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

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.11 \[ \int \text {erfc}(b x) \sinh \left (c+b^2 x^2\right ) \, dx=\frac {-4 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right ) (\cosh (c)+\sinh (c))+\pi \left (-2 \text {erf}(b x) (\cosh (c)-\sinh (c))+\text {erf}(b x)^2 (\cosh (c)-\sinh (c))+2 \text {erfi}(b x) (\cosh (c)+\sinh (c))\right )}{8 b \sqrt {\pi }} \] Input:

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

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6965, 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.

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

\(\Big \downarrow \) 6965

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

\(\Big \downarrow \) 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[Erfc[b*x]*Sinh[c + b^2*x^2],x]
 

Output:

(Sqrt[Pi]*Erfc[b*x]^2)/(8*b*E^c) + ((E^c*Sqrt[Pi]*Erfi[b*x])/(2*b) - (b*E^ 
c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/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 6965

Int[Erfc[(b_.)*(x_)]*Sinh[(c_.) + (d_.)*(x_)^2], 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sinh(b**2*x**2 + c),x) - int(erf(b*x)*sinh(b**2*x**2 + c),x)