3.105 \(\int e^x \cos (a+c x^2) \, dx\)

Optimal. Leaf size=115 \[ \frac {\sqrt [4]{-1} \sqrt {\pi } e^{-\frac {1}{4} i \left (4 a+\frac {1}{c}\right )} \text {erfi}\left (\frac {\sqrt [4]{-1} (1-2 i c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt [4]{-1} \sqrt {\pi } e^{\frac {1}{4} i \left (4 a+\frac {1}{c}\right )} \text {erf}\left (\frac {\sqrt [4]{-1} (1+2 i c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c}} \]

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4473, 2234, 2204, 2205} \[ \frac {\sqrt [4]{-1} \sqrt {\pi } e^{-\frac {1}{4} i \left (4 a+\frac {1}{c}\right )} \text {Erfi}\left (\frac {\sqrt [4]{-1} (1-2 i c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt [4]{-1} \sqrt {\pi } e^{\frac {1}{4} i \left (4 a+\frac {1}{c}\right )} \text {Erf}\left (\frac {\sqrt [4]{-1} (1+2 i c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c}} \]

Antiderivative was successfully verified.

[In]

Int[E^x*Cos[a + c*x^2],x]

[Out]

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

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 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[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 \cos \left (a+c x^2\right ) \, dx &=\int \left (\frac {1}{2} e^{-i a+x-i c x^2}+\frac {1}{2} e^{i a+x+i c x^2}\right ) \, dx\\ &=\frac {1}{2} \int e^{-i a+x-i c x^2} \, dx+\frac {1}{2} \int e^{i a+x+i c x^2} \, dx\\ &=\frac {1}{2} e^{-\frac {1}{4} i \left (4 a+\frac {1}{c}\right )} \int e^{\frac {i (1-2 i c x)^2}{4 c}} \, dx+\frac {1}{2} e^{\frac {1}{4} i \left (4 a+\frac {1}{c}\right )} \int e^{-\frac {i (1+2 i c x)^2}{4 c}} \, dx\\ &=-\frac {\sqrt [4]{-1} e^{\frac {1}{4} i \left (4 a+\frac {1}{c}\right )} \sqrt {\pi } \text {erf}\left (\frac {\sqrt [4]{-1} (1+2 i c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {\sqrt [4]{-1} e^{-\frac {1}{4} i \left (4 a+\frac {1}{c}\right )} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt [4]{-1} (1-2 i c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 109, normalized size = 0.95 \[ \frac {\sqrt [4]{-1} \sqrt {\pi } e^{\left .-\frac {i}{4}\right /c} \left (e^{\left .\frac {i}{2}\right /c} (\sin (a)-i \cos (a)) \text {erfi}\left (\frac {\sqrt [4]{-1} (2 c x-i)}{2 \sqrt {c}}\right )-(\cos (a)-i \sin (a)) \text {erfi}\left (\frac {(-1)^{3/4} (2 c x+i)}{2 \sqrt {c}}\right )\right )}{4 \sqrt {c}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Cos[a + c*x^2],x]

[Out]

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

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fricas [B]  time = 2.09, size = 193, normalized size = 1.68 \[ \frac {\sqrt {2} \pi \sqrt {\frac {c}{\pi }} e^{\left (\frac {-4 i \, a c - i}{4 \, c}\right )} \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, c x + i\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) - \sqrt {2} \pi \sqrt {\frac {c}{\pi }} e^{\left (\frac {4 i \, a c + i}{4 \, c}\right )} \operatorname {C}\left (-\frac {\sqrt {2} {\left (2 \, c x - i\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) - i \, \sqrt {2} \pi \sqrt {\frac {c}{\pi }} e^{\left (\frac {-4 i \, a c - i}{4 \, c}\right )} \operatorname {S}\left (\frac {\sqrt {2} {\left (2 \, c x + i\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) - i \, \sqrt {2} \pi \sqrt {\frac {c}{\pi }} e^{\left (\frac {4 i \, a c + i}{4 \, c}\right )} \operatorname {S}\left (-\frac {\sqrt {2} {\left (2 \, c x - i\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right )}{4 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.14, size = 127, normalized size = 1.10 \[ -\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} {\left (2 \, x + \frac {i}{c}\right )} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {4 i \, a c + i}{4 \, c}\right )}}{4 \, {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} {\left (2 \, x - \frac {i}{c}\right )} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {-4 i \, a c - i}{4 \, c}\right )}}{4 \, {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.13, size = 86, normalized size = 0.75 \[ \frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {i \left (4 a c +1\right )}{4 c}} \erf \left (\sqrt {i c}\, x -\frac {1}{2 \sqrt {i c}}\right )}{4 \sqrt {i c}}+\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {i \left (4 a c +1\right )}{4 c}} \erf \left (\sqrt {-i c}\, x -\frac {1}{2 \sqrt {-i c}}\right )}{4 \sqrt {-i c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.34, size = 100, normalized size = 0.87 \[ -\frac {\sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (\frac {4 \, a c + 1}{4 \, c}\right ) + \left (i + 1\right ) \, \sin \left (\frac {4 \, a c + 1}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 i \, c x - 1}{2 \, \sqrt {i \, c}}\right ) + {\left (\left (i + 1\right ) \, \cos \left (\frac {4 \, a c + 1}{4 \, c}\right ) + \left (i - 1\right ) \, \sin \left (\frac {4 \, a c + 1}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 i \, c x + 1}{2 \, \sqrt {-i \, c}}\right )\right )}}{8 \, \sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*cos(a + c*x^2),x)

[Out]

int(exp(x)*cos(a + c*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(x)*cos(a + c*x**2), x)

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