Optimal. Leaf size=64 \[ -\frac {e^c \text {Erfi}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {\pi } x\right )^2}{8 b}+\frac {1}{4} i b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right ) \]
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Rubi [A]
time = 0.05, 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 = {6571, 6511,
6510, 30} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 6510
Rule 6511
Rule 6571
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 \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 {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 )\\ \end {align*}
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Mathematica [F]
time = 0.02, size = 0, normalized size = 0.00 \begin {gather*} \int e^{c+\frac {1}{2} i b^2 \pi x^2} S(b x) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.15, 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.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{c} \int e^{\frac {i \pi b^{2} x^{2}}{2}} S\left (b x\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\mathrm {e}}^{\frac {1{}\mathrm {i}\,\Pi \,b^2\,x^2}{2}+c}\,\mathrm {FresnelS}\left (b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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