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

Optimal. Leaf size=18 \[ -\frac {e^{n \cos (a+b x)}}{b n} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4335, 2194} \[ -\frac {e^{n \cos (a+b x)}}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^(n*Cos[a + b*x])/(b*n))

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]

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])

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 18, normalized size = 1.00 \[ -\frac {e^{n \cos (a+b x)}}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^(n*Cos[a + b*x])/(b*n))

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fricas [A]  time = 1.04, size = 17, normalized size = 0.94 \[ -\frac {e^{\left (n \cos \left (b x + a\right )\right )}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(n*cos(b*x + a))/(b*n)

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giac [A]  time = 0.14, size = 17, normalized size = 0.94 \[ -\frac {e^{\left (n \cos \left (b x + a\right )\right )}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(n*cos(b*x + a))/(b*n)

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maple [A]  time = 0.01, size = 18, normalized size = 1.00 \[ -\frac {{\mathrm e}^{n \cos \left (b x +a \right )}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.31, size = 17, normalized size = 0.94 \[ -\frac {e^{\left (n \cos \left (b x + a\right )\right )}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(n*cos(b*x + a))/(b*n)

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mupad [B]  time = 0.10, size = 17, normalized size = 0.94 \[ -\frac {{\mathrm {e}}^{n\,\cos \left (a+b\,x\right )}}{b\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(n*cos(a + b*x))/(b*n)

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sympy [A]  time = 0.64, size = 39, normalized size = 2.17 \[ \begin {cases} x \sin {\relax (a )} & \text {for}\: b = 0 \wedge n = 0 \\x e^{n \cos {\relax (a )}} \sin {\relax (a )} & \text {for}\: b = 0 \\- \frac {\cos {\left (a + b x \right )}}{b} & \text {for}\: n = 0 \\- \frac {e^{n \cos {\left (a + b x \right )}}}{b n} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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