3.23 \(\int e^x \cos ^2(e^x) \sin (e^x) \, dx\)

Optimal. Leaf size=10 \[ -\frac{1}{3} \cos ^3\left (e^x\right ) \]

[Out]

-Cos[E^x]^3/3

________________________________________________________________________________________

Rubi [A]  time = 0.0274706, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2282, 2565, 30} \[ -\frac{1}{3} \cos ^3\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x*Cos[E^x]^2*Sin[E^x],x]

[Out]

-Cos[E^x]^3/3

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int e^x \cos ^2\left (e^x\right ) \sin \left (e^x\right ) \, dx &=\operatorname{Subst}\left (\int \cos ^2(x) \sin (x) \, dx,x,e^x\right )\\ &=-\operatorname{Subst}\left (\int x^2 \, dx,x,\cos \left (e^x\right )\right )\\ &=-\frac{1}{3} \cos ^3\left (e^x\right )\\ \end{align*}

Mathematica [A]  time = 0.0203797, size = 19, normalized size = 1.9 \[ -\frac{1}{4} \cos \left (e^x\right )-\frac{1}{12} \cos \left (3 e^x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Cos[E^x]^2*Sin[E^x],x]

[Out]

-Cos[E^x]/4 - Cos[3*E^x]/12

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 8, normalized size = 0.8 \begin{align*} -{\frac{ \left ( \cos \left ({{\rm e}^{x}} \right ) \right ) ^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*cos(exp(x))^2*sin(exp(x)),x)

[Out]

-1/3*cos(exp(x))^3

________________________________________________________________________________________

Maxima [A]  time = 0.922909, size = 9, normalized size = 0.9 \begin{align*} -\frac{1}{3} \, \cos \left (e^{x}\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cos(exp(x))^2*sin(exp(x)),x, algorithm="maxima")

[Out]

-1/3*cos(e^x)^3

________________________________________________________________________________________

Fricas [A]  time = 1.8898, size = 23, normalized size = 2.3 \begin{align*} -\frac{1}{3} \, \cos \left (e^{x}\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cos(exp(x))^2*sin(exp(x)),x, algorithm="fricas")

[Out]

-1/3*cos(e^x)^3

________________________________________________________________________________________

Sympy [A]  time = 2.08616, size = 8, normalized size = 0.8 \begin{align*} - \frac{\cos ^{3}{\left (e^{x} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cos(exp(x))**2*sin(exp(x)),x)

[Out]

-cos(exp(x))**3/3

________________________________________________________________________________________

Giac [A]  time = 1.08064, size = 9, normalized size = 0.9 \begin{align*} -\frac{1}{3} \, \cos \left (e^{x}\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*cos(exp(x))^2*sin(exp(x)),x, algorithm="giac")

[Out]

-1/3*cos(e^x)^3