3.1.9 \(\int e^x \sin ^4(x) \, dx\) [9]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4519, 2225} \begin {gather*} \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) \end {gather*}

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 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 4519

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]

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*}

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Mathematica [A]
time = 0.04, size = 33, normalized size = 0.61 \begin {gather*} \frac {1}{680} e^x (255-68 \cos (2 x)+5 \cos (4 x)-136 \sin (2 x)+20 \sin (4 x)) \end {gather*}

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 = 0.09, size = 34, normalized size = 0.63

method	result	size
default	\(\frac {\left (\sin \left (x \right )-4 \cos \left (x \right )\right ) {\mathrm e}^{x} \left (\sin ^{3}\left (x \right )\right )}{17}+\frac {12 \left (\sin \left (x \right )-2 \cos \left (x \right )\right ) {\mathrm e}^{x} \sin \left (x \right )}{85}+\frac {24 \,{\mathrm e}^{x}}{85}\)	\(34\)
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	\(\frac {-\frac {48 \,{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )}{85}+\frac {144 \,{\mathrm e}^{x} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{85}-\frac {208 \,{\mathrm e}^{x} \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{85}+\frac {64 \,{\mathrm e}^{x} \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{17}+\frac {208 \,{\mathrm e}^{x} \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{85}+\frac {144 \,{\mathrm e}^{x} \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{85}+\frac {48 \,{\mathrm e}^{x} \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{85}+\frac {24 \,{\mathrm e}^{x} \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{85}+\frac {24 \,{\mathrm e}^{x}}{85}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{4}}\)	\(95\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.28, size = 37, normalized size = 0.69 \begin {gather*} \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 {gather*}

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 = 2.87, size = 36, normalized size = 0.67 \begin {gather*} \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 {gather*}

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 = 0.63, size = 70, normalized size = 1.30 \begin {gather*} \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 {gather*}

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 = 0.41, size = 35, normalized size = 0.65 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 0.05, size = 41, normalized size = 0.76 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*sin(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

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