3.308 \(\int \text{Erfi}(b x) \sinh (c+b^2 x^2) \, dx\)

Optimal. Leaf size=57 \[ \frac{\sqrt{\pi } e^c \text{Erfi}(b x)^2}{8 b}-\frac{b e^{-c} x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-b^2 x^2\right )}{2 \sqrt{\pi }} \]

[Out]

(E^c*Sqrt[Pi]*Erfi[b*x]^2)/(8*b) - (b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, -(b^2*x^2)])/(2*E^c*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.

[In]

Int[Erfi[b*x]*Sinh[c + b^2*x^2],x]

[Out]

(E^c*Sqrt[Pi]*Erfi[b*x]^2)/(8*b) - (b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, -(b^2*x^2)])/(2*E^c*Sqrt[Pi])

Rule 6412

Int[Erfi[(b_.)*(x_)]*Sinh[(c_.) + (d_.)*(x_)^2], x_Symbol] :> Dist[1/2, Int[E^(c + d*x^2)*Erfi[b*x], x], x] -
Dist[1/2, Int[E^(-c - d*x^2)*Erfi[b*x], x], x] /; FreeQ[{b, c, d}, x] && EqQ[d^2, b^4]

Rule 6375

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[(E^c*Sqrt[Pi])/(2*b), Subst[Int[x^n, x]
, x, Erfi[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, b^2]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6378

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(b_.)*(x_)], x_Symbol] :> Simp[(b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}
, -(b^2*x^2)])/Sqrt[Pi], x] /; FreeQ[{b, c, d}, x] && EqQ[d, -b^2]

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.

[In]

Integrate[Erfi[b*x]*Sinh[c + b^2*x^2],x]

[Out]

(4*b^2*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2]*(Cosh[c] - Sinh[c]) + Pi*Erfi[b*x]*(-2*Erf[b*x]*(Cosh[
c] - Sinh[c]) + Erfi[b*x]*(Cosh[c] + Sinh[c])))/(8*b*Sqrt[Pi])

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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.

[In]

int(erfi(b*x)*sinh(b^2*x^2+c),x)

[Out]

int(erfi(b*x)*sinh(b^2*x^2+c),x)

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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.

[In]

integrate(erfi(b*x)*sinh(b^2*x^2+c),x, algorithm="maxima")

[Out]

integrate(erfi(b*x)*sinh(b^2*x^2 + c), x)

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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.

[In]

integrate(erfi(b*x)*sinh(b^2*x^2+c),x, algorithm="fricas")

[Out]

integral(erfi(b*x)*sinh(b^2*x^2 + c), x)

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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.

[In]

integrate(erfi(b*x)*sinh(b**2*x**2+c),x)

[Out]

Integral(sinh(b**2*x**2 + c)*erfi(b*x), x)

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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.

[In]

integrate(erfi(b*x)*sinh(b^2*x^2+c),x, algorithm="giac")

[Out]

integrate(erfi(b*x)*sinh(b^2*x^2 + c), x)