Integrand size = 22, antiderivative size = 64 \[ \int e^{c+\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=-\frac {i e^c \text {erfi}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )^2}{8 b}+\frac {1}{4} b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right ) \]
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Time = 0.06 (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, 6511, 6510, 30} \[ \int e^{c+\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=\frac {1}{4} b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )-\frac {i e^c \text {erfi}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\pi } b x\right )^2}{8 b} \]
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Rule 30
Rule 6510
Rule 6511
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 {erfi}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )\right )}{4 b} \\ & = -\frac {i e^c \text {erfi}\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*}
\[ \int e^{c+\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=\int e^{c+\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx \]
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\[\int {\mathrm e}^{c +\frac {i b^{2} \pi \,x^{2}}{2}} \operatorname {FresnelC}\left (b x \right )d x\]
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\[ \int e^{c+\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=\int { e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + c\right )} \operatorname {C}\left (b x\right ) \,d x } \]
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\[ \int e^{c+\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=e^{c} \int e^{\frac {i \pi b^{2} x^{2}}{2}} C\left (b x\right )\, dx \]
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\[ \int e^{c+\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=\int { e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + c\right )} \operatorname {C}\left (b x\right ) \,d x } \]
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\[ \int e^{c+\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=\int { e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + c\right )} \operatorname {C}\left (b x\right ) \,d x } \]
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Timed out. \[ \int e^{c+\frac {1}{2} i b^2 \pi x^2} \operatorname {FresnelC}(b x) \, dx=\int {\mathrm {e}}^{\frac {1{}\mathrm {i}\,\Pi \,b^2\,x^2}{2}+c}\,\mathrm {FresnelC}\left (b\,x\right ) \,d x \]
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