\(\int e^{c+\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelS}(b x) \, dx\) [61]

Optimal result

Integrand size = 22, antiderivative size = 64 \[ \int e^{c+\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelS}(b x) \, dx=-\frac {e^c \text {erfi}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )^2}{8 b}+\frac {1}{4} i b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right ) \] Output:

-1/8*exp(c)*erfi((1/2+1/2*I)*b*Pi^(1/2)*x)^2/b+1/4*I*b*exp(c)*x^2*hypergeo 
m([1, 1],[3/2, 2],1/2*I*b^2*Pi*x^2)
 

Mathematica [F]

\[ \int e^{c+\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelS}(b x) \, dx=\int e^{c+\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelS}(b x) \, dx \] Input:

Integrate[E^(c + (I/2)*b^2*Pi*x^2)*FresnelS[b*x],x]
 

Output:

Integrate[E^(c + (I/2)*b^2*Pi*x^2)*FresnelS[b*x], x]
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6990, 26, 6929, 15, 6930}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int e^{c+\frac {1}{2} i \pi b^2 x^2} \operatorname {FresnelS}(b x) \, dx\)

\(\Big \downarrow \) 6990

\(\displaystyle \left (\frac {1}{4}+\frac {i}{4}\right ) \int e^{\frac {1}{2} i b^2 \pi x^2+c} \text {erf}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )dx+\left (\frac {1}{4}-\frac {i}{4}\right ) \int -i e^{\frac {1}{2} i b^2 \pi x^2+c} \text {erfi}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )dx\)

\(\Big \downarrow \) 26

\(\displaystyle \left (\frac {1}{4}+\frac {i}{4}\right ) \int e^{\frac {1}{2} i b^2 \pi x^2+c} \text {erf}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )dx-\left (\frac {1}{4}+\frac {i}{4}\right ) \int e^{\frac {1}{2} i b^2 \pi x^2+c} \text {erfi}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )dx\)

\(\Big \downarrow \) 6929

\(\displaystyle \left (\frac {1}{4}+\frac {i}{4}\right ) \int e^{\frac {1}{2} i b^2 \pi x^2+c} \text {erf}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )dx-\frac {e^c \int \text {erfi}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )d\text {erfi}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )}{4 b}\)

\(\Big \downarrow \) 15

\(\displaystyle \left (\frac {1}{4}+\frac {i}{4}\right ) \int e^{\frac {1}{2} i b^2 \pi x^2+c} \text {erf}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )dx-\frac {e^c \text {erfi}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\pi } b x\right )^2}{8 b}\)

\(\Big \downarrow \) 6930

\(\displaystyle \frac {1}{4} i b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )-\frac {e^c \text {erfi}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\pi } b x\right )^2}{8 b}\)

Input:

Int[E^(c + (I/2)*b^2*Pi*x^2)*FresnelS[b*x],x]
 

Output:

-1/8*(E^c*Erfi[(1/2 + I/2)*b*Sqrt[Pi]*x]^2)/b + (I/4)*b*E^c*x^2*Hypergeome 
tricPFQ[{1, 1}, {3/2, 2}, (I/2)*b^2*Pi*x^2]
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 6929

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[E^c* 
(Sqrt[Pi]/(2*b))   Subst[Int[x^n, x], x, Erfi[b*x]], x] /; FreeQ[{b, c, d, 
n}, x] && EqQ[d, b^2]
 

rule 6930

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)], x_Symbol] :> Simp[b*E^c*(x^2/ 
Sqrt[Pi])*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2], x] /; FreeQ[{b, c, 
d}, x] && EqQ[d, b^2]
 

rule 6990

Int[E^((c_.) + (d_.)*(x_)^2)*FresnelS[(b_.)*(x_)], x_Symbol] :> Simp[(1 + I 
)/4   Int[E^(c + d*x^2)*Erf[(Sqrt[Pi]/2)*(1 + I)*b*x], x], x] + Simp[(1 - I 
)/4   Int[E^(c + d*x^2)*Erf[(Sqrt[Pi]/2)*(1 - I)*b*x], x], x] /; FreeQ[{b, 
c, d}, x] && EqQ[d^2, (-Pi^2/4)*b^4]
 

Maple [F]

\[\int {\mathrm e}^{c +\frac {i b^{2} \pi \,x^{2}}{2}} \operatorname {FresnelS}\left (b x \right )d x\]

Input:

int(exp(c+1/2*I*b^2*Pi*x^2)*FresnelS(b*x),x)
 

Output:

int(exp(c+1/2*I*b^2*Pi*x^2)*FresnelS(b*x),x)
 

Fricas [F]

\[ \int e^{c+\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelS}(b x) \, dx=\int { e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + c\right )} \operatorname {S}\left (b x\right ) \,d x } \] Input:

integrate(exp(c+1/2*I*b^2*pi*x^2)*fresnel_sin(b*x),x, algorithm="fricas")
 

Output:

integral(e^(1/2*I*pi*b^2*x^2 + c)*fresnel_sin(b*x), x)
 

Sympy [F]

\[ \int e^{c+\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelS}(b x) \, dx=e^{c} \int e^{\frac {i \pi b^{2} x^{2}}{2}} S\left (b x\right )\, dx \] Input:

integrate(exp(c+1/2*I*b**2*pi*x**2)*fresnels(b*x),x)
 

Output:

exp(c)*Integral(exp(I*pi*b**2*x**2/2)*fresnels(b*x), x)
 

Maxima [F]

\[ \int e^{c+\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelS}(b x) \, dx=\int { e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + c\right )} \operatorname {S}\left (b x\right ) \,d x } \] Input:

integrate(exp(c+1/2*I*b^2*pi*x^2)*fresnel_sin(b*x),x, algorithm="maxima")
 

Output:

integrate(e^(1/2*I*pi*b^2*x^2 + c)*fresnel_sin(b*x), x)
 

Giac [F]

\[ \int e^{c+\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelS}(b x) \, dx=\int { e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + c\right )} \operatorname {S}\left (b x\right ) \,d x } \] Input:

integrate(exp(c+1/2*I*b^2*pi*x^2)*fresnel_sin(b*x),x, algorithm="giac")
 

Output:

integrate(e^(1/2*I*pi*b^2*x^2 + c)*fresnel_sin(b*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{c+\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelS}(b x) \, dx=\int {\mathrm {e}}^{\frac {1{}\mathrm {i}\,\Pi \,b^2\,x^2}{2}+c}\,\mathrm {FresnelS}\left (b\,x\right ) \,d x \] Input:

int(exp(c + (Pi*b^2*x^2*1i)/2)*FresnelS(b*x),x)
 

Output:

int(exp(c + (Pi*b^2*x^2*1i)/2)*FresnelS(b*x), x)
 

Reduce [F]

\[ \int e^{c+\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelS}(b x) \, dx=\int {\mathrm e}^{c +\frac {i b^{2} \pi \,x^{2}}{2}} \mathrm {FresnelS}\left (b x \right )d x \] Input:

int(exp(c+1/2*I*b^2*Pi*x^2)*FresnelS(b*x),x)
 

Output:

int(exp(c+1/2*I*b^2*Pi*x^2)*FresnelS(b*x),x)