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]

E^(n*Sin[a + b*x])/(b*n)

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Rubi [A]  time = 0.0126711, antiderivative size = 17, normalized size of antiderivative = 1., 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 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])

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 \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*}

Mathematica [A]  time = 0.0167467, size = 17, normalized size = 1. \[ \frac{e^{n \sin (a+b x)}}{b n} \]

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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Maple [A]  time = 0.006, size = 17, normalized size = 1. \begin{align*}{\frac{{{\rm e}^{n\sin \left ( bx+a \right ) }}}{bn}} \end{align*}

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 = 0.962259, size = 22, normalized size = 1.29 \begin{align*} \frac{e^{\left (n \sin \left (b x + a\right )\right )}}{b n} \end{align*}

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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Fricas [A]  time = 2.02746, size = 35, normalized size = 2.06 \begin{align*} \frac{e^{\left (n \sin \left (b x + a\right )\right )}}{b n} \end{align*}

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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Sympy [A]  time = 0.460416, size = 36, normalized size = 2.12 \begin{align*} \begin{cases} x \cos{\left (a \right )} & \text{for}\: b = 0 \wedge n = 0 \\x e^{n \sin{\left (a \right )}} \cos{\left (a \right )} & \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} \end{align*}

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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Giac [A]  time = 1.11749, size = 22, normalized size = 1.29 \begin{align*} \frac{e^{\left (n \sin \left (b x + a\right )\right )}}{b n} \end{align*}

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)