3.656 \(\int e^{\cos (1+3 x)} \cos (1+3 x) \sin (1+3 x) \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{3} e^{\cos (3 x+1)}-\frac{1}{3} e^{\cos (3 x+1)} \cos (3 x+1) \]

[Out]

E^Cos[1 + 3*x]/3 - (E^Cos[1 + 3*x]*Cos[1 + 3*x])/3

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Rubi [A]  time = 0.0226498, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4335, 2176, 2194} \[ \frac{1}{3} e^{\cos (3 x+1)}-\frac{1}{3} e^{\cos (3 x+1)} \cos (3 x+1) \]

Antiderivative was successfully verified.

[In]

Int[E^Cos[1 + 3*x]*Cos[1 + 3*x]*Sin[1 + 3*x],x]

[Out]

E^Cos[1 + 3*x]/3 - (E^Cos[1 + 3*x]*Cos[1 + 3*x])/3

Rule 4335

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[d/(
b*c), Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a
+ b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])

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

Mathematica [A]  time = 0.121238, size = 24, normalized size = 0.77 \[ \frac{2}{3} \sin ^2\left (\frac{1}{2} (3 x+1)\right ) e^{\cos (3 x+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[E^Cos[1 + 3*x]*Cos[1 + 3*x]*Sin[1 + 3*x],x]

[Out]

(2*E^Cos[1 + 3*x]*Sin[(1 + 3*x)/2]^2)/3

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Maple [A]  time = 0.005, size = 26, normalized size = 0.8 \begin{align*}{\frac{{{\rm e}^{\cos \left ( 1+3\,x \right ) }}}{3}}-{\frac{{{\rm e}^{\cos \left ( 1+3\,x \right ) }}\cos \left ( 1+3\,x \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(cos(1+3*x))*cos(1+3*x)*sin(1+3*x),x)

[Out]

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

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Maxima [A]  time = 0.976038, size = 23, normalized size = 0.74 \begin{align*} -\frac{1}{3} \,{\left (\cos \left (3 \, x + 1\right ) - 1\right )} e^{\left (\cos \left (3 \, x + 1\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(cos(1+3*x))*cos(1+3*x)*sin(1+3*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.90291, size = 57, normalized size = 1.84 \begin{align*} -\frac{1}{3} \,{\left (\cos \left (3 \, x + 1\right ) - 1\right )} e^{\left (\cos \left (3 \, x + 1\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(cos(1+3*x))*cos(1+3*x)*sin(1+3*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.819937, size = 26, normalized size = 0.84 \begin{align*} - \frac{e^{\cos{\left (3 x + 1 \right )}} \cos{\left (3 x + 1 \right )}}{3} + \frac{e^{\cos{\left (3 x + 1 \right )}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(cos(1+3*x))*cos(1+3*x)*sin(1+3*x),x)

[Out]

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

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Giac [A]  time = 1.10971, size = 23, normalized size = 0.74 \begin{align*} -\frac{1}{3} \,{\left (\cos \left (3 \, x + 1\right ) - 1\right )} e^{\left (\cos \left (3 \, x + 1\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(cos(1+3*x))*cos(1+3*x)*sin(1+3*x),x, algorithm="giac")

[Out]

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