\(\int \cosh (c-b^2 x^2) \text {erfc}(b x) \, dx\) [206]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.31 \[ \int \cosh \left (c-b^2 x^2\right ) \text {erfc}(b x) \, dx=\frac {4 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right ) (\cosh (c)-\sinh (c))-\pi \left (2 \text {erf}(b x) (-\cosh (c)+\text {erfi}(b x) (\cosh (c)-\sinh (c))-\sinh (c))+2 \text {erfi}(b x) (-\cosh (c)+\sinh (c))+\text {erf}(b x)^2 (\cosh (c)+\sinh (c))\right )}{8 b \sqrt {\pi }} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6968, 6928, 15, 6931, 2633, 6930}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \text {erfc}(b x) \cosh \left (c-b^2 x^2\right ) \, dx\)

\(\Big \downarrow \) 6968

\(\displaystyle \frac {1}{2} \int e^{c-b^2 x^2} \text {erfc}(b x)dx+\frac {1}{2} \int e^{b^2 x^2-c} \text {erfc}(b x)dx\)

\(\Big \downarrow \) 6928

\(\displaystyle \frac {1}{2} \int e^{b^2 x^2-c} \text {erfc}(b x)dx-\frac {\sqrt {\pi } e^c \int \text {erfc}(b x)d\text {erfc}(b x)}{4 b}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{2} \int e^{b^2 x^2-c} \text {erfc}(b x)dx-\frac {\sqrt {\pi } e^c \text {erfc}(b x)^2}{8 b}\)

\(\Big \downarrow \) 6931

\(\displaystyle \frac {1}{2} \left (\int e^{b^2 x^2-c}dx-\int e^{b^2 x^2-c} \text {erf}(b x)dx\right )-\frac {\sqrt {\pi } e^c \text {erfc}(b x)^2}{8 b}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {1}{2} \left (\frac {\sqrt {\pi } e^{-c} \text {erfi}(b x)}{2 b}-\int e^{b^2 x^2-c} \text {erf}(b x)dx\right )-\frac {\sqrt {\pi } e^c \text {erfc}(b x)^2}{8 b}\)

\(\Big \downarrow \) 6930

\(\displaystyle \frac {1}{2} \left (\frac {\sqrt {\pi } e^{-c} \text {erfi}(b x)}{2 b}-\frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }}\right )-\frac {\sqrt {\pi } e^c \text {erfc}(b x)^2}{8 b}\)

Input:

Int[Cosh[c - b^2*x^2]*Erfc[b*x],x]
 

Output:

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

Defintions of rubi rules used

rule 15

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

rule 2633

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 6928

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[(-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 6930

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 6931

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 6968

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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