3.58 \(\int e^{x^2} \cos (b x) \, dx\)

Optimal. Leaf size=65 \[ \frac{1}{4} \sqrt{\pi } e^{\frac{b^2}{4}} \text{Erfi}\left (\frac{1}{2} (2 x-i b)\right )+\frac{1}{4} \sqrt{\pi } e^{\frac{b^2}{4}} \text{Erfi}\left (\frac{1}{2} (2 x+i b)\right ) \]

[Out]

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

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Rubi [A]  time = 0.0473719, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4473, 2234, 2204} \[ \frac{1}{4} \sqrt{\pi } e^{\frac{b^2}{4}} \text{Erfi}\left (\frac{1}{2} (2 x-i b)\right )+\frac{1}{4} \sqrt{\pi } e^{\frac{b^2}{4}} \text{Erfi}\left (\frac{1}{2} (2 x+i b)\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x^2*Cos[b*x],x]

[Out]

(E^(b^2/4)*Sqrt[Pi]*Erfi[((-I)*b + 2*x)/2])/4 + (E^(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 (b x) \, dx &=\int \left (\frac{1}{2} e^{-i b x+x^2}+\frac{1}{2} e^{i b x+x^2}\right ) \, dx\\ &=\frac{1}{2} \int e^{-i b x+x^2} \, dx+\frac{1}{2} \int e^{i b x+x^2} \, dx\\ &=\frac{1}{2} e^{\frac{b^2}{4}} \int e^{\frac{1}{4} (-i b+2 x)^2} \, dx+\frac{1}{2} e^{\frac{b^2}{4}} \int e^{\frac{1}{4} (i b+2 x)^2} \, dx\\ &=\frac{1}{4} e^{\frac{b^2}{4}} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} (-i b+2 x)\right )+\frac{1}{4} e^{\frac{b^2}{4}} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} (i b+2 x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0203367, size = 47, normalized size = 0.72 \[ \frac{1}{4} \sqrt{\pi } e^{\frac{b^2}{4}} \left (\text{Erfi}\left (\frac{1}{2} (2 x-i b)\right )+\text{Erfi}\left (\frac{1}{2} (2 x+i b)\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x^2*Cos[b*x],x]

[Out]

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

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Maple [A]  time = 0.039, size = 44, normalized size = 0.7 \begin{align*} -{\frac{i}{4}}\sqrt{\pi }{{\rm e}^{{\frac{{b}^{2}}{4}}}}{\it Erf} \left ( ix+{\frac{b}{2}} \right ) +{\frac{i}{4}}\sqrt{\pi }{{\rm e}^{{\frac{{b}^{2}}{4}}}}{\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),x)

[Out]

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

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Maxima [A]  time = 1.0258, size = 51, normalized size = 0.78 \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}\right )} + i \, \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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.473565, size = 95, normalized size = 1.46 \begin{align*} \frac{1}{4} \, \sqrt{\pi }{\left (-i \, \operatorname{erf}\left (\frac{1}{2} \, b + i \, x\right ) - i \, \operatorname{erf}\left (-\frac{1}{2} \, b + i \, x\right )\right )} e^{\left (\frac{1}{4} \, b^{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(x**2)*cos(b*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (b x\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),x, algorithm="giac")

[Out]

integrate(cos(b*x)*e^(x^2), x)