\(\int e^{n \sin (a+b x)} \cot (a+b x) \, dx\) [687]

Optimal result

Integrand size = 17, antiderivative size = 13 \[ \int e^{n \sin (a+b x)} \cot (a+b x) \, dx=\frac {\operatorname {ExpIntegralEi}(n \sin (a+b x))}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4423, 2209} \[ \int e^{n \sin (a+b x)} \cot (a+b x) \, dx=\frac {\operatorname {ExpIntegralEi}(n \sin (a+b x))}{b} \]

[In]

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

[Out]

ExpIntegralEi[n*Sin[a + b*x]]/b

Rule 2209

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

Rule 4423

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int e^{n \sin (a+b x)} \cot (a+b x) \, dx=\frac {\operatorname {ExpIntegralEi}(n \sin (a+b x))}{b} \]

[In]

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

[Out]

ExpIntegralEi[n*Sin[a + b*x]]/b

Maple [F]

\[\int {\mathrm e}^{n \sin \left (x b +a \right )} \cot \left (x b +a \right )d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int e^{n \sin (a+b x)} \cot (a+b x) \, dx=\frac {{\rm Ei}\left (n \sin \left (b x + a\right )\right )}{b} \]

[In]

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

[Out]

Ei(n*sin(b*x + a))/b

Sympy [F]

\[ \int e^{n \sin (a+b x)} \cot (a+b x) \, dx=\int e^{n \sin {\left (a + b x \right )}} \cot {\left (a + b x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int e^{n \sin (a+b x)} \cot (a+b x) \, dx=\frac {{\rm Ei}\left (n \sin \left (b x + a\right )\right )}{b} \]

[In]

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

[Out]

Ei(n*sin(b*x + a))/b

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int e^{n \sin (a+b x)} \cot (a+b x) \, dx=\frac {{\rm Ei}\left (n \sin \left (b x + a\right )\right )}{b} \]

[In]

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

[Out]

Ei(n*sin(b*x + a))/b

Mupad [F(-1)]

Timed out. \[ \int e^{n \sin (a+b x)} \cot (a+b x) \, dx=\int \mathrm {cot}\left (a+b\,x\right )\,{\mathrm {e}}^{n\,\sin \left (a+b\,x\right )} \,d x \]

[In]

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

[Out]

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