\left ((d e-c f)^3 \cos (a)\right ) \operatorname{Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,c+d x\right )}{d^4}+\frac{\left ((d e-c f)^3 \sin (a)\right ) \operatorname{Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,c+d x\right )}{d^4}\\ &=-\frac{3 f (d e-c f)^2 \cos \left (a+b (c+d x)^2\right )}{2 b d^4}-\frac{3 f^2 (d e-c f) (c+d x) \cos \left (a+b (c+d x)^2\right )}{2 b d^4}-\frac{f^3 (c+d x)^2 \cos \left (a+b (c+d x)^2\right )}{2 b d^4}+\frac{(d e-c f)^3 \sqrt{\frac{\pi }{2}} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right )}{\sqrt{b} d^4}+\frac{(d e-c f)^3 \sqrt{\frac{\pi }{2}} C\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right ) \sin (a)}{\sqrt{b} d^4}+\frac{f^3 \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,(c+d x)^2\right )}{2 b d^4}+\frac{\left (3 f^2 (d e-c f) \cos (a)\right ) \operatorname{Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,c+d x\right )}{2 b d^4}-\frac{\left (3 f^2 (d e-c f) \sin (a)\right ) \operatorname{Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,c+d x\right )}{2 b d^4}\\ &=-\frac{3 f (d e-c f)^2 \cos \left (a+b (c+d x)^2\right )}{2 b d^4}-\frac{3 f^2 (d e-c f) (c+d x) \cos \left (a+b (c+d x)^2\right )}{2 b d^4}-\frac{f^3 (c+d x)^2 \cos \left (a+b (c+d x)^2\right )}{2 b d^4}+\frac{3 f^2 (d e-c f) \sqrt{\frac{\pi }{2}} \cos (a) C\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right )}{2 b^{3/2} d^4}+\frac{(d e-c f)^3 \sqrt{\frac{\pi }{2}} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right )}{\sqrt{b} d^4}+\frac{(d e-c f)^3 \sqrt{\frac{\pi }{2}} C\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right ) \sin (a)}{\sqrt{b} d^4}-\frac{3 f^2 (d e-c f) \sqrt{\frac{\pi }{2}} S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right ) \sin (a)}{2 b^{3/2} d^4}+\frac{f^3 \sin \left (a+b (c+d x)^2\right )}{2 b^2 d^4}\\ \end{align*}
Mathematica [A] time = 2.94714, size = 218, normalized size = 0.64 \[ \frac{-4 b f \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right ) \cos \left (a+b (c+d x)^2\right )+2 \sqrt{2 \pi } \sqrt{b} (d e-c f) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} (c+d x)\right ) \left (2 b \sin (a) (d e-c f)^2+3 f^2 \cos (a)\right )+2 \sqrt{2 \pi } \sqrt{b} (d e-c f) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right ) \left (2 b \cos (a) (d e-c f)^2-3 f^2 \sin (a)\right )+4 f^3 \sin \left (a+b (c+d x)^2\right )}{8 b^2 d^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(e + f*x)^3*Sin[a + b*(c + d*x)^2],x]
[Out]
(-4*b*f*(c^2*f^2 - c*d*f*(3*e + f*x) + d^2*(3*e^2 + 3*e*f*x + f^2*x^2))*Cos[a + b*(c + d*x)^2] + 2*Sqrt[b]*(d*
e - c*f)*Sqrt[2*Pi]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*(2*b*(d*e - c*f)^2*Cos[a] - 3*f^2*Sin[a]) + 2*Sqrt[
b]*(d*e - c*f)*Sqrt[2*Pi]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*(3*f^2*Cos[a] + 2*b*(d*e - c*f)^2*Sin[a]) + 4
*f^3*Sin[a + b*(c + d*x)^2])/(8*b^2*d^4)
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Maple [B] time = 0.013, size = 1248, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f*x+e)^3*sin(a+(d*x+c)^2*b),x)
[Out]
-1/2*f^3/d^2/b*x^2*cos(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-f^3*c/d*(-1/2/d^2/b*x*cos(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-c/d
*(-1/2/d^2/b*cos(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-1/2*c/d*2^(1/2)*Pi^(1/2)/(d^2*b)^(1/2)*(cos((b^2*c^2*d^2-d^2*b*(
b*c^2+a))/d^2/b)*FresnelS(2^(1/2)/Pi^(1/2)/(d^2*b)^(1/2)*(b*d^2*x+b*c*d))-sin((b^2*c^2*d^2-d^2*b*(b*c^2+a))/d^
2/b)*FresnelC(2^(1/2)/Pi^(1/2)/(d^2*b)^(1/2)*(b*d^2*x+b*c*d))))+1/4/d^2/b*2^(1/2)*Pi^(1/2)/(d^2*b)^(1/2)*(cos(
(b^2*c^2*d^2-d^2*b*(b*c^2+a))/d^2/b)*FresnelC(2^(1/2)/Pi^(1/2)/(d^2*b)^(1/2)*(b*d^2*x+b*c*d))+sin((b^2*c^2*d^2
-d^2*b*(b*c^2+a))/d^2/b)*FresnelS(2^(1/2)/Pi^(1/2)/(d^2*b)^(1/2)*(b*d^2*x+b*c*d))))+f^3/d^2/b*(1/2/d^2/b*sin(b
*d^2*x^2+2*b*c*d*x+b*c^2+a)-1/2*c/d*2^(1/2)*Pi^(1/2)/(d^2*b)^(1/2)*(cos((b^2*c^2*d^2-d^2*b*(b*c^2+a))/d^2/b)*F
resnelC(2^(1/2)/Pi^(1/2)/(d^2*b)^(1/2)*(b*d^2*x+b*c*d))+sin((b^2*c^2*d^2-d^2*b*(b*c^2+a))/d^2/b)*FresnelS(2^(1
/2)/Pi^(1/2)/(d^2*b)^(1/2)*(b*d^2*x+b*c*d))))-3/2*e*f^2/d^2/b*x*cos(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-3*e*f^2*c/d*(
-1/2/d^2/b*cos(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-1/2*c/d*2^(1/2)*Pi^(1/2)/(d^2*b)^(1/2)*(cos((b^2*c^2*d^2-d^2*b*(b*
c^2+a))/d^2/b)*FresnelS(2^(1/2)/Pi^(1/2)/(d^2*b)^(1/2)*(b*d^2*x+b*c*d))-sin((b^2*c^2*d^2-d^2*b*(b*c^2+a))/d^2/
b)*FresnelC(2^(1/2)/Pi^(1/2)/(d^2*b)^(1/2)*(b*d^2*x+b*c*d))))+3/4*e*f^2/d^2/b*2^(1/2)*Pi^(1/2)/(d^2*b)^(1/2)*(
cos((b^2*c^2*d^2-d^2*b*(b*c^2+a))/d^2/b)*FresnelC(2^(1/2)/Pi^(1/2)/(d^2*b)^(1/2)*(b*d^2*x+b*c*d))+sin((b^2*c^2
*d^2-d^2*b*(b*c^2+a))/d^2/b)*FresnelS(2^(1/2)/Pi^(1/2)/(d^2*b)^(1/2)*(b*d^2*x+b*c*d)))-3/2*e^2*f/d^2/b*cos(b*d
^2*x^2+2*b*c*d*x+b*c^2+a)-3/2*e^2*f*c/d*2^(1/2)*Pi^(1/2)/(d^2*b)^(1/2)*(cos((b^2*c^2*d^2-d^2*b*(b*c^2+a))/d^2/
b)*FresnelS(2^(1/2)/Pi^(1/2)/(d^2*b)^(1/2)*(b*d^2*x+b*c*d))-sin((b^2*c^2*d^2-d^2*b*(b*c^2+a))/d^2/b)*FresnelC(
2^(1/2)/Pi^(1/2)/(d^2*b)^(1/2)*(b*d^2*x+b*c*d)))+1/2*2^(1/2)*Pi^(1/2)/(d^2*b)^(1/2)*e^3*(cos((b^2*c^2*d^2-d^2*
b*(b*c^2+a))/d^2/b)*FresnelS(2^(1/2)/Pi^(1/2)/(d^2*b)^(1/2)*(b*d^2*x+b*c*d))-sin((b^2*c^2*d^2-d^2*b*(b*c^2+a))
/d^2/b)*FresnelC(2^(1/2)/Pi^(1/2)/(d^2*b)^(1/2)*(b*d^2*x+b*c*d)))
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Maxima [C] time = 4.86107, size = 6666, normalized size = 19.55 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^3*sin(a+b*(d*x+c)^2),x, algorithm="maxima")
[Out]
-1/8*sqrt(pi)*(((-I*cos(1/4*pi + 1/2*arctan2(0, b)) - I*cos(-1/4*pi + 1/2*arctan2(0, b)) - sin(1/4*pi + 1/2*ar
ctan2(0, b)) + sin(-1/4*pi + 1/2*arctan2(0, b)))*cos(a) - (cos(1/4*pi + 1/2*arctan2(0, b)) + cos(-1/4*pi + 1/2
*arctan2(0, b)) - I*sin(1/4*pi + 1/2*arctan2(0, b)) + I*sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(a))*erf((I*b*d*x
+ I*b*c)/sqrt(I*b)) + ((-I*cos(1/4*pi + 1/2*arctan2(0, b)) - I*cos(-1/4*pi + 1/2*arctan2(0, b)) + sin(1/4*pi
+ 1/2*arctan2(0, b)) - sin(-1/4*pi + 1/2*arctan2(0, b)))*cos(a) + (cos(1/4*pi + 1/2*arctan2(0, b)) + cos(-1/4*
pi + 1/2*arctan2(0, b)) + I*sin(1/4*pi + 1/2*arctan2(0, b)) - I*sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(a))*erf(
(I*b*d*x + I*b*c)/sqrt(-I*b)))*e^3/(d*sqrt(abs(b))) - 3/4*(((e^(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + e^(-I*b
*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) + (I*e^(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) - I*e^(-I*b*d^2*x^2 - 2
*I*b*c*d*x - I*b*c^2))*sin(a))*abs(4*b*d^2*x^2 + 8*b*c*d*x + 4*b*c^2) + ((((2*I*sqrt(pi)*(erf(sqrt(I*b*d^2*x^2
+ 2*I*b*c*d*x + I*b*c^2)) - 1) - 2*I*sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*cos(a) +
2*(sqrt(pi)*(erf(sqrt(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2)) - 1) + sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*
d*x - I*b*c^2)) - 1))*sin(a))*b*c*d*x + ((2*I*sqrt(pi)*(erf(sqrt(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2)) - 1) -
2*I*sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*cos(a) + 2*(sqrt(pi)*(erf(sqrt(I*b*d^2*x^2
+ 2*I*b*c*d*x + I*b*c^2)) - 1) + sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*sin(a))*b*c^
2)*cos(1/2*arctan2(4*b*d^2*x^2 + 8*b*c*d*x + 4*b*c^2, 0)) + ((2*(sqrt(pi)*(erf(sqrt(I*b*d^2*x^2 + 2*I*b*c*d*x
+ I*b*c^2)) - 1) + sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*cos(a) + (-2*I*sqrt(pi)*(er
f(sqrt(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2)) - 1) + 2*I*sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^
2)) - 1))*sin(a))*b*c*d*x + (2*(sqrt(pi)*(erf(sqrt(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2)) - 1) + sqrt(pi)*(erf(
sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*cos(a) + (-2*I*sqrt(pi)*(erf(sqrt(I*b*d^2*x^2 + 2*I*b*c*d*x
+ I*b*c^2)) - 1) + 2*I*sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*sin(a))*b*c^2)*sin(1/2*
arctan2(4*b*d^2*x^2 + 8*b*c*d*x + 4*b*c^2, 0)))*sqrt(abs(4*b*d^2*x^2 + 8*b*c*d*x + 4*b*c^2)))*e^2*f/(b*d^2*abs
(4*b*d^2*x^2 + 8*b*c*d*x + 4*b*c^2)) + 3/32*(16*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)^2*((e^(I*b*d^2*x^2 + 2*I*b*c*d
*x + I*b*c^2) + e^(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) + (I*e^(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2)
- I*e^(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*sin(a))*c + ((((I*sqrt(pi)*(erf(sqrt(I*b*d^2*x^2 + 2*I*b*c*d*x +
I*b*c^2)) - 1) - I*sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*cos(a) + (sqrt(pi)*(erf(sq
rt(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2)) - 1) + sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1)
)*sin(a))*b*c^2*d*x + ((I*sqrt(pi)*(erf(sqrt(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2)) - 1) - I*sqrt(pi)*(erf(sqrt
(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*cos(a) + (sqrt(pi)*(erf(sqrt(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2
)) - 1) + sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*sin(a))*b*c^3)*abs(4*b*d^2*x^2 + 8*b
*c*d*x + 4*b*c^2)*cos(1/2*arctan2(4*b*d^2*x^2 + 8*b*c*d*x + 4*b*c^2, 0)) + (((sqrt(pi)*(erf(sqrt(I*b*d^2*x^2 +
2*I*b*c*d*x + I*b*c^2)) - 1) + sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*cos(a) + (-I*s
qrt(pi)*(erf(sqrt(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2)) - 1) + I*sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x
- I*b*c^2)) - 1))*sin(a))*b*c^2*d*x + ((sqrt(pi)*(erf(sqrt(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2)) - 1) + sqrt(
pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*cos(a) + (-I*sqrt(pi)*(erf(sqrt(I*b*d^2*x^2 + 2*I*b
*c*d*x + I*b*c^2)) - 1) + I*sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*sin(a))*b*c^3)*abs
(4*b*d^2*x^2 + 8*b*c*d*x + 4*b*c^2)*sin(1/2*arctan2(4*b*d^2*x^2 + 8*b*c*d*x + 4*b*c^2, 0)) + (((-4*I*gamma(3/2
, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + 4*I*gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) - 4*(gam
ma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*sin(a))*b*d^3
*x^3 + ((-12*I*gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + 12*I*gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x -
I*b*c^2))*cos(a) - 12*(gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x
- I*b*c^2))*sin(a))*b*c*d^2*x^2 + ((-12*I*gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + 12*I*gamma(3/2, -
I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) - 12*(gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + gamma(3/2
, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*sin(a))*b*c^2*d*x + ((-4*I*gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b
*c^2) + 4*I*gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) - 4*(gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x
+ I*b*c^2) + gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*sin(a))*b*c^3)*cos(3/2*arctan2(4*b*d^2*x^2 + 8
*b*c*d*x + 4*b*c^2, 0)) - ((4*(gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + gamma(3/2, -I*b*d^2*x^2 - 2*I
*b*c*d*x - I*b*c^2))*cos(a) - (4*I*gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) - 4*I*gamma(3/2, -I*b*d^2*x
^2 - 2*I*b*c*d*x - I*b*c^2))*sin(a))*b*d^3*x^3 + (12*(gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + gamma(
3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) - (12*I*gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) - 1
2*I*gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*sin(a))*b*c*d^2*x^2 + (12*(gamma(3/2, I*b*d^2*x^2 + 2*I*
b*c*d*x + I*b*c^2) + gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) - (12*I*gamma(3/2, I*b*d^2*x^2 +
2*I*b*c*d*x + I*b*c^2) - 12*I*gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*sin(a))*b*c^2*d*x + (4*(gamma
(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) - (4*I*g
amma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) - 4*I*gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*sin(a))
*b*c^3)*sin(3/2*arctan2(4*b*d^2*x^2 + 8*b*c*d*x + 4*b*c^2, 0)))*sqrt(abs(4*b*d^2*x^2 + 8*b*c*d*x + 4*b*c^2)))*
e*f^2/((b*d^2*x^2 + 2*b*c*d*x + b*c^2)^2*b*d^3) - 1/64*(16*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)^2*((3*(e^(I*b*d^2*x
^2 + 2*I*b*c*d*x + I*b*c^2) + e^(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) + (3*I*e^(I*b*d^2*x^2 + 2*I*b*c
*d*x + I*b*c^2) - 3*I*e^(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*sin(a))*b*c^2 + (-I*gamma(2, I*b*d^2*x^2 + 2*I
*b*c*d*x + I*b*c^2) + I*gamma(2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) - (gamma(2, I*b*d^2*x^2 + 2*I*b
*c*d*x + I*b*c^2) + gamma(2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*sin(a)) + ((((2*I*sqrt(pi)*(erf(sqrt(I*b*d
^2*x^2 + 2*I*b*c*d*x + I*b*c^2)) - 1) - 2*I*sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*co
s(a) + 2*(sqrt(pi)*(erf(sqrt(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2)) - 1) + sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*
I*b*c*d*x - I*b*c^2)) - 1))*sin(a))*b^2*c^3*d*x + ((2*I*sqrt(pi)*(erf(sqrt(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2
)) - 1) - 2*I*sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*cos(a) + 2*(sqrt(pi)*(erf(sqrt(I
*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2)) - 1) + sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*si
n(a))*b^2*c^4)*abs(4*b*d^2*x^2 + 8*b*c*d*x + 4*b*c^2)*cos(1/2*arctan2(4*b*d^2*x^2 + 8*b*c*d*x + 4*b*c^2, 0)) +
((2*(sqrt(pi)*(erf(sqrt(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2)) - 1) + sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*
c*d*x - I*b*c^2)) - 1))*cos(a) + (-2*I*sqrt(pi)*(erf(sqrt(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2)) - 1) + 2*I*sqr
t(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*sin(a))*b^2*c^3*d*x + (2*(sqrt(pi)*(erf(sqrt(I*b*
d^2*x^2 + 2*I*b*c*d*x + I*b*c^2)) - 1) + sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*cos(a
) + (-2*I*sqrt(pi)*(erf(sqrt(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2)) - 1) + 2*I*sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2
- 2*I*b*c*d*x - I*b*c^2)) - 1))*sin(a))*b^2*c^4)*abs(4*b*d^2*x^2 + 8*b*c*d*x + 4*b*c^2)*sin(1/2*arctan2(4*b*d^
2*x^2 + 8*b*c*d*x + 4*b*c^2, 0)) + (((-24*I*gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + 24*I*gamma(3/2,
-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) - 24*(gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + gamma(3/
2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*sin(a))*b^2*c*d^3*x^3 + ((-72*I*gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x
+ I*b*c^2) + 72*I*gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) - 72*(gamma(3/2, I*b*d^2*x^2 + 2*I
*b*c*d*x + I*b*c^2) + gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*sin(a))*b^2*c^2*d^2*x^2 + ((-72*I*gamm
a(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + 72*I*gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) -
72*(gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*sin(a)
)*b^2*c^3*d*x + ((-24*I*gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + 24*I*gamma(3/2, -I*b*d^2*x^2 - 2*I*b
*c*d*x - I*b*c^2))*cos(a) - 24*(gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + gamma(3/2, -I*b*d^2*x^2 - 2*
I*b*c*d*x - I*b*c^2))*sin(a))*b^2*c^4)*cos(3/2*arctan2(4*b*d^2*x^2 + 8*b*c*d*x + 4*b*c^2, 0)) - ((24*(gamma(3/
2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) - (24*I*gam
ma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) - 24*I*gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*sin(a))*
b^2*c*d^3*x^3 + (72*(gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x -
I*b*c^2))*cos(a) - (72*I*gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) - 72*I*gamma(3/2, -I*b*d^2*x^2 - 2*I
*b*c*d*x - I*b*c^2))*sin(a))*b^2*c^2*d^2*x^2 + (72*(gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + gamma(3/
2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) - (72*I*gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) - 72*
I*gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*sin(a))*b^2*c^3*d*x + (24*(gamma(3/2, I*b*d^2*x^2 + 2*I*b*
c*d*x + I*b*c^2) + gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) - (24*I*gamma(3/2, I*b*d^2*x^2 + 2
*I*b*c*d*x + I*b*c^2) - 24*I*gamma(3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*sin(a))*b^2*c^4)*sin(3/2*arctan
2(4*b*d^2*x^2 + 8*b*c*d*x + 4*b*c^2, 0)))*sqrt(abs(4*b*d^2*x^2 + 8*b*c*d*x + 4*b*c^2)))*f^3/((b*d^2*x^2 + 2*b*
c*d*x + b*c^2)^2*b^2*d^4)
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Fricas [A] time = 1.95853, size = 760, normalized size = 2.23 \begin{align*} \frac{2 \, d f^{3} \sin \left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right ) + \sqrt{2}{\left (3 \, \pi{\left (d e f^{2} - c f^{3}\right )} \cos \left (a\right ) + 2 \, \pi{\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2} - b c^{3} f^{3}\right )} \sin \left (a\right )\right )} \sqrt{\frac{b d^{2}}{\pi }} \operatorname{C}\left (\frac{\sqrt{2} \sqrt{\frac{b d^{2}}{\pi }}{\left (d x + c\right )}}{d}\right ) + \sqrt{2}{\left (2 \, \pi{\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2} - b c^{3} f^{3}\right )} \cos \left (a\right ) - 3 \, \pi{\left (d e f^{2} - c f^{3}\right )} \sin \left (a\right )\right )} \sqrt{\frac{b d^{2}}{\pi }} \operatorname{S}\left (\frac{\sqrt{2} \sqrt{\frac{b d^{2}}{\pi }}{\left (d x + c\right )}}{d}\right ) - 2 \,{\left (b d^{3} f^{3} x^{2} + 3 \, b d^{3} e^{2} f - 3 \, b c d^{2} e f^{2} + b c^{2} d f^{3} +{\left (3 \, b d^{3} e f^{2} - b c d^{2} f^{3}\right )} x\right )} \cos \left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}{4 \, b^{2} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^3*sin(a+b*(d*x+c)^2),x, algorithm="fricas")
[Out]
1/4*(2*d*f^3*sin(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a) + sqrt(2)*(3*pi*(d*e*f^2 - c*f^3)*cos(a) + 2*pi*(b*d^3*e^3
- 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*sin(a))*sqrt(b*d^2/pi)*fresnel_cos(sqrt(2)*sqrt(b*d^2/pi)*(d
*x + c)/d) + sqrt(2)*(2*pi*(b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*cos(a) - 3*pi*(d*e*f^2
- c*f^3)*sin(a))*sqrt(b*d^2/pi)*fresnel_sin(sqrt(2)*sqrt(b*d^2/pi)*(d*x + c)/d) - 2*(b*d^3*f^3*x^2 + 3*b*d^3*e
^2*f - 3*b*c*d^2*e*f^2 + b*c^2*d*f^3 + (3*b*d^3*e*f^2 - b*c*d^2*f^3)*x)*cos(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a)
)/(b^2*d^5)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e + f x\right )^{3} \sin{\left (a + b c^{2} + 2 b c d x + b d^{2} x^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)**3*sin(a+b*(d*x+c)**2),x)
[Out]
Integral((e + f*x)**3*sin(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2), x)
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Giac [C] time = 1.21802, size = 1449, normalized size = 4.25 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^3*sin(a+b*(d*x+c)^2),x, algorithm="giac")
[Out]
1/4*I*sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*sqrt(b*d^2)*(-I*b*d^2/sqrt(b^2*d^4) + 1)*(x + c/d))*e^(I*a + 3)/(sqrt(
b*d^2)*(-I*b*d^2/sqrt(b^2*d^4) + 1)) - 1/4*I*sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*sqrt(b*d^2)*(I*b*d^2/sqrt(b^2*d
^4) + 1)*(x + c/d))*e^(-I*a + 3)/(sqrt(b*d^2)*(I*b*d^2/sqrt(b^2*d^4) + 1)) - 1/4*(3*I*sqrt(2)*sqrt(pi)*c*f*erf
(-1/2*sqrt(2)*sqrt(b*d^2)*(-I*b*d^2/sqrt(b^2*d^4) + 1)*(x + c/d))*e^(I*a + 2)/(sqrt(b*d^2)*(-I*b*d^2/sqrt(b^2*
d^4) + 1)) + 3*f*e^(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2 + I*a + 2)/(b*d))/d - 1/4*(-3*I*sqrt(2)*sqrt(pi)*c*f*e
rf(-1/2*sqrt(2)*sqrt(b*d^2)*(I*b*d^2/sqrt(b^2*d^4) + 1)*(x + c/d))*e^(-I*a + 2)/(sqrt(b*d^2)*(I*b*d^2/sqrt(b^2
*d^4) + 1)) + 3*f*e^(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2 - I*a + 2)/(b*d))/d - 1/8*(-I*sqrt(2)*sqrt(pi)*(6*b*
c^2*f^2 + 3*I*f^2)*erf(-1/2*sqrt(2)*sqrt(b*d^2)*(-I*b*d^2/sqrt(b^2*d^4) + 1)*(x + c/d))*e^(I*a + 1)/(sqrt(b*d^
2)*(-I*b*d^2/sqrt(b^2*d^4) + 1)*b) + 2*I*(d*f^2*(-3*I*x - 3*I*c/d) + 6*I*c*f^2)*e^(I*b*d^2*x^2 + 2*I*b*c*d*x +
I*b*c^2 + I*a + 1)/(b*d))/d^2 - 1/8*(I*sqrt(2)*sqrt(pi)*(6*b*c^2*f^2 - 3*I*f^2)*erf(-1/2*sqrt(2)*sqrt(b*d^2)*
(I*b*d^2/sqrt(b^2*d^4) + 1)*(x + c/d))*e^(-I*a + 1)/(sqrt(b*d^2)*(I*b*d^2/sqrt(b^2*d^4) + 1)*b) + 2*I*(d*f^2*(
-3*I*x - 3*I*c/d) + 6*I*c*f^2)*e^(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2 - I*a + 1)/(b*d))/d^2 + 1/8*(sqrt(2)*sq
rt(pi)*(-2*I*b*c^3*f^3 + 3*c*f^3)*erf(-1/2*sqrt(2)*sqrt(b*d^2)*(-I*b*d^2/sqrt(b^2*d^4) + 1)*(x + c/d))*e^(I*a)
/(sqrt(b*d^2)*(-I*b*d^2/sqrt(b^2*d^4) + 1)*b) - 2*(b*d^2*f^3*(x + c/d)^2 - 3*b*c*d*f^3*(x + c/d) + 3*b*c^2*f^3
+ I*f^3)*e^(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2 + I*a)/(b^2*d))/d^3 + 1/8*(sqrt(2)*sqrt(pi)*(2*I*b*c^3*f^3 +
3*c*f^3)*erf(-1/2*sqrt(2)*sqrt(b*d^2)*(I*b*d^2/sqrt(b^2*d^4) + 1)*(x + c/d))*e^(-I*a)/(sqrt(b*d^2)*(I*b*d^2/sq
rt(b^2*d^4) + 1)*b) - 2*(b*d^2*f^3*(x + c/d)^2 - 3*b*c*d*f^3*(x + c/d) + 3*b*c^2*f^3 - I*f^3)*e^(-I*b*d^2*x^2
- 2*I*b*c*d*x - I*b*c^2 - I*a)/(b^2*d))/d^3