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

Optimal. Leaf size=17 \[ \frac {e^{n \sin (a+b x)}}{b n} \]

[Out]

exp(n*sin(b*x+a))/b/n

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Rubi [A]  time = 0.01, antiderivative size = 17, 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 = {4334, 2194} \[ \frac {e^{n \sin (a+b x)}}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^(n*Sin[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 4334

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 17, normalized size = 1.00 \[ \frac {e^{n \sin (a+b x)}}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^(n*Sin[a + b*x])/(b*n)

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fricas [A]  time = 0.78, size = 16, normalized size = 0.94 \[ \frac {e^{\left (n \sin \left (b x + a\right )\right )}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(n*sin(b*x + a))/(b*n)

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giac [A]  time = 0.13, size = 16, normalized size = 0.94 \[ \frac {e^{\left (n \sin \left (b x + a\right )\right )}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(n*sin(b*x + a))/(b*n)

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maple [A]  time = 0.01, size = 17, normalized size = 1.00 \[ \frac {{\mathrm e}^{n \sin \left (b x +a \right )}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(n*sin(b*x+a))/b/n

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maxima [A]  time = 0.31, size = 16, normalized size = 0.94 \[ \frac {e^{\left (n \sin \left (b x + a\right )\right )}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(n*sin(b*x + a))/(b*n)

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mupad [B]  time = 0.11, size = 16, normalized size = 0.94 \[ \frac {{\mathrm {e}}^{n\,\sin \left (a+b\,x\right )}}{b\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(n*sin(a + b*x))/(b*n)

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sympy [A]  time = 0.43, size = 36, normalized size = 2.12 \[ \begin {cases} x \cos {\relax (a )} & \text {for}\: b = 0 \wedge n = 0 \\x e^{n \sin {\relax (a )}} \cos {\relax (a )} & \text {for}\: b = 0 \\\frac {\sin {\left (a + b x \right )}}{b} & \text {for}\: n = 0 \\\frac {e^{n \sin {\left (a + b x \right )}}}{b n} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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