\(\int \text {erfc}(b x) \sinh (c+b^2 x^2) \, dx\) [203]

Optimal result

Integrand size = 15, antiderivative size = 75 \[ \int \text {erfc}(b x) \sinh \left (c+b^2 x^2\right ) \, dx=\frac {e^{-c} \sqrt {\pi } \text {erfc}(b x)^2}{8 b}+\frac {e^c \sqrt {\pi } \text {erfi}(b x)}{4 b}-\frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6546, 6512, 2235, 6511, 6509, 30} \[ \int \text {erfc}(b x) \sinh \left (c+b^2 x^2\right ) \, dx=-\frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }}+\frac {\sqrt {\pi } e^{-c} \text {erfc}(b x)^2}{8 b}+\frac {\sqrt {\pi } e^c \text {erfi}(b x)}{4 b} \]

[In]

Int[Erfc[b*x]*Sinh[c + b^2*x^2],x]

[Out]

(Sqrt[Pi]*Erfc[b*x]^2)/(8*b*E^c) + (E^c*Sqrt[Pi]*Erfi[b*x])/(4*b) - (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 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}& = -\left (\frac {1}{2} \int e^{-c-b^2 x^2} \text {erfc}(b x) \, dx\right )+\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} \, dx-\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 = 0.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.11 \[ \int \text {erfc}(b x) \sinh \left (c+b^2 x^2\right ) \, dx=\frac {-4 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right ) (\cosh (c)+\sinh (c))+\pi \left (-2 \text {erf}(b x) (\cosh (c)-\sinh (c))+\text {erf}(b x)^2 (\cosh (c)-\sinh (c))+2 \text {erfi}(b x) (\cosh (c)+\sinh (c))\right )}{8 b \sqrt {\pi }} \]

[In]

Integrate[Erfc[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*(-2*Erf[b*x]*(Cosh[c] - Sinh
[c]) + Erf[b*x]^2*(Cosh[c] - Sinh[c]) + 2*Erfi[b*x]*(Cosh[c] + Sinh[c])))/(8*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. \[ \int \text {erfc}(b x) \sinh \left (c+b^2 x^2\right ) \, dx=\int \mathrm {sinh}\left (b^2\,x^2+c\right )\,\mathrm {erfc}\left (b\,x\right ) \,d x \]

[In]

int(sinh(c + b^2*x^2)*erfc(b*x),x)

[Out]

int(sinh(c + b^2*x^2)*erfc(b*x), x)