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]

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

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 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 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*}

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Mathematica [A]  time = 0.14, 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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fricas [A]  time = 1.05, size = 22, normalized size = 1.00 \[ \frac {e^{\left (n \sin \left (b c x + a c\right )\right )}}{b c n} \]

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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giac [A]  time = 0.13, size = 22, normalized size = 1.00 \[ \frac {e^{\left (n \sin \left (b c x + a c\right )\right )}}{b c n} \]

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]

e^(n*sin(b*c*x + a*c))/(b*c*n)

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maple [A]  time = 0.06, size = 23, normalized size = 1.05 \[ \frac {{\mathrm e}^{n \sin \left (b c x +a c \right )}}{b c n} \]

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 = 0.32, size = 22, normalized size = 1.00 \[ \frac {e^{\left (n \sin \left (b c x + a c\right )\right )}}{b c n} \]

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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mupad [B]  time = 3.08, size = 22, normalized size = 1.00 \[ \frac {{\mathrm {e}}^{n\,\sin \left (a\,c+b\,c\,x\right )}}{b\,c\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(n*sin(a*c + b*c*x))/(b*c*n)

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sympy [A]  time = 9.32, size = 51, normalized size = 2.32 \[ \begin {cases} x e^{n \sin {\left (a c \right )}} \cos {\left (a c \right )} & \text {for}\: b = 0 \\x & \text {for}\: c = 0 \\\begin {cases} x \cos {\left (a c \right )} & \text {for}\: b = 0 \\x & \text {for}\: c = 0 \\\frac {\sin {\left (a c + b c x \right )}}{b c} & \text {otherwise} \end {cases} & \text {for}\: n = 0 \\\frac {e^{n \sin {\left (a c + b c x \right )}}}{b c n} & \text {otherwise} \end {cases} \]

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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