Optimal. Leaf size=85 \[ -\frac{b e^{-i c} x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},b^2 x^2\right )}{2 \sqrt{\pi }}-\frac{\sqrt{\pi } e^{i c} \text{Erfc}(b x)^2}{8 b}+\frac{\sqrt{\pi } e^{-i c} \text{Erfi}(b x)}{4 b} \]
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Rubi [A] time = 0.0734826, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6408, 6377, 2204, 6376, 6374, 30} \[ -\frac{b e^{-i c} x^2 \, _2F_2\left (1,1;\frac{3}{2},2;b^2 x^2\right )}{2 \sqrt{\pi }}-\frac{\sqrt{\pi } e^{i c} \text{Erfc}(b x)^2}{8 b}+\frac{\sqrt{\pi } e^{-i c} \text{Erfi}(b x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 6408
Rule 6377
Rule 2204
Rule 6376
Rule 6374
Rule 30
Rubi steps
\begin{align*} \int \cos \left (c+i b^2 x^2\right ) \text{erfc}(b x) \, dx &=\frac{1}{2} \int e^{i c-b^2 x^2} \text{erfc}(b x) \, dx+\frac{1}{2} \int e^{-i c+b^2 x^2} \text{erfc}(b x) \, dx\\ &=\frac{1}{2} \int e^{-i c+b^2 x^2} \, dx-\frac{1}{2} \int e^{-i c+b^2 x^2} \text{erf}(b x) \, dx-\frac{\left (e^{i c} \sqrt{\pi }\right ) \operatorname{Subst}(\int x \, dx,x,\text{erfc}(b x))}{4 b}\\ &=-\frac{e^{i c} \sqrt{\pi } \text{erfc}(b x)^2}{8 b}+\frac{e^{-i c} \sqrt{\pi } \text{erfi}(b x)}{4 b}-\frac{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 = 1.61014, size = 0, normalized size = 0. \[ \int \cos \left (c+i b^2 x^2\right ) \text{Erfc}(b x) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.41, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( c+i{b}^{2}{x}^{2} \right ){\it erfc} \left ( bx \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{\sqrt{\pi } \cos \left (c\right ) \operatorname{erfc}\left (b x\right )^{2}}{8 \, b} - \frac{i \, \sqrt{\pi } \operatorname{erfc}\left (b x\right )^{2} \sin \left (c\right )}{8 \, b} + \frac{1}{2} \, \cos \left (c\right ) \int \operatorname{erfc}\left (b x\right ) e^{\left (b^{2} x^{2}\right )}\,{d x} - \frac{1}{2} i \, \int \operatorname{erfc}\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 ({\left (\operatorname{erf}\left (b x\right ) - 1\right )} e^{\left (-2 \, b^{2} x^{2} + 2 i \, c\right )} + \operatorname{erf}\left (b x\right ) - 1\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 \cos{\left (i b^{2} x^{2} + c \right )} \operatorname{erfc}{\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 \cos \left (i \, b^{2} x^{2} + c\right ) \operatorname{erfc}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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