Optimal. Leaf size=67 \[ \frac{i b e^{-i c} x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-b^2 x^2\right )}{2 \sqrt{\pi }}-\frac{i \sqrt{\pi } e^{i c} \text{Erfi}(b x)^2}{8 b} \]
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Rubi [A] time = 0.0553932, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6406, 6378, 6375, 30} \[ \frac{i b e^{-i c} x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-b^2 x^2\right )}{2 \sqrt{\pi }}-\frac{i \sqrt{\pi } e^{i c} \text{Erfi}(b x)^2}{8 b} \]
Antiderivative was successfully verified.
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Rule 6406
Rule 6378
Rule 6375
Rule 30
Rubi steps
\begin{align*} \int \text{erfi}(b x) \sin \left (c-i b^2 x^2\right ) \, dx &=\frac{1}{2} i \int e^{-i c-b^2 x^2} \text{erfi}(b x) \, dx-\frac{1}{2} i \int e^{i c+b^2 x^2} \text{erfi}(b x) \, dx\\ &=\frac{i b e^{-i c} x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-b^2 x^2\right )}{2 \sqrt{\pi }}-\frac{\left (i e^{i c} \sqrt{\pi }\right ) \operatorname{Subst}(\int x \, dx,x,\text{erfi}(b x))}{4 b}\\ &=-\frac{i e^{i c} \sqrt{\pi } \text{erfi}(b x)^2}{8 b}+\frac{i b e^{-i c} x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-b^2 x^2\right )}{2 \sqrt{\pi }}\\ \end{align*}
Mathematica [F] time = 0.532506, size = 0, normalized size = 0. \[ \int \text{Erfi}(b x) \sin \left (c-i b^2 x^2\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int -{\it erfi} \left ( bx \right ) \sin \left ( -c+i{b}^{2}{x}^{2} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{i \, \sqrt{\pi } \cos \left (c\right ) \operatorname{erfi}\left (b x\right )^{2}}{8 \, b} + \frac{\sqrt{\pi } \operatorname{erfi}\left (b x\right )^{2} \sin \left (c\right )}{8 \, b} + \frac{1}{2} i \, \cos \left (c\right ) \int \operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} + \frac{1}{2} \, \int \operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \sin \left (c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{2} \,{\left (i \, \operatorname{erfi}\left (b x\right ) e^{\left (-2 \, b^{2} x^{2} - 2 i \, c\right )} - i \, \operatorname{erfi}\left (b x\right )\right )} e^{\left (b^{2} x^{2} + i \, c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \sin{\left (i b^{2} x^{2} - c \right )} \operatorname{erfi}{\left (b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\operatorname{erfi}\left (b x\right ) \sin \left (i \, b^{2} x^{2} - c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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