\(\int e^{\sin (x)} \cos (x) \sin (x) \, dx\) [678]

Optimal result

Integrand size = 9, antiderivative size = 14 \[ \int e^{\sin (x)} \cos (x) \sin (x) \, dx=-e^{\sin (x)}+e^{\sin (x)} \sin (x) \]

[Out]

-exp(sin(x))+exp(sin(x))*sin(x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4419, 2207, 2225} \[ \int e^{\sin (x)} \cos (x) \sin (x) \, dx=e^{\sin (x)} \sin (x)-e^{\sin (x)} \]

[In]

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

[Out]

-E^Sin[x] + E^Sin[x]*Sin[x]

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.64 \[ \int e^{\sin (x)} \cos (x) \sin (x) \, dx=e^{\sin (x)} (-1+\sin (x)) \]

[In]

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

[Out]

E^Sin[x]*(-1 + Sin[x])

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.64

method	result	size
parallelrisch	\(\left (\sin \left (x \right )-1\right ) {\mathrm e}^{\sin \left (x \right )}\)	\(9\)
derivativedivides	\(-{\mathrm e}^{\sin \left (x \right )}+{\mathrm e}^{\sin \left (x \right )} \sin \left (x \right )\)	\(13\)
default	\(-{\mathrm e}^{\sin \left (x \right )}+{\mathrm e}^{\sin \left (x \right )} \sin \left (x \right )\)	\(13\)
risch	\(-{\mathrm e}^{\sin \left (x \right )}+{\mathrm e}^{\sin \left (x \right )} \sin \left (x \right )\)	\(13\)
norman	\(\frac {2 \tan \left (\frac {x}{2}\right ) {\mathrm e}^{\frac {2 \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}}-2 \tan \left (\frac {x}{2}\right )^{2} {\mathrm e}^{\frac {2 \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}}+2 \tan \left (\frac {x}{2}\right )^{3} {\mathrm e}^{\frac {2 \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}}-\tan \left (\frac {x}{2}\right )^{4} {\mathrm e}^{\frac {2 \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}}-{\mathrm e}^{\frac {2 \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{2}}\)	\(130\)

[In]

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

[Out]

(sin(x)-1)*exp(sin(x))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int e^{\sin (x)} \cos (x) \sin (x) \, dx={\left (\sin \left (x\right ) - 1\right )} e^{\sin \left (x\right )} \]

[In]

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

[Out]

(sin(x) - 1)*e^sin(x)

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int e^{\sin (x)} \cos (x) \sin (x) \, dx=e^{\sin {\left (x \right )}} \sin {\left (x \right )} - e^{\sin {\left (x \right )}} \]

[In]

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

[Out]

exp(sin(x))*sin(x) - exp(sin(x))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int e^{\sin (x)} \cos (x) \sin (x) \, dx={\left (\sin \left (x\right ) - 1\right )} e^{\sin \left (x\right )} \]

[In]

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

[Out]

(sin(x) - 1)*e^sin(x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int e^{\sin (x)} \cos (x) \sin (x) \, dx={\left (\sin \left (x\right ) - 1\right )} e^{\sin \left (x\right )} \]

[In]

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

[Out]

(sin(x) - 1)*e^sin(x)

Mupad [B] (verification not implemented)

Time = 26.69 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int e^{\sin (x)} \cos (x) \sin (x) \, dx={\mathrm {e}}^{\sin \left (x\right )}\,\left (\sin \left (x\right )-1\right ) \]

[In]

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

[Out]

exp(sin(x))*(sin(x) - 1)