Integrand size = 22, antiderivative size = 64 \[ \int e^{c-\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=-\frac {i e^c \text {erf}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )^2}{8 b}+\frac {1}{4} b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right ) \]
[Out]
Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6572, 6508, 30, 6513} \[ \int e^{c-\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=\frac {1}{4} b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )-\frac {i e^c \text {erf}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\pi } b x\right )^2}{8 b} \]
[In]
[Out]
Rule 30
Rule 6508
Rule 6513
Rule 6572
Rubi steps \begin{align*} \text {integral}& = \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 \\ & = \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 {\left (i e^c\right ) \text {Subst}\left (\int x \, dx,x,\text {erf}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )\right )}{4 b} \\ & = -\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 ) \\ \end{align*}
\[ \int e^{c-\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=\int e^{c-\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx \]
[In]
[Out]
\[\int {\mathrm e}^{c -\frac {i b^{2} \pi \,x^{2}}{2}} \operatorname {FresnelC}\left (b x \right )d x\]
[In]
[Out]
\[ \int e^{c-\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=\int { e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} + c\right )} \operatorname {C}\left (b x\right ) \,d x } \]
[In]
[Out]
\[ \int e^{c-\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=e^{c} \int e^{- \frac {i \pi b^{2} x^{2}}{2}} C\left (b x\right )\, dx \]
[In]
[Out]
\[ \int e^{c-\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=\int { e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} + c\right )} \operatorname {C}\left (b x\right ) \,d x } \]
[In]
[Out]
\[ \int e^{c-\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=\int { e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} + c\right )} \operatorname {C}\left (b x\right ) \,d x } \]
[In]
[Out]
Timed out. \[ \int e^{c-\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=\int {\mathrm {e}}^{c-\frac {\Pi \,b^2\,x^2\,1{}\mathrm {i}}{2}}\,\mathrm {FresnelC}\left (b\,x\right ) \,d x \]
[In]
[Out]