3.746 \(\int e^{n \cos (a+b x)} \sin (2 (a+b x)) \, dx\)

Optimal. Leaf size=43 \[ \frac{2 e^{n \cos (a+b x)}}{b n^2}-\frac{2 \cos (a+b x) e^{n \cos (a+b x)}}{b n} \]

[Out]

(2*E^(n*Cos[a + b*x]))/(b*n^2) - (2*E^(n*Cos[a + b*x])*Cos[a + b*x])/(b*n)

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Rubi [A]  time = 0.0349637, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {12, 2176, 2194} \[ \frac{2 e^{n \cos (a+b x)}}{b n^2}-\frac{2 \cos (a+b x) e^{n \cos (a+b x)}}{b n} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*Cos[a + b*x])*Sin[2*(a + b*x)],x]

[Out]

(2*E^(n*Cos[a + b*x]))/(b*n^2) - (2*E^(n*Cos[a + b*x])*Cos[a + b*x])/(b*n)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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^{n \cos (a+b x)} \sin (2 (a+b x)) \, dx &=-\frac{\operatorname{Subst}\left (\int 2 e^{n x} x \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{2 \operatorname{Subst}\left (\int e^{n x} x \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{2 e^{n \cos (a+b x)} \cos (a+b x)}{b n}+\frac{2 \operatorname{Subst}\left (\int e^{n x} \, dx,x,\cos (a+b x)\right )}{b n}\\ &=\frac{2 e^{n \cos (a+b x)}}{b n^2}-\frac{2 e^{n \cos (a+b x)} \cos (a+b x)}{b n}\\ \end{align*}

Mathematica [A]  time = 0.0346088, size = 28, normalized size = 0.65 \[ -\frac{2 e^{n \cos (a+b x)} (n \cos (a+b x)-1)}{b n^2} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(n*Cos[a + b*x])*Sin[2*(a + b*x)],x]

[Out]

(-2*E^(n*Cos[a + b*x])*(-1 + n*Cos[a + b*x]))/(b*n^2)

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Maple [C]  time = 0., size = 106, normalized size = 2.5 \begin{align*} -{\frac{{{\rm e}^{n\cos \left ( bx \right ) \cos \left ( a \right ) -n\sin \left ( bx \right ) \sin \left ( a \right ) }}{{\rm e}^{-ibx}}{{\rm e}^{-ia}}}{bn}}-{\frac{{{\rm e}^{n\cos \left ( bx \right ) \cos \left ( a \right ) -n\sin \left ( bx \right ) \sin \left ( a \right ) }}{{\rm e}^{ibx}}{{\rm e}^{ia}}}{bn}}+2\,{\frac{{{\rm e}^{-n \left ( \sin \left ( bx \right ) \sin \left ( a \right ) -\cos \left ( bx \right ) \cos \left ( a \right ) \right ) }}}{b{n}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*cos(b*x+a))*sin(2*b*x+2*a),x)

[Out]

-1/b/n*exp(n*cos(b*x)*cos(a)-n*sin(b*x)*sin(a))*exp(-I*b*x)*exp(-I*a)-1/b/n*exp(n*cos(b*x)*cos(a)-n*sin(b*x)*s
in(a))*exp(I*b*x)*exp(I*a)+2/b/n^2*exp(-n*(sin(b*x)*sin(a)-cos(b*x)*cos(a)))

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Maxima [A]  time = 1.05626, size = 50, normalized size = 1.16 \begin{align*} -\frac{2 \,{\left (n \cos \left (b x + a\right ) e^{\left (n \cos \left (b x + a\right )\right )} - e^{\left (n \cos \left (b x + a\right )\right )}\right )}}{b n^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*cos(b*x+a))*sin(2*b*x+2*a),x, algorithm="maxima")

[Out]

-2*(n*cos(b*x + a)*e^(n*cos(b*x + a)) - e^(n*cos(b*x + a)))/(b*n^2)

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Fricas [A]  time = 2.03885, size = 70, normalized size = 1.63 \begin{align*} -\frac{2 \,{\left (n \cos \left (b x + a\right ) - 1\right )} e^{\left (n \cos \left (b x + a\right )\right )}}{b n^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*cos(b*x+a))*sin(2*b*x+2*a),x, algorithm="fricas")

[Out]

-2*(n*cos(b*x + a) - 1)*e^(n*cos(b*x + a))/(b*n^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{n \cos{\left (a + b x \right )}} \sin{\left (2 a + 2 b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*cos(b*x+a))*sin(2*b*x+2*a),x)

[Out]

Integral(exp(n*cos(a + b*x))*sin(2*a + 2*b*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (n \cos \left (b x + a\right )\right )} \sin \left (2 \, b x + 2 \, a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*cos(b*x+a))*sin(2*b*x+2*a),x, algorithm="giac")

[Out]

integrate(e^(n*cos(b*x + a))*sin(2*b*x + 2*a), x)