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

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

[Out]

E^(4 + Sin[x])

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Rubi [A]  time = 0.0095759, antiderivative size = 6, normalized size of antiderivative = 1., 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 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])

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^{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 = 0.0105102, size = 6, normalized size = 1. \[ e^{\sin (x)+4} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(4 + Sin[x])*Cos[x],x]

[Out]

E^(4 + Sin[x])

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Maple [A]  time = 0.006, size = 6, normalized size = 1. \begin{align*}{{\rm e}^{4+\sin \left ( x \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(4+sin(x))

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Maxima [A]  time = 0.957979, size = 7, normalized size = 1.17 \begin{align*} e^{\left (\sin \left (x\right ) + 4\right )} \end{align*}

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)

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Fricas [A]  time = 1.95384, size = 22, normalized size = 3.67 \begin{align*} e^{\left (\sin \left (x\right ) + 4\right )} \end{align*}

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)

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Sympy [A]  time = 0.725248, size = 7, normalized size = 1.17 \begin{align*} e^{4} e^{\sin{\left (x \right )}} \end{align*}

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))

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Giac [A]  time = 1.11814, size = 7, normalized size = 1.17 \begin{align*} e^{\left (\sin \left (x\right ) + 4\right )} \end{align*}

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)