3.107 \(\int e^{x^2} \cos (a+b x) \, dx\)

Optimal. Leaf size=77 \[ \frac{1}{4} \sqrt{\pi } e^{\frac{b^2}{4}-i a} \text{Erfi}\left (\frac{1}{2} (2 x-i b)\right )+\frac{1}{4} \sqrt{\pi } e^{\frac{b^2}{4}+i a} \text{Erfi}\left (\frac{1}{2} (2 x+i b)\right ) \]

[Out]

(E^((-I)*a + b^2/4)*Sqrt[Pi]*Erfi[((-I)*b + 2*x)/2])/4 + (E^(I*a + b^2/4)*Sqrt[Pi]*Erfi[(I*b + 2*x)/2])/4

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Rubi [A]  time = 0.0567589, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4473, 2234, 2204} \[ \frac{1}{4} \sqrt{\pi } e^{\frac{b^2}{4}-i a} \text{Erfi}\left (\frac{1}{2} (2 x-i b)\right )+\frac{1}{4} \sqrt{\pi } e^{\frac{b^2}{4}+i a} \text{Erfi}\left (\frac{1}{2} (2 x+i b)\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x^2*Cos[a + b*x],x]

[Out]

(E^((-I)*a + b^2/4)*Sqrt[Pi]*Erfi[((-I)*b + 2*x)/2])/4 + (E^(I*a + b^2/4)*Sqrt[Pi]*Erfi[(I*b + 2*x)/2])/4

Rule 4473

Int[Cos[v_]^(n_.)*(F_)^(u_), x_Symbol] :> Int[ExpandTrigToExp[F^u, Cos[v]^n, x], x] /; FreeQ[F, x] && (LinearQ
[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2204

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]

Rubi steps

\begin{align*} \int e^{x^2} \cos (a+b x) \, dx &=\int \left (\frac{1}{2} e^{-i a-i b x+x^2}+\frac{1}{2} e^{i a+i b x+x^2}\right ) \, dx\\ &=\frac{1}{2} \int e^{-i a-i b x+x^2} \, dx+\frac{1}{2} \int e^{i a+i b x+x^2} \, dx\\ &=\frac{1}{2} e^{-i a+\frac{b^2}{4}} \int e^{\frac{1}{4} (-i b+2 x)^2} \, dx+\frac{1}{2} e^{i a+\frac{b^2}{4}} \int e^{\frac{1}{4} (i b+2 x)^2} \, dx\\ &=\frac{1}{4} e^{-i a+\frac{b^2}{4}} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} (-i b+2 x)\right )+\frac{1}{4} e^{i a+\frac{b^2}{4}} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} (i b+2 x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0826598, size = 82, normalized size = 1.06 \[ \frac{1}{4} \sqrt{\pi } e^{\frac{b^2}{4}} \left (-\sin (a) \left (\text{Erf}\left (\frac{b}{2}-i x\right )+\text{Erf}\left (\frac{b}{2}+i x\right )\right )+\cos (a) \text{Erfi}\left (\frac{1}{2} (2 x-i b)\right )+\cos (a) \text{Erfi}\left (\frac{1}{2} (2 x+i b)\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x^2*Cos[a + b*x],x]

[Out]

(E^(b^2/4)*Sqrt[Pi]*(Cos[a]*Erfi[((-I)*b + 2*x)/2] + Cos[a]*Erfi[(I*b + 2*x)/2] - (Erf[b/2 - I*x] + Erf[b/2 +
I*x])*Sin[a]))/4

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Maple [A]  time = 0.002, size = 54, normalized size = 0.7 \begin{align*} -{\frac{i}{4}}\sqrt{\pi }{{\rm e}^{{\frac{{b}^{2}}{4}}}}{{\rm e}^{-ia}}{\it Erf} \left ( ix+{\frac{b}{2}} \right ) +{\frac{i}{4}}\sqrt{\pi }{{\rm e}^{{\frac{{b}^{2}}{4}}}}{{\rm e}^{ia}}{\it Erf} \left ( -ix+{\frac{b}{2}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^2)*cos(b*x+a),x)

[Out]

-1/4*I*Pi^(1/2)*exp(1/4*b^2)*exp(-I*a)*erf(I*x+1/2*b)+1/4*I*Pi^(1/2)*exp(1/4*b^2)*exp(I*a)*erf(-I*x+1/2*b)

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Maxima [A]  time = 1.16329, size = 70, normalized size = 0.91 \begin{align*} -\frac{1}{4} \, \sqrt{\pi }{\left ({\left (i \, \cos \left (a\right ) + \sin \left (a\right )\right )} \operatorname{erf}\left (\frac{1}{2} \, b + i \, x\right ) e^{\left (\frac{1}{4} \, b^{2}\right )} +{\left (i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \operatorname{erf}\left (-\frac{1}{2} \, b + i \, x\right ) e^{\left (\frac{1}{4} \, b^{2}\right )}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^2)*cos(b*x+a),x, algorithm="maxima")

[Out]

-1/4*sqrt(pi)*((I*cos(a) + sin(a))*erf(1/2*b + I*x)*e^(1/4*b^2) + (I*cos(a) - sin(a))*erf(-1/2*b + I*x)*e^(1/4
*b^2))

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Fricas [A]  time = 0.475381, size = 127, normalized size = 1.65 \begin{align*} \frac{1}{4} \, \sqrt{\pi }{\left (-i \, \operatorname{erf}\left (-\frac{1}{2} \, b + i \, x\right ) e^{\left (\frac{1}{4} \, b^{2} + i \, a\right )} - i \, \operatorname{erf}\left (\frac{1}{2} \, b + i \, x\right ) e^{\left (\frac{1}{4} \, b^{2} - i \, a\right )}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^2)*cos(b*x+a),x, algorithm="fricas")

[Out]

1/4*sqrt(pi)*(-I*erf(-1/2*b + I*x)*e^(1/4*b^2 + I*a) - I*erf(1/2*b + I*x)*e^(1/4*b^2 - I*a))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{x^{2}} \cos{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x**2)*cos(b*x+a),x)

[Out]

Integral(exp(x**2)*cos(a + b*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (b x + a\right ) e^{\left (x^{2}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^2)*cos(b*x+a),x, algorithm="giac")

[Out]

integrate(cos(b*x + a)*e^(x^2), x)