3.9 \(\int e^x \sin ^4(x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0256499, 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 = {4434, 2194} \[ \frac{24 e^x}{85}+\frac{1}{17} e^x \sin ^4(x)+\frac{12}{85} e^x \sin ^2(x)-\frac{4}{17} e^x \sin ^3(x) \cos (x)-\frac{24}{85} e^x \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4434

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)]^(n_), x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*
x))*Sin[d + e*x]^n)/(e^2*n^2 + b^2*c^2*Log[F]^2), x] + (Dist[(n*(n - 1)*e^2)/(e^2*n^2 + b^2*c^2*Log[F]^2), Int
[F^(c*(a + b*x))*Sin[d + e*x]^(n - 2), x], x] - Simp[(e*n*F^(c*(a + b*x))*Cos[d + e*x]*Sin[d + e*x]^(n - 1))/(
e^2*n^2 + b^2*c^2*Log[F]^2), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 + b^2*c^2*Log[F]^2, 0] && GtQ[
n, 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 \sin ^4(x) \, dx &=-\frac{4}{17} e^x \cos (x) \sin ^3(x)+\frac{1}{17} e^x \sin ^4(x)+\frac{12}{17} \int e^x \sin ^2(x) \, dx\\ &=-\frac{24}{85} e^x \cos (x) \sin (x)+\frac{12}{85} e^x \sin ^2(x)-\frac{4}{17} e^x \cos (x) \sin ^3(x)+\frac{1}{17} e^x \sin ^4(x)+\frac{24 \int e^x \, dx}{85}\\ &=\frac{24 e^x}{85}-\frac{24}{85} e^x \cos (x) \sin (x)+\frac{12}{85} e^x \sin ^2(x)-\frac{4}{17} e^x \cos (x) \sin ^3(x)+\frac{1}{17} e^x \sin ^4(x)\\ \end{align*}

Mathematica [A]  time = 0.0376441, 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*Sin[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 = 0.01, size = 34, normalized size = 0.6 \begin{align*}{\frac{ \left ( \sin \left ( x \right ) -4\,\cos \left ( x \right ) \right ){{\rm e}^{x}} \left ( \sin \left ( x \right ) \right ) ^{3}}{17}}+{\frac{ \left ( 12\,\sin \left ( x \right ) -24\,\cos \left ( x \right ) \right ){{\rm e}^{x}}\sin \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)*sin(x)^4,x)

[Out]

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

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Maxima [A]  time = 1.02925, 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)*sin(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 = 0.462996, size = 115, normalized size = 2.13 \begin{align*} \frac{4}{85} \,{\left (5 \, \cos \left (x\right )^{3} - 11 \, \cos \left (x\right )\right )} e^{x} \sin \left (x\right ) + \frac{1}{85} \,{\left (5 \, \cos \left (x\right )^{4} - 22 \, \cos \left (x\right )^{2} + 41\right )} e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/85*(5*cos(x)^3 - 11*cos(x))*e^x*sin(x) + 1/85*(5*cos(x)^4 - 22*cos(x)^2 + 41)*e^x

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Sympy [A]  time = 6.4161, size = 70, normalized size = 1.3 \begin{align*} \frac{41 e^{x} \sin ^{4}{\left (x \right )}}{85} - \frac{44 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{24 e^{x} \sin{\left (x \right )} \cos ^{3}{\left (x \right )}}{85} + \frac{24 e^{x} \cos ^{4}{\left (x \right )}}{85} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14763, 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)*sin(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