\(\int e^x \cos ^4(x) \, dx\) [18]

Optimal result

Integrand size = 8, antiderivative size = 54 \[ \int e^x \cos ^4(x) \, dx=\frac {24 e^x}{85}+\frac {12}{85} e^x \cos ^2(x)+\frac {1}{17} e^x \cos ^4(x)+\frac {24}{85} e^x \cos (x) \sin (x)+\frac {4}{17} e^x \cos ^3(x) \sin (x) \]

[Out]

24/85*exp(x)+12/85*exp(x)*cos(x)^2+1/17*exp(x)*cos(x)^4+24/85*exp(x)*cos(x)*sin(x)+4/17*exp(x)*cos(x)^3*sin(x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4520, 2225} \[ \int e^x \cos ^4(x) \, dx=\frac {24 e^x}{85}+\frac {1}{17} e^x \cos ^4(x)+\frac {12}{85} e^x \cos ^2(x)+\frac {4}{17} e^x \sin (x) \cos ^3(x)+\frac {24}{85} e^x \sin (x) \cos (x) \]

[In]

Int[E^x*Cos[x]^4,x]

[Out]

(24*E^x)/85 + (12*E^x*Cos[x]^2)/85 + (E^x*Cos[x]^4)/17 + (24*E^x*Cos[x]*Sin[x])/85 + (4*E^x*Cos[x]^3*Sin[x])/1
7

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 4520

Int[Cos[(d_.) + (e_.)*(x_)]^(m_)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x
))*(Cos[d + e*x]^m/(e^2*m^2 + b^2*c^2*Log[F]^2)), x] + (Dist[(m*(m - 1)*e^2)/(e^2*m^2 + b^2*c^2*Log[F]^2), Int
[F^(c*(a + b*x))*Cos[d + e*x]^(m - 2), x], x] + Simp[e*m*F^(c*(a + b*x))*Sin[d + e*x]*(Cos[d + e*x]^(m - 1)/(e
^2*m^2 + b^2*c^2*Log[F]^2)), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*m^2 + b^2*c^2*Log[F]^2, 0] && GtQ[
m, 1]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{17} e^x \cos ^4(x)+\frac {4}{17} e^x \cos ^3(x) \sin (x)+\frac {12}{17} \int e^x \cos ^2(x) \, dx \\ & = \frac {12}{85} e^x \cos ^2(x)+\frac {1}{17} e^x \cos ^4(x)+\frac {24}{85} e^x \cos (x) \sin (x)+\frac {4}{17} e^x \cos ^3(x) \sin (x)+\frac {24 \int e^x \, dx}{85} \\ & = \frac {24 e^x}{85}+\frac {12}{85} e^x \cos ^2(x)+\frac {1}{17} e^x \cos ^4(x)+\frac {24}{85} e^x \cos (x) \sin (x)+\frac {4}{17} e^x \cos ^3(x) \sin (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.61 \[ \int e^x \cos ^4(x) \, dx=\frac {1}{680} e^x (255+68 \cos (2 x)+5 \cos (4 x)+136 \sin (2 x)+20 \sin (4 x)) \]

[In]

Integrate[E^x*Cos[x]^4,x]

[Out]

(E^x*(255 + 68*Cos[2*x] + 5*Cos[4*x] + 136*Sin[2*x] + 20*Sin[4*x]))/680

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.57

method	result	size
parallelrisch	\(\frac {{\mathrm e}^{x} \left (255+5 \cos \left (4 x \right )+68 \cos \left (2 x \right )+136 \sin \left (2 x \right )+20 \sin \left (4 x \right )\right )}{680}\)	\(31\)
default	\(\frac {\left (\cos \left (x \right )+4 \sin \left (x \right )\right ) {\mathrm e}^{x} \cos \left (x \right )^{3}}{17}+\frac {12 \left (\cos \left (x \right )+2 \sin \left (x \right )\right ) {\mathrm e}^{x} \cos \left (x \right )}{85}+\frac {24 \,{\mathrm e}^{x}}{85}\)	\(34\)
risch	\(\frac {3 \,{\mathrm e}^{x}}{8}+\frac {{\mathrm e}^{\left (1+4 i\right ) x}}{272}-\frac {i {\mathrm e}^{\left (1+4 i\right ) x}}{68}+\frac {{\mathrm e}^{\left (1+2 i\right ) x}}{20}-\frac {i {\mathrm e}^{\left (1+2 i\right ) x}}{10}+\frac {{\mathrm e}^{\left (1-2 i\right ) x}}{20}+\frac {i {\mathrm e}^{\left (1-2 i\right ) x}}{10}+\frac {{\mathrm e}^{\left (1-4 i\right ) x}}{272}+\frac {i {\mathrm e}^{\left (1-4 i\right ) x}}{68}\)	\(74\)
norman	\(\frac {\frac {88 \,{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )}{85}+\frac {76 \,{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )^{2}}{85}-\frac {72 \,{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )^{3}}{85}+\frac {30 \,{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )^{4}}{17}+\frac {72 \,{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )^{5}}{85}+\frac {76 \,{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )^{6}}{85}-\frac {88 \,{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )^{7}}{85}+\frac {41 \,{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )^{8}}{85}+\frac {41 \,{\mathrm e}^{x}}{85}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{4}}\)	\(95\)

[In]

int(exp(x)*cos(x)^4,x,method=_RETURNVERBOSE)

[Out]

1/680*exp(x)*(255+5*cos(4*x)+68*cos(2*x)+136*sin(2*x)+20*sin(4*x))

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.67 \[ \int e^x \cos ^4(x) \, dx=\frac {4}{85} \, {\left (5 \, \cos \left (x\right )^{3} + 6 \, \cos \left (x\right )\right )} e^{x} \sin \left (x\right ) + \frac {1}{85} \, {\left (5 \, \cos \left (x\right )^{4} + 12 \, \cos \left (x\right )^{2} + 24\right )} e^{x} \]

[In]

integrate(exp(x)*cos(x)^4,x, algorithm="fricas")

[Out]

4/85*(5*cos(x)^3 + 6*cos(x))*e^x*sin(x) + 1/85*(5*cos(x)^4 + 12*cos(x)^2 + 24)*e^x

Sympy [A] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.30 \[ \int e^x \cos ^4(x) \, dx=\frac {24 e^{x} \sin ^{4}{\left (x \right )}}{85} + \frac {24 e^{x} \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{85} + \frac {12 e^{x} \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{17} + \frac {44 e^{x} \sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{85} + \frac {41 e^{x} \cos ^{4}{\left (x \right )}}{85} \]

[In]

integrate(exp(x)*cos(x)**4,x)

[Out]

24*exp(x)*sin(x)**4/85 + 24*exp(x)*sin(x)**3*cos(x)/85 + 12*exp(x)*sin(x)**2*cos(x)**2/17 + 44*exp(x)*sin(x)*c
os(x)**3/85 + 41*exp(x)*cos(x)**4/85

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.69 \[ \int e^x \cos ^4(x) \, dx=\frac {1}{136} \, \cos \left (4 \, x\right ) e^{x} + \frac {1}{10} \, \cos \left (2 \, x\right ) e^{x} + \frac {1}{34} \, e^{x} \sin \left (4 \, x\right ) + \frac {1}{5} \, e^{x} \sin \left (2 \, x\right ) + \frac {3}{8} \, e^{x} \]

[In]

integrate(exp(x)*cos(x)^4,x, algorithm="maxima")

[Out]

1/136*cos(4*x)*e^x + 1/10*cos(2*x)*e^x + 1/34*e^x*sin(4*x) + 1/5*e^x*sin(2*x) + 3/8*e^x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.65 \[ \int e^x \cos ^4(x) \, dx=\frac {1}{136} \, {\left (\cos \left (4 \, x\right ) + 4 \, \sin \left (4 \, x\right )\right )} e^{x} + \frac {1}{10} \, {\left (\cos \left (2 \, x\right ) + 2 \, \sin \left (2 \, x\right )\right )} e^{x} + \frac {3}{8} \, e^{x} \]

[In]

integrate(exp(x)*cos(x)^4,x, algorithm="giac")

[Out]

1/136*(cos(4*x) + 4*sin(4*x))*e^x + 1/10*(cos(2*x) + 2*sin(2*x))*e^x + 3/8*e^x

Mupad [B] (verification not implemented)

Time = 27.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.76 \[ \int e^x \cos ^4(x) \, dx=\frac {3\,{\mathrm {e}}^x}{8}+\frac {{\mathrm {e}}^x\,\left (\frac {4\,\cos \left (2\,x\right )}{5}+\frac {8\,\sin \left (2\,x\right )}{5}+\frac {2\,{\cos \left (2\,x\right )}^2}{17}+\frac {8\,\cos \left (2\,x\right )\,\sin \left (2\,x\right )}{17}-\frac {1}{17}\right )}{8} \]

[In]

int(exp(x)*cos(x)^4,x)

[Out]

(3*exp(x))/8 + (exp(x)*((4*cos(2*x))/5 + (8*sin(2*x))/5 + (2*cos(2*x)^2)/17 + (8*cos(2*x)*sin(2*x))/17 - 1/17)
)/8