3.681 \(\int e^{4+\sin (x)} \cos (x) \, dx\)

Optimal. Leaf size=6 \[ e^{\sin (x)+4} \]

[Out]

exp(4+sin(x))

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4334, 2194} \[ e^{\sin (x)+4} \]

Antiderivative was successfully verified.

[In]

Int[E^(4 + Sin[x])*Cos[x],x]

[Out]

E^(4 + Sin[x])

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]

Rule 4334

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b
*c), Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a +
b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])

Rubi steps

\begin {align*} \int e^{4+\sin (x)} \cos (x) \, dx &=\operatorname {Subst}\left (\int e^{4+x} \, dx,x,\sin (x)\right )\\ &=e^{4+\sin (x)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 6, normalized size = 1.00 \[ e^{\sin (x)+4} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(4 + Sin[x])*Cos[x],x]

[Out]

E^(4 + Sin[x])

________________________________________________________________________________________

fricas [A]  time = 0.94, size = 5, normalized size = 0.83 \[ e^{\left (\sin \relax (x) + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(sin(x) + 4)

________________________________________________________________________________________

giac [A]  time = 0.12, size = 5, normalized size = 0.83 \[ e^{\left (\sin \relax (x) + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(sin(x) + 4)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 6, normalized size = 1.00 \[ {\mathrm e}^{4+\sin \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(4+sin(x))*cos(x),x)

[Out]

exp(4+sin(x))

________________________________________________________________________________________

maxima [A]  time = 0.32, size = 5, normalized size = 0.83 \[ e^{\left (\sin \relax (x) + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(sin(x) + 4)

________________________________________________________________________________________

mupad [B]  time = 2.91, size = 6, normalized size = 1.00 \[ {\mathrm {e}}^{\sin \relax (x)}\,{\mathrm {e}}^4 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(sin(x) + 4)*cos(x),x)

[Out]

exp(sin(x))*exp(4)

________________________________________________________________________________________

sympy [A]  time = 0.54, size = 7, normalized size = 1.17 \[ e^{4} e^{\sin {\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(4)*exp(sin(x))

________________________________________________________________________________________