Optimal. Leaf size=57 \[ \frac{\sqrt{\pi } e^c \text{Erfi}(b x)^2}{8 b}-\frac{b e^{-c} x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-b^2 x^2\right )}{2 \sqrt{\pi }} \]
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Rubi [A] time = 0.0508234, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {6412, 6375, 30, 6378} \[ \frac{\sqrt{\pi } e^c \text{Erfi}(b x)^2}{8 b}-\frac{b e^{-c} x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-b^2 x^2\right )}{2 \sqrt{\pi }} \]
Antiderivative was successfully verified.
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Rule 6412
Rule 6375
Rule 30
Rule 6378
Rubi steps
\begin{align*} \int \text{erfi}(b x) \sinh \left (c+b^2 x^2\right ) \, dx &=-\left (\frac{1}{2} \int e^{-c-b^2 x^2} \text{erfi}(b x) \, dx\right )+\frac{1}{2} \int e^{c+b^2 x^2} \text{erfi}(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{erfi}(b x))}{4 b}\\ &=\frac{e^c \sqrt{\pi } \text{erfi}(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 = 1.22833, size = 74, normalized size = 1.3 \[ \frac{4 b^2 x^2 (\cosh (c)-\sinh (c)) \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},b^2 x^2\right )+\pi \text{Erfi}(b x) (\text{Erfi}(b x) (\sinh (c)+\cosh (c))-2 \text{Erf}(b x) (\cosh (c)-\sinh (c)))}{8 \sqrt{\pi } b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\it erfi} \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*} \int \operatorname{erfi}\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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\operatorname{erfi}\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{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 ) \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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