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

3.7.87.1 Optimal result

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

output

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

3.7.87.2 Mathematica [A] (verified)

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

input

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

output

ExpIntegralEi[n*Sin[a + b*x]]/b
 

3.7.87.3 Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4838, 2609}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 4838

\(\displaystyle \frac {\int e^{n \sin (a+b x)} \csc (a+b x)d\sin (a+b x)}{b}\)

\(\Big \downarrow \) 2609

\(\displaystyle \frac {\operatorname {ExpIntegralEi}(n \sin (a+b x))}{b}\)

input

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

output

ExpIntegralEi[n*Sin[a + b*x]]/b
 

3.7.87.3.1 Defintions of rubi rules used

rule 2609

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

rule 4838

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFacto 
rs[Sin[c*(a + b*x)], x]}, Simp[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])
 

3.7.87.4 Maple [F]

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

input

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

output

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

3.7.87.5 Fricas [A] (verification not implemented)

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

input

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

output

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

3.7.87.6 Sympy [F]

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

input

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

output

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

3.7.87.7 Maxima [A] (verification not implemented)

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

input

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

output

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

3.7.87.8 Giac [A] (verification not implemented)

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

input

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

output

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

3.7.87.9 Mupad [F(-1)]

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

input

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

output

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