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

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

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 19, 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 = {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 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

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])

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*}

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Mathematica [A]  time = 0.06, size = 18, normalized size = 0.95 \[ \frac {\text {Ei}(n \sin (c (a+b x)))}{b c} \]

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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fricas [A]  time = 0.92, size = 19, normalized size = 1.00 \[ \frac {{\rm Ei}\left (n \sin \left (b c x + a c\right )\right )}{b c} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cot \left (b c x + a c\right ) e^{\left (n \sin \left ({\left (b x + a\right )} c\right )\right )}\,{d x} \]

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)

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maple [A]  time = 0.07, size = 23, normalized size = 1.21 \[ -\frac {\Ei \left (1, -n \sin \left (b c x +a c \right )\right )}{c b} \]

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 = 0.41, size = 19, normalized size = 1.00 \[ \frac {{\rm Ei}\left (n \sin \left (b c x + a c\right )\right )}{b c} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \[ \int {\mathrm {e}}^{n\,\sin \left (c\,\left (a+b\,x\right )\right )}\,\mathrm {cot}\left (a\,c+b\,c\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{n \sin {\left (a c + b c x \right )}} \cot {\left (a c + b c x \right )}\, dx \]

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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