3.18 \(\int e^x \cos ^4(x) \, dx\)

Optimal. Leaf size=54 \[ \frac{24 e^x}{85}+\frac{1}{17} e^x \cos ^4(x)+\frac{12}{85} e^x \cos ^2(x)+\frac{4}{17} e^x \sin (x) \cos ^3(x)+\frac{24}{85} e^x \sin (x) \cos (x) \]

[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

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Rubi [A]  time = 0.0289402, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4435, 2194} \[ \frac{24 e^x}{85}+\frac{1}{17} e^x \cos ^4(x)+\frac{12}{85} e^x \cos ^2(x)+\frac{4}{17} e^x \sin (x) \cos ^3(x)+\frac{24}{85} e^x \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[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 4435

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]

Rule 2194

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]

Rubi steps

\begin{align*} \int e^x \cos ^4(x) \, dx &=\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]  time = 0.026743, size = 33, normalized size = 0.61 \[ \frac{1}{680} e^x (136 \sin (2 x)+20 \sin (4 x)+68 \cos (2 x)+5 \cos (4 x)+255) \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.007, size = 34, normalized size = 0.6 \begin{align*}{\frac{ \left ( \cos \left ( x \right ) +4\,\sin \left ( x \right ) \right ){{\rm e}^{x}} \left ( \cos \left ( x \right ) \right ) ^{3}}{17}}+{\frac{ \left ( 12\,\cos \left ( x \right ) +24\,\sin \left ( x \right ) \right ){{\rm e}^{x}}\cos \left ( x \right ) }{85}}+{\frac{24\,{{\rm e}^{x}}}{85}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02456, size = 50, normalized size = 0.93 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 0.468187, size = 113, normalized size = 2.09 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 8.32891, size = 70, normalized size = 1.3 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.15466, size = 47, normalized size = 0.87 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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