\(\int \cosh (c-b^2 x^2) \text {erf}(b x) \, dx\) [103]

Optimal result

Integrand size = 16, antiderivative size = 56 \[ \int \cosh \left (c-b^2 x^2\right ) \text {erf}(b x) \, dx=\frac {e^c \sqrt {\pi } \text {erf}(b x)^2}{8 b}+\frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }} \]

[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)*erf(b*x)^2*Pi^(1/2)/b

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6548, 6508, 30, 6511} \[ \int \cosh \left (c-b^2 x^2\right ) \text {erf}(b x) \, dx=\frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }}+\frac {\sqrt {\pi } e^c \text {erf}(b x)^2}{8 b} \]

[In]

Int[Cosh[c - b^2*x^2]*Erf[b*x],x]

[Out]

(E^c*Sqrt[Pi]*Erf[b*x]^2)/(8*b) + (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 6508

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[E^c*(Sqrt[Pi]/(2*b)), Subst[Int[x^n, x],
x, Erf[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 6548

Int[Cosh[(c_.) + (d_.)*(x_)^2]*Erf[(b_.)*(x_)], x_Symbol] :> Dist[1/2, Int[E^(c + d*x^2)*Erf[b*x], x], x] + Di
st[1/2, Int[E^(-c - d*x^2)*Erf[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 {erf}(b x) \, dx+\frac {1}{2} \int e^{-c+b^2 x^2} \text {erf}(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 ) \text {Subst}(\int x \, dx,x,\text {erf}(b x))}{4 b} \\ & = \frac {e^c \sqrt {\pi } \text {erf}(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] (verified)

Time = 0.35 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.34 \[ \int \cosh \left (c-b^2 x^2\right ) \text {erf}(b x) \, dx=\frac {4 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right ) (-\cosh (c)+\sinh (c))+\pi \text {erf}(b x) (2 \text {erfi}(b x) (\cosh (c)-\sinh (c))+\text {erf}(b x) (\cosh (c)+\sinh (c)))}{8 b \sqrt {\pi }} \]

[In]

Integrate[Cosh[c - b^2*x^2]*Erf[b*x],x]

[Out]

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

Maple [F]

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

[In]

int(cosh(b^2*x^2-c)*erf(b*x),x)

[Out]

int(cosh(b^2*x^2-c)*erf(b*x),x)

Fricas [F]

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

[In]

integrate(cosh(b^2*x^2-c)*erf(b*x),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate(cosh(b**2*x**2-c)*erf(b*x),x)

[Out]

Integral(cosh(b**2*x**2 - c)*erf(b*x), x)

Maxima [F]

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

[In]

integrate(cosh(b^2*x^2-c)*erf(b*x),x, algorithm="maxima")

[Out]

1/8*sqrt(pi)*erf(b*x)^2*e^c/b + 1/2*integrate(erf(b*x)*e^(b^2*x^2 - c), x)

Giac [F]

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

[In]

integrate(cosh(b^2*x^2-c)*erf(b*x),x, algorithm="giac")

[Out]

integrate(cosh(b^2*x^2 - c)*erf(b*x), x)

Mupad [F(-1)]

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

[In]

int(cosh(c - b^2*x^2)*erf(b*x),x)

[Out]

int(cosh(c - b^2*x^2)*erf(b*x), x)