\(\int \text {erfc}(b x) \sin (c+i b^2 x^2) \, dx\) [199]

Optimal result

Integrand size = 18, antiderivative size = 91 \[ \int \text {erfc}(b x) \sin \left (c+i b^2 x^2\right ) \, dx=\frac {i e^{i c} \sqrt {\pi } \text {erfc}(b x)^2}{8 b}+\frac {i e^{-i c} \sqrt {\pi } \text {erfi}(b x)}{4 b}-\frac {i b e^{-i c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }} \] Output:

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.03 \[ \int \text {erfc}(b x) \sin \left (c+i b^2 x^2\right ) \, dx=\frac {(i \cos (c)+\sin (c)) \left (-4 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )+\pi \left (2 \text {erfi}(b x)-2 \text {erf}(b x) (\cos (2 c)+i \sin (2 c))+\text {erf}(b x)^2 (\cos (2 c)+i \sin (2 c))\right )\right )}{8 b \sqrt {\pi }} \] Input:

Integrate[Erfc[b*x]*Sin[c + I*b^2*x^2],x]
 

Output:

((I*Cos[c] + Sin[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]*(Cos[2*c] + I*Sin[2*c]) + Erf[b*x]^2*(C 
os[2*c] + I*Sin[2*c]))))/(8*b*Sqrt[Pi])
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6959, 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.

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

\(\Big \downarrow \) 6959

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

\(\Big \downarrow \) 6928

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6931

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 6930

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

Input:

Int[Erfc[b*x]*Sin[c + I*b^2*x^2],x]
 

Output:

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

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 6959

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(1/2*((I*erf(b*x) - I)*e^(-2*b^2*x^2 + 2*I*c) - I*erf(b*x) + I)*e^ 
(b^2*x^2 - I*c), x)
 

Sympy [F]

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

integrate(erfc(b*x)*sin(c+I*b**2*x**2),x)
 

Output:

Integral(sin(I*b**2*x**2 + c)*erfc(b*x), x)
 

Maxima [F]

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

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

Output:

1/8*I*sqrt(pi)*cos(c)*erfc(b*x)^2/b - 1/8*sqrt(pi)*erfc(b*x)^2*sin(c)/b + 
1/2*I*cos(c)*integrate(erfc(b*x)*e^(b^2*x^2), x) + 1/2*integrate(erfc(b*x) 
*e^(b^2*x^2), x)*sin(c)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \text {erfc}(b x) \sin \left (c+i b^2 x^2\right ) \, dx=\int \sin \left (b^2\,x^2\,1{}\mathrm {i}+c\right )\,\mathrm {erfc}\left (b\,x\right ) \,d x \] Input:

int(sin(c + b^2*x^2*1i)*erfc(b*x),x)
 

Output:

int(sin(c + b^2*x^2*1i)*erfc(b*x), x)
 

Reduce [F]

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

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

Output:

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