\(\int e^{\sin (x)} \cos (x) \, dx\) [122]

Optimal result

Integrand size = 7, antiderivative size = 4 \[ \int e^{\sin (x)} \cos (x) \, dx=e^{\sin (x)} \]

[Out]

exp(sin(x))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4419, 2225} \[ \int e^{\sin (x)} \cos (x) \, dx=e^{\sin (x)} \]

[In]

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

[Out]

E^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^x \, dx,x,\sin (x)\right ) \\ & = e^{\sin (x)} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

Integrate[E^Sin[x]*Cos[x],x]

[Out]

E^Sin[x]

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\({\mathrm e}^{\sin \left (x \right )}\)	\(4\)
default	\({\mathrm e}^{\sin \left (x \right )}\)	\(4\)
risch	\({\mathrm e}^{\sin \left (x \right )}\)	\(4\)

[In]

int(exp(sin(x))/tan(x)/csc(x),x,method=_RETURNVERBOSE)

[Out]

exp(sin(x))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.75 \[ \int e^{\sin (x)} \cos (x) \, dx=e^{\sin \left (x\right )} \]

[In]

integrate(exp(sin(x))/tan(x)/csc(x),x, algorithm="fricas")

[Out]

e^sin(x)

Sympy [F]

\[ \int e^{\sin (x)} \cos (x) \, dx=\int \frac {e^{\sin {\left (x \right )}}}{\tan {\left (x \right )} \csc {\left (x \right )}}\, dx \]

[In]

integrate(exp(sin(x))/tan(x)/csc(x),x)

[Out]

Integral(exp(sin(x))/(tan(x)*csc(x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.75 \[ \int e^{\sin (x)} \cos (x) \, dx=e^{\sin \left (x\right )} \]

[In]

integrate(exp(sin(x))/tan(x)/csc(x),x, algorithm="maxima")

[Out]

e^sin(x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.75 \[ \int e^{\sin (x)} \cos (x) \, dx=e^{\sin \left (x\right )} \]

[In]

integrate(exp(sin(x))/tan(x)/csc(x),x, algorithm="giac")

[Out]

e^sin(x)

Mupad [B] (verification not implemented)

Time = 17.48 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.75 \[ \int e^{\sin (x)} \cos (x) \, dx={\mathrm {e}}^{\sin \left (x\right )} \]

[In]

int((exp(sin(x))*sin(x))/tan(x),x)

[Out]

exp(sin(x))