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

Optimal. Leaf size=19 \[ \frac{\text{ExpIntegralEi}(n \sin (a c+b c x))}{b c} \]

[Out]

ExpIntegralEi[n*Sin[a*c + b*c*x]]/(b*c)

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Rubi [A]  time = 0.0203697, antiderivative size = 19, 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 = {4338, 2178} \[ \frac{\text{Ei}(n \sin (a c+b x c))}{b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ExpIntegralEi[n*Sin[a*c + b*c*x]]/(b*c)

Rule 4338

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[1/(b
*c), Subst[Int[SubstFor[1/x, 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, Cot] || EqQ[F, cot])

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

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

Mathematica [A]  time = 0.0600483, size = 18, normalized size = 0.95 \[ \frac{\text{ExpIntegralEi}(n \sin (c (a+b x)))}{b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ExpIntegralEi[n*Sin[c*(a + b*x)]]/(b*c)

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Maple [A]  time = 0.014, size = 23, normalized size = 1.2 \begin{align*} -{\frac{{\it Ei} \left ( 1,-n\sin \left ( bcx+ac \right ) \right ) }{cb}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/c/b*Ei(1,-n*sin(b*c*x+a*c))

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Maxima [A]  time = 1.06385, size = 26, normalized size = 1.37 \begin{align*} \frac{{\rm Ei}\left (n \sin \left (b c x + a c\right )\right )}{b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(n*sin(b*c*x + a*c))/(b*c)

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Fricas [A]  time = 2.09454, size = 41, normalized size = 2.16 \begin{align*} \frac{{\rm Ei}\left (n \sin \left (b c x + a c\right )\right )}{b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(n*sin(b*c*x + a*c))/(b*c)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{n \sin{\left (a c + b c x \right )}} \cot{\left (a c + b c x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(n*sin(a*c + b*c*x))*cot(a*c + b*c*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cot \left (b c x + a c\right ) e^{\left (n \sin \left ({\left (b x + a\right )} c\right )\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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