Integrand size = 12, antiderivative size = 115 \[ \int e^x \cos \left (a+c x^2\right ) \, dx=-\frac {\sqrt [4]{-1} e^{\frac {1}{4} i \left (4 a+\frac {1}{c}\right )} \sqrt {\pi } \text {erf}\left (\frac {\sqrt [4]{-1} (1+2 i c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {\sqrt [4]{-1} e^{-\frac {1}{4} i \left (4 a+\frac {1}{c}\right )} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt [4]{-1} (1-2 i c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c}} \] Output:
-1/4*(-1)^(1/4)*exp(1/4*I*(4*a+1/c))*Pi^(1/2)*erf(1/2*(-1)^(1/4)*(1+2*I*c* x)/c^(1/2))/c^(1/2)+1/4*(-1)^(1/4)*Pi^(1/2)*erfi(1/2*(-1)^(1/4)*(1-2*I*c*x )/c^(1/2))/c^(1/2)/exp(1/4*I*(4*a+1/c))
Time = 0.12 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95 \[ \int e^x \cos \left (a+c x^2\right ) \, dx=\frac {\sqrt [4]{-1} e^{\left .-\frac {i}{4}\right /c} \sqrt {\pi } \left (-\text {erfi}\left (\frac {(-1)^{3/4} (i+2 c x)}{2 \sqrt {c}}\right ) (\cos (a)-i \sin (a))+e^{\left .\frac {i}{2}\right /c} \text {erfi}\left (\frac {\sqrt [4]{-1} (-i+2 c x)}{2 \sqrt {c}}\right ) (-i \cos (a)+\sin (a))\right )}{4 \sqrt {c}} \] Input:
Integrate[E^x*Cos[a + c*x^2],x]
Output:
((-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))
Time = 0.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4976, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int e^x \cos \left (a+c x^2\right ) \, dx\) |
\(\Big \downarrow \) 4976 |
\(\displaystyle \int \left (\frac {1}{2} e^{-i a-i c x^2+x}+\frac {1}{2} e^{i a+i c x^2+x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt [4]{-1} \sqrt {\pi } e^{-\frac {1}{4} i \left (4 a+\frac {1}{c}\right )} \text {erfi}\left (\frac {\sqrt [4]{-1} (1-2 i c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt [4]{-1} \sqrt {\pi } e^{\frac {1}{4} i \left (4 a+\frac {1}{c}\right )} \text {erf}\left (\frac {\sqrt [4]{-1} (1+2 i c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\) |
Input:
Int[E^x*Cos[a + c*x^2],x]
Output:
-1/4*((-1)^(1/4)*E^((I/4)*(4*a + c^(-1)))*Sqrt[Pi]*Erf[((-1)^(1/4)*(1 + (2 *I)*c*x))/(2*Sqrt[c])])/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))))
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]
Time = 0.51 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.75
method | result | size |
risch | \(\frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {i \left (4 a c +1\right )}{4 c}} \operatorname {erf}\left (\sqrt {i c}\, x -\frac {1}{2 \sqrt {i c}}\right )}{4 \sqrt {i c}}+\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {i \left (4 a c +1\right )}{4 c}} \operatorname {erf}\left (\sqrt {-i c}\, x -\frac {1}{2 \sqrt {-i c}}\right )}{4 \sqrt {-i c}}\) | \(86\) |
Input:
int(exp(x)*cos(c*x^2+a),x,method=_RETURNVERBOSE)
Output:
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))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (73) = 146\).
Time = 0.08 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.26 \[ \int e^x \cos \left (a+c x^2\right ) \, dx=\frac {\sqrt {2} {\left (\pi \cos \left (\frac {4 \, a c + 1}{4 \, c}\right ) - i \, \pi \sin \left (\frac {4 \, a c + 1}{4 \, c}\right )\right )} \sqrt {\frac {c}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, c x + i\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) - \sqrt {2} {\left (\pi \cos \left (\frac {4 \, a c + 1}{4 \, c}\right ) + i \, \pi \sin \left (\frac {4 \, a c + 1}{4 \, c}\right )\right )} \sqrt {\frac {c}{\pi }} \operatorname {C}\left (-\frac {\sqrt {2} {\left (2 \, c x - i\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) + \sqrt {2} {\left (-i \, \pi \cos \left (\frac {4 \, a c + 1}{4 \, c}\right ) - \pi \sin \left (\frac {4 \, a c + 1}{4 \, c}\right )\right )} \sqrt {\frac {c}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} {\left (2 \, c x + i\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) + \sqrt {2} {\left (-i \, \pi \cos \left (\frac {4 \, a c + 1}{4 \, c}\right ) + \pi \sin \left (\frac {4 \, a c + 1}{4 \, c}\right )\right )} \sqrt {\frac {c}{\pi }} \operatorname {S}\left (-\frac {\sqrt {2} {\left (2 \, c x - i\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right )}{4 \, c} \] Input:
integrate(exp(x)*cos(c*x^2+a),x, algorithm="fricas")
Output:
1/4*(sqrt(2)*(pi*cos(1/4*(4*a*c + 1)/c) - I*pi*sin(1/4*(4*a*c + 1)/c))*sqr t(c/pi)*fresnel_cos(1/2*sqrt(2)*(2*c*x + I)*sqrt(c/pi)/c) - sqrt(2)*(pi*co s(1/4*(4*a*c + 1)/c) + I*pi*sin(1/4*(4*a*c + 1)/c))*sqrt(c/pi)*fresnel_cos (-1/2*sqrt(2)*(2*c*x - I)*sqrt(c/pi)/c) + sqrt(2)*(-I*pi*cos(1/4*(4*a*c + 1)/c) - pi*sin(1/4*(4*a*c + 1)/c))*sqrt(c/pi)*fresnel_sin(1/2*sqrt(2)*(2*c *x + I)*sqrt(c/pi)/c) + sqrt(2)*(-I*pi*cos(1/4*(4*a*c + 1)/c) + pi*sin(1/4 *(4*a*c + 1)/c))*sqrt(c/pi)*fresnel_sin(-1/2*sqrt(2)*(2*c*x - I)*sqrt(c/pi )/c))/c
\[ \int e^x \cos \left (a+c x^2\right ) \, dx=\int e^{x} \cos {\left (a + c x^{2} \right )}\, dx \] Input:
integrate(exp(x)*cos(c*x**2+a),x)
Output:
Integral(exp(x)*cos(a + c*x**2), x)
Time = 0.04 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.87 \[ \int e^x \cos \left (a+c x^2\right ) \, dx=-\frac {\sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (\frac {4 \, a c + 1}{4 \, c}\right ) + \left (i + 1\right ) \, \sin \left (\frac {4 \, a c + 1}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 i \, c x - 1}{2 \, \sqrt {i \, c}}\right ) + {\left (\left (i + 1\right ) \, \cos \left (\frac {4 \, a c + 1}{4 \, c}\right ) + \left (i - 1\right ) \, \sin \left (\frac {4 \, a c + 1}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 i \, c x + 1}{2 \, \sqrt {-i \, c}}\right )\right )}}{8 \, \sqrt {c}} \] Input:
integrate(exp(x)*cos(c*x^2+a),x, algorithm="maxima")
Output:
-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)
Time = 0.11 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.10 \[ \int e^x \cos \left (a+c x^2\right ) \, dx=-\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} {\left (2 \, x + \frac {i}{c}\right )} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {4 i \, a c + i}{4 \, c}\right )}}{4 \, {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} {\left (2 \, x - \frac {i}{c}\right )} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {-4 i \, a c - i}{4 \, c}\right )}}{4 \, {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} \] Input:
integrate(exp(x)*cos(c*x^2+a),x, algorithm="giac")
Output:
-1/4*sqrt(2)*sqrt(pi)*erf(-1/4*sqrt(2)*(2*x + I/c)*(I*c/abs(c) + 1)*sqrt(a bs(c)))*e^(-1/4*(4*I*a*c + I)/c)/((I*c/abs(c) + 1)*sqrt(abs(c))) - 1/4*sqr t(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)))
Timed out. \[ \int e^x \cos \left (a+c x^2\right ) \, dx=\int {\mathrm {e}}^x\,\cos \left (c\,x^2+a\right ) \,d x \] Input:
int(exp(x)*cos(a + c*x^2),x)
Output:
int(exp(x)*cos(a + c*x^2), x)
\[ \int e^x \cos \left (a+c x^2\right ) \, dx=-e^{x}+2 \left (\int \frac {e^{x}}{\tan \left (\frac {c \,x^{2}}{2}+\frac {a}{2}\right )^{2}+1}d x \right ) \] Input:
int(exp(x)*cos(c*x^2+a),x)
Output:
- e**x + 2*int(e**x/(tan((a + c*x**2)/2)**2 + 1),x)