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

Optimal. Leaf size=24 \[ -\frac{e^{n \cos (a c+b c x)}}{b c n} \]

[Out]

-(E^(n*Cos[a*c + b*c*x])/(b*c*n))

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Rubi [A]  time = 0.0144949, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4335, 2194} \[ -\frac{e^{n \cos (a c+b c x)}}{b c n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^(n*Cos[a*c + b*c*x])/(b*c*n))

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

Mathematica [A]  time = 0.0425428, size = 23, normalized size = 0.96 \[ -\frac{e^{n \cos (c (a+b x))}}{b c n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^(n*Cos[c*(a + b*x)])/(b*c*n))

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Maple [A]  time = 0.01, size = 24, normalized size = 1. \begin{align*} -{\frac{{{\rm e}^{n\cos \left ( bcx+ac \right ) }}}{cbn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(n*cos(b*c*x+a*c))/b/c/n

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Maxima [A]  time = 0.960272, size = 31, normalized size = 1.29 \begin{align*} -\frac{e^{\left (n \cos \left (b c x + a c\right )\right )}}{b c n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(n*cos(b*c*x + a*c))/(b*c*n)

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Fricas [A]  time = 2.0295, size = 45, normalized size = 1.88 \begin{align*} -\frac{e^{\left (n \cos \left (b c x + a c\right )\right )}}{b c n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(n*cos(b*c*x + a*c))/(b*c*n)

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Sympy [A]  time = 2.76905, size = 51, normalized size = 2.12 \begin{align*} \begin{cases} 0 & \text{for}\: b = 0 \wedge c = 0 \wedge n = 0 \\x e^{n \cos{\left (a c \right )}} \sin{\left (a c \right )} & \text{for}\: b = 0 \\0 & \text{for}\: c = 0 \\- \frac{\cos{\left (a c + b c x \right )}}{b c} & \text{for}\: n = 0 \\- \frac{e^{n \cos{\left (a c + b c x \right )}}}{b c n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((0, Eq(b, 0) & Eq(c, 0) & Eq(n, 0)), (x*exp(n*cos(a*c))*sin(a*c), Eq(b, 0)), (0, Eq(c, 0)), (-cos(a*
c + b*c*x)/(b*c), Eq(n, 0)), (-exp(n*cos(a*c + b*c*x))/(b*c*n), True))

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Giac [A]  time = 1.14893, size = 31, normalized size = 1.29 \begin{align*} -\frac{e^{\left (n \cos \left (b c x + a c\right )\right )}}{b c n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(n*cos(b*c*x + a*c))/(b*c*n)