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

Optimal. Leaf size=22 \[ \frac{e^{n \sin (c (a+b x))}}{b c n} \]

[Out]

E^(n*Sin[c*(a + b*x)])/(b*c*n)

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Rubi [A]  time = 0.0132438, antiderivative size = 22, 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 = {4334, 2194} \[ \frac{e^{n \sin (c (a+b x))}}{b c n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.145826, size = 23, normalized size = 1.05 \[ \frac{e^{n \sin (a c+b c x)}}{b c n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^(n*Sin[a*c + b*c*x])/(b*c*n)

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Maple [A]  time = 0.015, size = 23, normalized size = 1.1 \begin{align*}{\frac{{{\rm e}^{n\sin \left ( bcx+ac \right ) }}}{cbn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.956044, size = 30, normalized size = 1.36 \begin{align*} \frac{e^{\left (n \sin \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*sin(b*c*x+a*c))*cos(c*(b*x+a)),x, algorithm="maxima")

[Out]

e^(n*sin(b*c*x + a*c))/(b*c*n)

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Fricas [A]  time = 2.10694, size = 43, normalized size = 1.95 \begin{align*} \frac{e^{\left (n \sin \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*sin(b*c*x+a*c))*cos(c*(b*x+a)),x, algorithm="fricas")

[Out]

e^(n*sin(b*c*x + a*c))/(b*c*n)

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Sympy [A]  time = 8.67409, size = 51, normalized size = 2.32 \begin{align*} \begin{cases} x e^{n \sin{\left (a c \right )}} \cos{\left (a c \right )} & \text{for}\: b = 0 \\x & \text{for}\: c = 0 \\\begin{cases} x \cos{\left (a c \right )} & \text{for}\: b = 0 \\x & \text{for}\: c = 0 \\\frac{\sin{\left (a c + b c x \right )}}{b c} & \text{otherwise} \end{cases} & \text{for}\: n = 0 \\\frac{e^{n \sin{\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*sin(b*c*x+a*c))*cos(c*(b*x+a)),x)

[Out]

Piecewise((x*exp(n*sin(a*c))*cos(a*c), Eq(b, 0)), (x, Eq(c, 0)), (Piecewise((x*cos(a*c), Eq(b, 0)), (x, Eq(c,
0)), (sin(a*c + b*c*x)/(b*c), True)), Eq(n, 0)), (exp(n*sin(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 \cos \left ({\left (b x + a\right )} c\right ) e^{\left (n \sin \left (b c x + a c\right )\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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