\(\int \cos (c-i b^2 x^2) \text {erfc}(b x) \, dx\) [202]

Optimal result

Integrand size = 18, antiderivative size = 85 \[ \int \cos \left (c-i b^2 x^2\right ) \text {erfc}(b x) \, dx=-\frac {e^{-i c} \sqrt {\pi } \text {erfc}(b x)^2}{8 b}+\frac {e^{i c} \sqrt {\pi } \text {erfi}(b x)}{4 b}-\frac {b e^{i c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }} \]

[Out]

-1/2*b*exp(I*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(I*c)+1/4*exp(I*
c)*erfi(b*x)*Pi^(1/2)/b

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6543, 6509, 30, 6512, 2235, 6511} \[ \int \cos \left (c-i b^2 x^2\right ) \text {erfc}(b x) \, dx=-\frac {b e^{i c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }}-\frac {\sqrt {\pi } e^{-i c} \text {erfc}(b x)^2}{8 b}+\frac {\sqrt {\pi } e^{i c} \text {erfi}(b x)}{4 b} \]

[In]

Int[Cos[c - I*b^2*x^2]*Erfc[b*x],x]

[Out]

-1/8*(Sqrt[Pi]*Erfc[b*x]^2)/(b*E^(I*c)) + (E^(I*c)*Sqrt[Pi]*Erfi[b*x])/(4*b) - (b*E^(I*c)*x^2*HypergeometricPF
Q[{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 6543

Int[Cos[(c_.) + (d_.)*(x_)^2]*Erfc[(b_.)*(x_)], x_Symbol] :> Dist[1/2, Int[E^((-I)*c - I*d*x^2)*Erfc[b*x], x],
x] + Dist[1/2, Int[E^(I*c + I*d*x^2)*Erfc[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^{-i c-b^2 x^2} \text {erfc}(b x) \, dx+\frac {1}{2} \int e^{i c+b^2 x^2} \text {erfc}(b x) \, dx \\ & = \frac {1}{2} \int e^{i c+b^2 x^2} \, dx-\frac {1}{2} \int e^{i c+b^2 x^2} \text {erf}(b x) \, dx-\frac {\left (e^{-i c} \sqrt {\pi }\right ) \text {Subst}(\int x \, dx,x,\text {erfc}(b x))}{4 b} \\ & = -\frac {e^{-i c} \sqrt {\pi } \text {erfc}(b x)^2}{8 b}+\frac {e^{i c} \sqrt {\pi } \text {erfi}(b x)}{4 b}-\frac {b e^{i c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }} \\ \end{align*}

Mathematica [F]

\[ \int \cos \left (c-i b^2 x^2\right ) \text {erfc}(b x) \, dx=\int \cos \left (c-i b^2 x^2\right ) \text {erfc}(b x) \, dx \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(cos(I*b**2*x**2 - c)*erfc(b*x), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(cos(c - b^2*x^2*1i)*erfc(b*x),x)

[Out]

int(cos(c - b^2*x^2*1i)*erfc(b*x), x)