3.678 \(\int e^{\sin (x)} \cos (x) \sin (x) \, dx\)

Optimal. Leaf size=14 \[ e^{\sin (x)} \sin (x)-e^{\sin (x)} \]

[Out]

-E^Sin[x] + E^Sin[x]*Sin[x]

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Rubi [A]  time = 0.0163624, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4334, 2176, 2194} \[ e^{\sin (x)} \sin (x)-e^{\sin (x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-E^Sin[x] + E^Sin[x]*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 2176

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] &&  !$UseGamma === True

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

Mathematica [A]  time = 0.0098843, size = 9, normalized size = 0.64 \[ e^{\sin (x)} (\sin (x)-1) \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^Sin[x]*(-1 + Sin[x])

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Maple [A]  time = 0.004, size = 13, normalized size = 0.9 \begin{align*} -{{\rm e}^{\sin \left ( x \right ) }}+{{\rm e}^{\sin \left ( x \right ) }}\sin \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.964944, size = 11, normalized size = 0.79 \begin{align*}{\left (\sin \left (x\right ) - 1\right )} e^{\sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sin(x) - 1)*e^sin(x)

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Fricas [A]  time = 2.16587, size = 31, normalized size = 2.21 \begin{align*}{\left (\sin \left (x\right ) - 1\right )} e^{\sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sin(x) - 1)*e^sin(x)

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Sympy [A]  time = 0.837246, size = 12, normalized size = 0.86 \begin{align*} e^{\sin{\left (x \right )}} \sin{\left (x \right )} - e^{\sin{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(sin(x))*sin(x) - exp(sin(x))

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Giac [A]  time = 1.08536, size = 11, normalized size = 0.79 \begin{align*}{\left (\sin \left (x\right ) - 1\right )} e^{\sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sin(x) - 1)*e^sin(x)