Optimal. Leaf size=56 \[ \frac{\sqrt{\pi } e^c \text{Erf}(b x)^2}{8 b}-\frac{b e^{-c} x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},b^2 x^2\right )}{2 \sqrt{\pi }} \]
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Rubi [A] time = 0.0561107, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6410, 6373, 30, 6376} \[ \frac{\sqrt{\pi } e^c \text{Erf}(b x)^2}{8 b}-\frac{b e^{-c} x^2 \, _2F_2\left (1,1;\frac{3}{2},2;b^2 x^2\right )}{2 \sqrt{\pi }} \]
Antiderivative was successfully verified.
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Rule 6410
Rule 6373
Rule 30
Rule 6376
Rubi steps
\begin{align*} \int \text{erf}(b x) \sinh \left (c-b^2 x^2\right ) \, dx &=\frac{1}{2} \int e^{c-b^2 x^2} \text{erf}(b x) \, dx-\frac{1}{2} \int e^{-c+b^2 x^2} \text{erf}(b x) \, dx\\ &=-\frac{b e^{-c} x^2 \, _2F_2\left (1,1;\frac{3}{2},2;b^2 x^2\right )}{2 \sqrt{\pi }}+\frac{\left (e^c \sqrt{\pi }\right ) \operatorname{Subst}(\int x \, dx,x,\text{erf}(b x))}{4 b}\\ &=\frac{e^c \sqrt{\pi } \text{erf}(b x)^2}{8 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 = 0.0394838, size = 61, normalized size = 1.09 \[ \frac{(\cosh (c)-\sinh (c)) \left (\pi \text{Erf}(b x)^2 (\sinh (2 c)+\cosh (2 c))-4 b^2 x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},b^2 x^2\right )\right )}{8 \sqrt{\pi } b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.065, size = 0, normalized size = 0. \begin{align*} \int -{\it Erf} \left ( bx \right ) \sinh \left ({b}^{2}{x}^{2}-c \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 } \operatorname{erf}\left (b x\right )^{2} e^{c}}{8 \, b} - \frac{1}{2} \, \int \operatorname{erf}\left (b x\right ) e^{\left (b^{2} x^{2} - c\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 (-\operatorname{erf}\left (b x\right ) \sinh \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 \sinh{\left (b^{2} x^{2} - c \right )} \operatorname{erf}{\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{erf}\left (b x\right ) \sinh \left (b^{2} x^{2} - c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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