Optimal. Leaf size=77 \[ -\frac{b e^{-c} x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},b^2 x^2\right )}{2 \sqrt{\pi }}-\frac{\sqrt{\pi } e^c \text{Erfc}(b x)^2}{8 b}+\frac{\sqrt{\pi } e^{-c} \text{Erfi}(b x)}{4 b} \]
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Rubi [A] time = 0.0714556, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6414, 6374, 30, 6377, 2204, 6376} \[ -\frac{b e^{-c} x^2 \, _2F_2\left (1,1;\frac{3}{2},2;b^2 x^2\right )}{2 \sqrt{\pi }}-\frac{\sqrt{\pi } e^c \text{Erfc}(b x)^2}{8 b}+\frac{\sqrt{\pi } e^{-c} \text{Erfi}(b x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 6414
Rule 6374
Rule 30
Rule 6377
Rule 2204
Rule 6376
Rubi steps
\begin{align*} \int \cosh \left (c-b^2 x^2\right ) \text{erfc}(b x) \, dx &=\frac{1}{2} \int e^{c-b^2 x^2} \text{erfc}(b x) \, dx+\frac{1}{2} \int e^{-c+b^2 x^2} \text{erfc}(b x) \, dx\\ &=\frac{1}{2} \int e^{-c+b^2 x^2} \, dx-\frac{1}{2} \int e^{-c+b^2 x^2} \text{erf}(b x) \, dx-\frac{\left (e^c \sqrt{\pi }\right ) \operatorname{Subst}(\int x \, dx,x,\text{erfc}(b x))}{4 b}\\ &=-\frac{e^c \sqrt{\pi } \text{erfc}(b x)^2}{8 b}+\frac{e^{-c} \sqrt{\pi } \text{erfi}(b x)}{4 b}-\frac{b e^{-c} x^2 \, _2F_2\left (1,1;\frac{3}{2},2;b^2 x^2\right )}{2 \sqrt{\pi }}\\ \end{align*}
Mathematica [A] time = 0.11046, size = 117, normalized size = 1.52 \[ \frac{4 b^2 x^2 \sinh (c) \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},b^2 x^2\right )+4 b^2 x^2 \cosh (c) \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-b^2 x^2\right )-\pi \left (2 \text{Erf}(b x) (\cosh (c) \text{Erfi}(b x)-\sinh (c)-\cosh (c))+\text{Erf}(b x)^2 (\sinh (c)+\cosh (c))+2 \text{Erfi}(b x) (\sinh (c)-\cosh (c))\right )}{8 \sqrt{\pi } b} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.342, size = 0, normalized size = 0. \begin{align*} \int \cosh \left ({b}^{2}{x}^{2}-c \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*} \int \cosh \left (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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\cosh \left (b^{2} x^{2} - c\right ) \operatorname{erf}\left (b x\right ) + \cosh \left (b^{2} x^{2} - 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 \cosh{\left (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 \cosh \left (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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