\(\int \text {erf}(b x) \sin (c-i b^2 x^2) \, dx\) [97]

Optimal result

Integrand size = 18, antiderivative size = 66 \[ \int \text {erf}(b x) \sin \left (c-i b^2 x^2\right ) \, dx=\frac {i e^{-i c} \sqrt {\pi } \text {erf}(b x)^2}{8 b}-\frac {i b e^{i c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6539, 6508, 30, 6511} \[ \int \text {erf}(b x) \sin \left (c-i b^2 x^2\right ) \, dx=\frac {i \sqrt {\pi } e^{-i c} \text {erf}(b x)^2}{8 b}-\frac {i b e^{i c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }} \]

[In]

Int[Erf[b*x]*Sin[c - I*b^2*x^2],x]

[Out]

((I/8)*Sqrt[Pi]*Erf[b*x]^2)/(b*E^(I*c)) - ((I/2)*b*E^(I*c)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/S
qrt[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 6539

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.02 \[ \int \text {erf}(b x) \sin \left (c-i b^2 x^2\right ) \, dx=\frac {(i \cos (c)+\sin (c)) \left (\pi \text {erf}(b x)^2-4 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right ) (\cos (2 c)+i \sin (2 c))\right )}{8 b \sqrt {\pi }} \]

[In]

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

[Out]

((I*Cos[c] + Sin[c])*(Pi*Erf[b*x]^2 - 4*b^2*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2]*(Cos[2*c] + I*Sin
[2*c])))/(8*b*Sqrt[Pi])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

-Integral(sin(I*b**2*x**2 - c)*erf(b*x), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(sin(c - b^2*x^2*1i)*erf(b*x),x)

[Out]

int(sin(c - b^2*x^2*1i)*erf(b*x), x)