Optimal. Leaf size=64 \[ \frac {e^c \text {erf}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\pi } b x\right )^2}{8 b}-\frac {1}{4} i b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6436, 6373, 30, 6378} \[ \frac {e^c \text {Erf}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\pi } b x\right )^2}{8 b}-\frac {1}{4} i b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 6373
Rule 6378
Rule 6436
Rubi steps
\begin {align*} \int e^{c-\frac {1}{2} i b^2 \pi x^2} S(b x) \, 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+\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\\ &=-\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 \operatorname {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 {e^c \text {erf}\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 )\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int e^{c-\frac {1}{2} i b^2 \pi x^2} S(b x) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} + c\right )} {\rm fresnels}\left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} + c\right )} {\rm fresnels}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{c -\frac {i b^{2} \pi \,x^{2}}{2}} \mathrm {S}\left (b x \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} + c\right )} {\rm fresnels}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {e}}^{c-\frac {\Pi \,b^2\,x^2\,1{}\mathrm {i}}{2}}\,\mathrm {FresnelS}\left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{c} \int e^{- \frac {i \pi b^{2} x^{2}}{2}} S\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________