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

Optimal. Leaf size=23 \[ -\frac{e^{n \cos (c (a+b x))}}{b c n} \]

[Out]

-(E^(n*Cos[c*(a + b*x)])/(b*c*n))

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Rubi [A]  time = 0.0149893, antiderivative size = 23, 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} \[ -\frac{e^{n \cos (c (a+b x))}}{b c n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.236888, size = 23, normalized size = 1. \[ -\frac{e^{n \cos (c (a+b x))}}{b c n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^(n*Cos[c*(a + b*x)])/(b*c*n))

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Maple [A]  time = 0.014, 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(b*c*x+a*c))*sin(c*(b*x+a)),x)

[Out]

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

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Maxima [A]  time = 0.98337, size = 31, normalized size = 1.35 \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(b*c*x+a*c))*sin(c*(b*x+a)),x, algorithm="maxima")

[Out]

-e^(n*cos(b*c*x + a*c))/(b*c*n)

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Fricas [A]  time = 1.88868, size = 45, normalized size = 1.96 \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(b*c*x+a*c))*sin(c*(b*x+a)),x, algorithm="fricas")

[Out]

-e^(n*cos(b*c*x + a*c))/(b*c*n)

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Sympy [A]  time = 9.43634, size = 54, normalized size = 2.35 \begin{align*} \begin{cases} x e^{n \cos{\left (a c \right )}} \sin{\left (a c \right )} & \text{for}\: b = 0 \\0 & \text{for}\: c = 0 \\\begin{cases} x \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{otherwise} \end{cases} & \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(b*c*x+a*c))*sin(c*(b*x+a)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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