3.305 \(\int \text {erfi}(b x) \sin (c-i b^2 x^2) \, dx\)

Optimal. Leaf size=67 \[ \frac {i b e^{-i c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{2 \sqrt {\pi }}-\frac {i \sqrt {\pi } e^{i c} \text {erfi}(b x)^2}{8 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6406, 6378, 6375, 30} \[ \frac {i b e^{-i c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{2 \sqrt {\pi }}-\frac {i \sqrt {\pi } e^{i c} \text {Erfi}(b x)^2}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[Erfi[b*x]*Sin[c - I*b^2*x^2],x]

[Out]

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

Rule 30

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

Rule 6375

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[(E^c*Sqrt[Pi])/(2*b), Subst[Int[x^n, x]
, x, Erfi[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, b^2]

Rule 6378

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(b_.)*(x_)], x_Symbol] :> Simp[(b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}
, -(b^2*x^2)])/Sqrt[Pi], x] /; FreeQ[{b, c, d}, x] && EqQ[d, -b^2]

Rule 6406

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

Rubi steps

\begin {align*} \int \text {erfi}(b x) \sin \left (c-i b^2 x^2\right ) \, dx &=\frac {1}{2} i \int e^{-i c-b^2 x^2} \text {erfi}(b x) \, dx-\frac {1}{2} i \int e^{i c+b^2 x^2} \text {erfi}(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 ) \operatorname {Subst}(\int x \, dx,x,\text {erfi}(b x))}{4 b}\\ &=-\frac {i e^{i c} \sqrt {\pi } \text {erfi}(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 [F]  time = 0.71, size = 0, normalized size = 0.00 \[ \int \text {erfi}(b x) \sin \left (c-i b^2 x^2\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{2} \, {\left (i \, \operatorname {erfi}\left (b x\right ) e^{\left (-2 \, b^{2} x^{2} - 2 i \, c\right )} - i \, \operatorname {erfi}\left (b x\right )\right )} e^{\left (b^{2} x^{2} + i \, c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\operatorname {erfi}\left (b x\right ) \sin \left (i \, b^{2} x^{2} - c\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[ \int -\erfi \left (b x \right ) \sin \left (i b^{2} x^{2}-c \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {i \, \sqrt {\pi } \cos \relax (c) \operatorname {erfi}\left (b x\right )^{2}}{8 \, b} + \frac {\sqrt {\pi } \operatorname {erfi}\left (b x\right )^{2} \sin \relax (c)}{8 \, b} + \frac {1}{2} i \, \cos \relax (c) \int \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} + \frac {1}{2} \, \int \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \sin \relax (c) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \sin \left (c-b^2\,x^2\,1{}\mathrm {i}\right )\,\mathrm {erfi}\left (b\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c - b^2*x^2*1i)*erfi(b*x),x)

[Out]

int(sin(c - b^2*x^2*1i)*erfi(b*x), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \sin {\left (i b^{2} x^{2} - c \right )} \operatorname {erfi}{\left (b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________