Integrand size = 22, antiderivative size = 64 \[ \int e^{c-\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=-\frac {i e^c \text {erf}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )^2}{8 b}+\frac {1}{4} b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right ) \]
-1/8*I*exp(c)*erf((1/2+1/2*I)*b*x*Pi^(1/2))^2/b+1/4*b*exp(c)*x^2*hypergeom ([1, 1],[3/2, 2],-1/2*I*b^2*Pi*x^2)
\[ \int e^{c-\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=\int e^{c-\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx \]
Time = 0.38 (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 = {6991, 26, 6927, 15, 6932}
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.
\(\displaystyle \int e^{c-\frac {1}{2} i \pi b^2 x^2} \operatorname {FresnelC}(b x) \, dx\) |
\(\Big \downarrow \) 6991 |
\(\displaystyle \left (\frac {1}{4}-\frac {i}{4}\right ) \int e^{c-\frac {1}{2} i b^2 \pi x^2} \text {erf}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )dx+\left (\frac {1}{4}+\frac {i}{4}\right ) \int -i e^{c-\frac {1}{2} i b^2 \pi x^2} \text {erfi}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \left (\frac {1}{4}-\frac {i}{4}\right ) \int e^{c-\frac {1}{2} i b^2 \pi x^2} \text {erf}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )dx+\left (\frac {1}{4}-\frac {i}{4}\right ) \int e^{c-\frac {1}{2} i b^2 \pi x^2} \text {erfi}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )dx\) |
\(\Big \downarrow \) 6927 |
\(\displaystyle \left (\frac {1}{4}-\frac {i}{4}\right ) \int e^{c-\frac {1}{2} i b^2 \pi x^2} \text {erfi}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )dx-\frac {i e^c \int \text {erf}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )d\text {erf}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )}{4 b}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \left (\frac {1}{4}-\frac {i}{4}\right ) \int e^{c-\frac {1}{2} i b^2 \pi x^2} \text {erfi}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )dx-\frac {i e^c \text {erf}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\pi } b x\right )^2}{8 b}\) |
\(\Big \downarrow \) 6932 |
\(\displaystyle \frac {1}{4} b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )-\frac {i e^c \text {erf}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\pi } b x\right )^2}{8 b}\) |
((-1/8*I)*E^c*Erf[(1/2 + I/2)*b*Sqrt[Pi]*x]^2)/b + (b*E^c*x^2*Hypergeometr icPFQ[{1, 1}, {3/2, 2}, (-1/2*I)*b^2*Pi*x^2])/4
3.2.71.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[E^c*( Sqrt[Pi]/(2*b)) Subst[Int[x^n, x], x, Erf[b*x]], x] /; FreeQ[{b, c, d, n} , x] && EqQ[d, -b^2]
Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(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]
Int[E^((c_.) + (d_.)*(x_)^2)*FresnelC[(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]
\[\int {\mathrm e}^{c -\frac {i b^{2} \pi \,x^{2}}{2}} \operatorname {FresnelC}\left (b x \right )d x\]
\[ \int e^{c-\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=\int { e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} + c\right )} \operatorname {C}\left (b x\right ) \,d x } \]
\[ \int e^{c-\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=e^{c} \int e^{- \frac {i \pi b^{2} x^{2}}{2}} C\left (b x\right )\, dx \]
\[ \int e^{c-\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=\int { e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} + c\right )} \operatorname {C}\left (b x\right ) \,d x } \]
\[ \int e^{c-\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=\int { e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} + c\right )} \operatorname {C}\left (b x\right ) \,d x } \]
Timed out. \[ \int e^{c-\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=\int {\mathrm {e}}^{c-\frac {\Pi \,b^2\,x^2\,1{}\mathrm {i}}{2}}\,\mathrm {FresnelC}\left (b\,x\right ) \,d x \]