\(\int e^{4+\sin (x)} \cos (x) \, dx\) [681]

Optimal result

Integrand size = 9, antiderivative size = 6 \[ \int e^{4+\sin (x)} \cos (x) \, dx=e^{4+\sin (x)} \]

[Out]

exp(4+sin(x))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4419, 2225} \[ \int e^{4+\sin (x)} \cos (x) \, dx=e^{\sin (x)+4} \]

[In]

Int[E^(4 + Sin[x])*Cos[x],x]

[Out]

E^(4 + Sin[x])

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 4419

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int e^{4+\sin (x)} \cos (x) \, dx=e^{4+\sin (x)} \]

[In]

Integrate[E^(4 + Sin[x])*Cos[x],x]

[Out]

E^(4 + Sin[x])

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\({\mathrm e}^{4+\sin \left (x \right )}\)	\(6\)
default	\({\mathrm e}^{4+\sin \left (x \right )}\)	\(6\)
risch	\({\mathrm e}^{4+\sin \left (x \right )}\)	\(6\)
parallelrisch	\({\mathrm e}^{4+\sin \left (x \right )}\)	\(6\)
norman	\(\frac {\tan \left (\frac {x}{2}\right )^{2} {\mathrm e}^{4+\frac {2 \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}}+{\mathrm e}^{4+\frac {2 \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}}}{1+\tan \left (\frac {x}{2}\right )^{2}}\)	\(58\)

[In]

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

[Out]

exp(4+sin(x))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83 \[ \int e^{4+\sin (x)} \cos (x) \, dx=e^{\left (\sin \left (x\right ) + 4\right )} \]

[In]

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

[Out]

e^(sin(x) + 4)

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.17 \[ \int e^{4+\sin (x)} \cos (x) \, dx=e^{4} e^{\sin {\left (x \right )}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83 \[ \int e^{4+\sin (x)} \cos (x) \, dx=e^{\left (\sin \left (x\right ) + 4\right )} \]

[In]

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

[Out]

e^(sin(x) + 4)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83 \[ \int e^{4+\sin (x)} \cos (x) \, dx=e^{\left (\sin \left (x\right ) + 4\right )} \]

[In]

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

[Out]

e^(sin(x) + 4)

Mupad [B] (verification not implemented)

Time = 26.31 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int e^{4+\sin (x)} \cos (x) \, dx={\mathrm {e}}^{\sin \left (x\right )}\,{\mathrm {e}}^4 \]

[In]

int(exp(sin(x) + 4)*cos(x),x)

[Out]

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