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

Optimal. Leaf size=151 \[ \frac {\sqrt {\pi } e^{i a+\frac {b^2}{4 (1+i c)}} \text {erfi}\left (\frac {i b+2 (1+i c) x}{2 \sqrt {1+i c}}\right )}{4 \sqrt {1+i c}}-\frac {\sqrt {\pi } e^{-i \left (a-\frac {b^2}{4 c+4 i}\right )} \text {erfi}\left (\frac {i b-2 (1-i c) x}{2 \sqrt {1-i c}}\right )}{4 \sqrt {1-i c}} \]

[Out]

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

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Rubi [A]  time = 0.17, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4473, 2234, 2204} \[ \frac {\sqrt {\pi } e^{i a+\frac {b^2}{4 (1+i c)}} \text {Erfi}\left (\frac {i b+2 (1+i c) x}{2 \sqrt {1+i c}}\right )}{4 \sqrt {1+i c}}-\frac {\sqrt {\pi } e^{-i \left (a-\frac {b^2}{4 c+4 i}\right )} \text {Erfi}\left (\frac {i b-2 (1-i c) x}{2 \sqrt {1-i c}}\right )}{4 \sqrt {1-i c}} \]

Antiderivative was successfully verified.

[In]

Int[E^x^2*Cos[a + b*x + c*x^2],x]

[Out]

-(Sqrt[Pi]*Erfi[(I*b - 2*(1 - I*c)*x)/(2*Sqrt[1 - I*c])])/(4*Sqrt[1 - I*c]*E^(I*(a - b^2/(4*I + 4*c)))) + (E^(
I*a + b^2/(4*(1 + I*c)))*Sqrt[Pi]*Erfi[(I*b + 2*(1 + I*c)*x)/(2*Sqrt[1 + I*c])])/(4*Sqrt[1 + I*c])

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]

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 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]

Rubi steps

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

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Mathematica [A]  time = 0.60, size = 166, normalized size = 1.10 \[ \frac {\sqrt [4]{-1} \sqrt {\pi } e^{\frac {i b^2}{-4 c+4 i}} \left (\sqrt {c-i} (c+i) (\sin (a)-i \cos (a)) \text {erfi}\left (\frac {\sqrt [4]{-1} (b+2 (c-i) x)}{2 \sqrt {c-i}}\right )-(c-i) \sqrt {c+i} e^{\frac {i b^2 c}{2 c^2+2}} (\cos (a)-i \sin (a)) \text {erfi}\left (\frac {(-1)^{3/4} (b+2 (c+i) x)}{2 \sqrt {c+i}}\right )\right )}{4 \left (c^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[E^x^2*Cos[a + b*x + c*x^2],x]

[Out]

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

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fricas [A]  time = 1.85, size = 164, normalized size = 1.09 \[ \frac {\sqrt {\pi } {\left (i \, c + 1\right )} \sqrt {i \, c - 1} \operatorname {erf}\left (-\frac {{\left (b c + 2 \, {\left (c^{2} + 1\right )} x - i \, b\right )} \sqrt {i \, c - 1}}{2 \, {\left (c^{2} + 1\right )}}\right ) e^{\left (\frac {i \, b^{2} c - 4 i \, a c^{2} + b^{2} - 4 i \, a}{4 \, {\left (c^{2} + 1\right )}}\right )} + \sqrt {\pi } {\left (i \, c - 1\right )} \sqrt {-i \, c - 1} \operatorname {erf}\left (\frac {{\left (b c + 2 \, {\left (c^{2} + 1\right )} x + i \, b\right )} \sqrt {-i \, c - 1}}{2 \, {\left (c^{2} + 1\right )}}\right ) e^{\left (\frac {-i \, b^{2} c + 4 i \, a c^{2} + b^{2} + 4 i \, a}{4 \, {\left (c^{2} + 1\right )}}\right )}}{4 \, {\left (c^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \left (c x^{2} + b x + a\right ) e^{\left (x^{2}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cos(c*x^2 + b*x + a)*e^(x^2), x)

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maple [A]  time = 0.12, size = 127, normalized size = 0.84 \[ \frac {\sqrt {\pi }\, {\mathrm e}^{\frac {4 a c +4 i a -b^{2}}{4 i c -4}} \erf \left (\sqrt {i c -1}\, x +\frac {i b}{2 \sqrt {i c -1}}\right )}{4 \sqrt {i c -1}}-\frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c -4 i a -b^{2}}{4 \left (i c +1\right )}} \erf \left (-\sqrt {-i c -1}\, x +\frac {i b}{2 \sqrt {-i c -1}}\right )}{4 \sqrt {-i c -1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.34, size = 474, normalized size = 3.14 \[ \frac {\sqrt {\pi } \sqrt {2 \, c^{2} + 2} {\left ({\left (-i \, \cos \left (-\frac {b^{2} c - 4 \, a c^{2} - 4 \, a}{4 \, {\left (c^{2} + 1\right )}}\right ) e^{\left (\frac {b^{2}}{4 \, {\left (c^{2} + 1\right )}}\right )} - e^{\left (\frac {b^{2}}{4 \, {\left (c^{2} + 1\right )}}\right )} \sin \left (-\frac {b^{2} c - 4 \, a c^{2} - 4 \, a}{4 \, {\left (c^{2} + 1\right )}}\right )\right )} \operatorname {erf}\left (-\frac {2 \, {\left (-i \, c + 1\right )} x - i \, b}{2 \, \sqrt {i \, c - 1}}\right ) + {\left (-i \, \cos \left (-\frac {b^{2} c - 4 \, a c^{2} - 4 \, a}{4 \, {\left (c^{2} + 1\right )}}\right ) e^{\left (\frac {b^{2}}{4 \, {\left (c^{2} + 1\right )}}\right )} + e^{\left (\frac {b^{2}}{4 \, {\left (c^{2} + 1\right )}}\right )} \sin \left (-\frac {b^{2} c - 4 \, a c^{2} - 4 \, a}{4 \, {\left (c^{2} + 1\right )}}\right )\right )} \operatorname {erf}\left (-\frac {2 \, {\left (-i \, c - 1\right )} x - i \, b}{2 \, \sqrt {-i \, c - 1}}\right )\right )} \sqrt {\sqrt {c^{2} + 1} + 1} + \sqrt {\pi } \sqrt {2 \, c^{2} + 2} {\left ({\left (\cos \left (-\frac {b^{2} c - 4 \, a c^{2} - 4 \, a}{4 \, {\left (c^{2} + 1\right )}}\right ) e^{\left (\frac {b^{2}}{4 \, {\left (c^{2} + 1\right )}}\right )} - i \, e^{\left (\frac {b^{2}}{4 \, {\left (c^{2} + 1\right )}}\right )} \sin \left (-\frac {b^{2} c - 4 \, a c^{2} - 4 \, a}{4 \, {\left (c^{2} + 1\right )}}\right )\right )} \operatorname {erf}\left (-\frac {2 \, {\left (-i \, c + 1\right )} x - i \, b}{2 \, \sqrt {i \, c - 1}}\right ) - {\left (\cos \left (-\frac {b^{2} c - 4 \, a c^{2} - 4 \, a}{4 \, {\left (c^{2} + 1\right )}}\right ) e^{\left (\frac {b^{2}}{4 \, {\left (c^{2} + 1\right )}}\right )} + i \, e^{\left (\frac {b^{2}}{4 \, {\left (c^{2} + 1\right )}}\right )} \sin \left (-\frac {b^{2} c - 4 \, a c^{2} - 4 \, a}{4 \, {\left (c^{2} + 1\right )}}\right )\right )} \operatorname {erf}\left (-\frac {2 \, {\left (-i \, c - 1\right )} x - i \, b}{2 \, \sqrt {-i \, c - 1}}\right )\right )} \sqrt {\sqrt {c^{2} + 1} - 1}}{8 \, {\left (c^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(sqrt(pi)*sqrt(2*c^2 + 2)*((-I*cos(-1/4*(b^2*c - 4*a*c^2 - 4*a)/(c^2 + 1))*e^(1/4*b^2/(c^2 + 1)) - e^(1/4*
b^2/(c^2 + 1))*sin(-1/4*(b^2*c - 4*a*c^2 - 4*a)/(c^2 + 1)))*erf(-1/2*(2*(-I*c + 1)*x - I*b)/sqrt(I*c - 1)) + (
-I*cos(-1/4*(b^2*c - 4*a*c^2 - 4*a)/(c^2 + 1))*e^(1/4*b^2/(c^2 + 1)) + e^(1/4*b^2/(c^2 + 1))*sin(-1/4*(b^2*c -
4*a*c^2 - 4*a)/(c^2 + 1)))*erf(-1/2*(2*(-I*c - 1)*x - I*b)/sqrt(-I*c - 1)))*sqrt(sqrt(c^2 + 1) + 1) + sqrt(pi
)*sqrt(2*c^2 + 2)*((cos(-1/4*(b^2*c - 4*a*c^2 - 4*a)/(c^2 + 1))*e^(1/4*b^2/(c^2 + 1)) - I*e^(1/4*b^2/(c^2 + 1)
)*sin(-1/4*(b^2*c - 4*a*c^2 - 4*a)/(c^2 + 1)))*erf(-1/2*(2*(-I*c + 1)*x - I*b)/sqrt(I*c - 1)) - (cos(-1/4*(b^2
*c - 4*a*c^2 - 4*a)/(c^2 + 1))*e^(1/4*b^2/(c^2 + 1)) + I*e^(1/4*b^2/(c^2 + 1))*sin(-1/4*(b^2*c - 4*a*c^2 - 4*a
)/(c^2 + 1)))*erf(-1/2*(2*(-I*c - 1)*x - I*b)/sqrt(-I*c - 1)))*sqrt(sqrt(c^2 + 1) - 1))/(c^2 + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^2)*cos(a + b*x + c*x^2),x)

[Out]

int(exp(x^2)*cos(a + b*x + c*x^2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{x^{2}} \cos {\left (a + b x + c x^{2} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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