3.108 \(\int e^{x^2} \cos (a+c x^2) \, dx\)

Optimal. Leaf size=83 \[ \frac{\sqrt{\pi } e^{-i a} \text{Erfi}\left (\sqrt{1-i c} x\right )}{4 \sqrt{1-i c}}+\frac{\sqrt{\pi } e^{i a} \text{Erfi}\left (\sqrt{1+i c} x\right )}{4 \sqrt{1+i c}} \]

[Out]

(Sqrt[Pi]*Erfi[Sqrt[1 - I*c]*x])/(4*Sqrt[1 - I*c]*E^(I*a)) + (E^(I*a)*Sqrt[Pi]*Erfi[Sqrt[1 + I*c]*x])/(4*Sqrt[
1 + I*c])

________________________________________________________________________________________

Rubi [A]  time = 0.078642, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4473, 2204} \[ \frac{\sqrt{\pi } e^{-i a} \text{Erfi}\left (\sqrt{1-i c} x\right )}{4 \sqrt{1-i c}}+\frac{\sqrt{\pi } e^{i a} \text{Erfi}\left (\sqrt{1+i c} x\right )}{4 \sqrt{1+i c}} \]

Antiderivative was successfully verified.

[In]

Int[E^x^2*Cos[a + c*x^2],x]

[Out]

(Sqrt[Pi]*Erfi[Sqrt[1 - I*c]*x])/(4*Sqrt[1 - I*c]*E^(I*a)) + (E^(I*a)*Sqrt[Pi]*Erfi[Sqrt[1 + I*c]*x])/(4*Sqrt[
1 + I*c])

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

Mathematica [A]  time = 0.193847, size = 107, normalized size = 1.29 \[ \frac{\sqrt [4]{-1} \sqrt{\pi } \left ((1-i c) \sqrt{c-i} (\cos (a)+i \sin (a)) \text{Erfi}\left (\sqrt [4]{-1} \sqrt{c-i} x\right )-(c-i) \sqrt{c+i} (\cos (a)-i \sin (a)) \text{Erfi}\left ((-1)^{3/4} \sqrt{c+i} x\right )\right )}{4 \left (c^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[E^x^2*Cos[a + c*x^2],x]

[Out]

((-1)^(1/4)*Sqrt[Pi]*(-((-I + c)*Sqrt[I + c]*Erfi[(-1)^(3/4)*Sqrt[I + c]*x]*(Cos[a] - I*Sin[a])) + (1 - I*c)*S
qrt[-I + c]*Erfi[(-1)^(1/4)*Sqrt[-I + c]*x]*(Cos[a] + I*Sin[a])))/(4*(1 + c^2))

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 60, normalized size = 0.7 \begin{align*}{\frac{\sqrt{\pi }{{\rm e}^{-ia}}}{4}{\it Erf} \left ( \sqrt{-1+ic}x \right ){\frac{1}{\sqrt{-1+ic}}}}+{\frac{\sqrt{\pi }{{\rm e}^{ia}}}{4}{\it Erf} \left ( \sqrt{-ic-1}x \right ){\frac{1}{\sqrt{-ic-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: IndexError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: IndexError

________________________________________________________________________________________

Fricas [A]  time = 0.479143, size = 198, normalized size = 2.39 \begin{align*} \frac{\sqrt{\pi }{\left (i \, c - 1\right )} \sqrt{-i \, c - 1} \operatorname{erf}\left (\sqrt{-i \, c - 1} x\right ) e^{\left (i \, a\right )} + \sqrt{\pi } \sqrt{i \, c - 1}{\left (-i \, c - 1\right )} \operatorname{erf}\left (\sqrt{i \, c - 1} x\right ) e^{\left (-i \, a\right )}}{4 \,{\left (c^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(x**2)*cos(a + c*x**2), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (c x^{2} + 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(c*x^2+a),x, algorithm="giac")

[Out]

integrate(cos(c*x^2 + a)*e^(x^2), x)