\(\int \text {erfc}(b x) \sinh (c-b^2 x^2) \, dx\) [204]

Optimal result

Integrand size = 16, antiderivative size = 77 \[ \int \text {erfc}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=-\frac {e^c \sqrt {\pi } \text {erfc}(b x)^2}{8 b}-\frac {e^{-c} \sqrt {\pi } \text {erfi}(b x)}{4 b}+\frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }} \]

[Out]

1/2*b*x^2*hypergeom([1, 1],[3/2, 2],b^2*x^2)/exp(c)/Pi^(1/2)-1/8*exp(c)*erfc(b*x)^2*Pi^(1/2)/b-1/4*erfi(b*x)*P
i^(1/2)/b/exp(c)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6546, 6509, 30, 6512, 2235, 6511} \[ \int \text {erfc}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=\frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }}-\frac {\sqrt {\pi } e^c \text {erfc}(b x)^2}{8 b}-\frac {\sqrt {\pi } e^{-c} \text {erfi}(b x)}{4 b} \]

[In]

Int[Erfc[b*x]*Sinh[c - b^2*x^2],x]

[Out]

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

Rule 30

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

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 6509

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

Rule 6511

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

Rule 6512

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

Rule 6546

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

Rubi steps \begin{align*} \text {integral}& = \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 \\ & = -\left (\frac {1}{2} \int e^{-c+b^2 x^2} \, dx\right )+\frac {1}{2} \int e^{-c+b^2 x^2} \text {erf}(b x) \, dx-\frac {\left (e^c \sqrt {\pi }\right ) \text {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] (verified)

Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.09 \[ \int \text {erfc}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=-\frac {(\cosh (c)-\sinh (c)) \left (-4 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )+\pi \left (2 \text {erfi}(b x)-2 \text {erf}(b x) (\cosh (2 c)+\sinh (2 c))+\text {erf}(b x)^2 (\cosh (2 c)+\sinh (2 c))\right )\right )}{8 b \sqrt {\pi }} \]

[In]

Integrate[Erfc[b*x]*Sinh[c - b^2*x^2],x]

[Out]

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

Maple [F]

\[\int -\operatorname {erfc}\left (b x \right ) \sinh \left (b^{2} x^{2}-c \right )d x\]

[In]

int(-erfc(b*x)*sinh(b^2*x^2-c),x)

[Out]

int(-erfc(b*x)*sinh(b^2*x^2-c),x)

Fricas [F]

\[ \int \text {erfc}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=\int { -\operatorname {erfc}\left (b x\right ) \sinh \left (b^{2} x^{2} - c\right ) \,d x } \]

[In]

integrate(-erfc(b*x)*sinh(b^2*x^2-c),x, algorithm="fricas")

[Out]

integral((erf(b*x) - 1)*sinh(b^2*x^2 - c), x)

Sympy [F]

\[ \int \text {erfc}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=- \int \sinh {\left (b^{2} x^{2} - c \right )} \operatorname {erfc}{\left (b x \right )}\, dx \]

[In]

integrate(-erfc(b*x)*sinh(b**2*x**2-c),x)

[Out]

-Integral(sinh(b**2*x**2 - c)*erfc(b*x), x)

Maxima [F]

\[ \int \text {erfc}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=\int { -\operatorname {erfc}\left (b x\right ) \sinh \left (b^{2} x^{2} - c\right ) \,d x } \]

[In]

integrate(-erfc(b*x)*sinh(b^2*x^2-c),x, algorithm="maxima")

[Out]

-integrate(erfc(b*x)*sinh(b^2*x^2 - c), x)

Giac [F]

\[ \int \text {erfc}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=\int { -\operatorname {erfc}\left (b x\right ) \sinh \left (b^{2} x^{2} - c\right ) \,d x } \]

[In]

integrate(-erfc(b*x)*sinh(b^2*x^2-c),x, algorithm="giac")

[Out]

integrate(-erfc(b*x)*sinh(b^2*x^2 - c), x)

Mupad [F(-1)]

Timed out. \[ \int \text {erfc}(b x) \sinh \left (c-b^2 x^2\right ) \, dx=\int \mathrm {sinh}\left (c-b^2\,x^2\right )\,\mathrm {erfc}\left (b\,x\right ) \,d x \]

[In]

int(sinh(c - b^2*x^2)*erfc(b*x),x)

[Out]

int(sinh(c - b^2*x^2)*erfc(b*x), x)