3.309 \(\int \text {erfi}(b x) \sinh (c-b^2 x^2) \, dx\)

Optimal. Leaf size=57 \[ \frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{2 \sqrt {\pi }}-\frac {\sqrt {\pi } e^{-c} \text {erfi}(b x)^2}{8 b} \]

[Out]

1/2*b*exp(c)*x^2*HypergeometricPFQ([1, 1],[3/2, 2],-b^2*x^2)/Pi^(1/2)-1/8*erfi(b*x)^2*Pi^(1/2)/b/exp(c)

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Rubi [A]  time = 0.05, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6412, 6378, 6375, 30} \[ \frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{2 \sqrt {\pi }}-\frac {\sqrt {\pi } e^{-c} \text {Erfi}(b x)^2}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[Erfi[b*x]*Sinh[c - b^2*x^2],x]

[Out]

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

Rule 30

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

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 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]

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]

Rubi steps

\begin {align*} \int \text {erfi}(b x) \sinh \left (c-b^2 x^2\right ) \, dx &=\frac {1}{2} \int e^{c-b^2 x^2} \text {erfi}(b x) \, dx-\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*}

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Mathematica [A]  time = 0.97, size = 72, normalized size = 1.26 \[ \frac {\pi \text {erfi}(b x) (2 \text {erf}(b x) (\sinh (c)+\cosh (c))+\text {erfi}(b x) (\sinh (c)-\cosh (c)))-4 b^2 x^2 (\sinh (c)+\cosh (c)) \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{8 \sqrt {\pi } b} \]

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]*(Erfi[b*x]*(-Cosh[
c] + Sinh[c]) + 2*Erf[b*x]*(Cosh[c] + Sinh[c])))/(8*b*Sqrt[Pi])

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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\operatorname {erfi}\left (b x\right ) \sinh \left (b^{2} x^{2} - c\right ), x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\operatorname {erfi}\left (b x\right ) \sinh \left (b^{2} x^{2} - c\right )\,{d x} \]

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)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[ \int -\erfi \left (b x \right ) \sinh \left (b^{2} x^{2}-c \right )\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ -\int \operatorname {erfi}\left (b x\right ) \sinh \left (b^{2} x^{2} - c\right )\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {sinh}\left (c-b^2\,x^2\right )\,\mathrm {erfi}\left (b\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c - b^2*x^2)*erfi(b*x),x)

[Out]

int(sinh(c - b^2*x^2)*erfi(b*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \sinh {\left (b^{2} x^{2} - c \right )} \operatorname {erfi}{\left (b x \right )}\, dx \]

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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