\(\int e^{n \cos (c (a+b x))} \tan (a c+b c x) \, dx\) [664]

Optimal result

Integrand size = 22, antiderivative size = 20 \[ \int e^{n \cos (c (a+b x))} \tan (a c+b c x) \, dx=-\frac {\operatorname {ExpIntegralEi}(n \cos (a c+b c x))}{b c} \]

[Out]

-Ei(n*cos(b*c*x+a*c))/b/c

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4424, 2209} \[ \int e^{n \cos (c (a+b x))} \tan (a c+b c x) \, dx=-\frac {\operatorname {ExpIntegralEi}(n \cos (a c+b x c))}{b c} \]

[In]

Int[E^(n*Cos[c*(a + b*x)])*Tan[a*c + b*c*x],x]

[Out]

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

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 4424

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, Dist[-(b*
c)^(-1), Subst[Int[SubstFor[1/x, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[
c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Tan] || EqQ[F, tan])

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int e^{n \cos (c (a+b x))} \tan (a c+b c x) \, dx=-\frac {\operatorname {ExpIntegralEi}(n \cos (c (a+b x)))}{b c} \]

[In]

Integrate[E^(n*Cos[c*(a + b*x)])*Tan[a*c + b*c*x],x]

[Out]

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

Maple [F]

\[\int {\mathrm e}^{n \cos \left (c \left (x b +a \right )\right )} \tan \left (b c x +a c \right )d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int e^{n \cos (c (a+b x))} \tan (a c+b c x) \, dx=-\frac {{\rm Ei}\left (n \cos \left (b c x + a c\right )\right )}{b c} \]

[In]

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

[Out]

-Ei(n*cos(b*c*x + a*c))/(b*c)

Sympy [F]

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

[In]

integrate(exp(n*cos(c*(b*x+a)))*tan(b*c*x+a*c),x)

[Out]

Integral(exp(n*cos(a*c + b*c*x))*tan(a*c + b*c*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int e^{n \cos (c (a+b x))} \tan (a c+b c x) \, dx=-\frac {{\rm Ei}\left (n \cos \left (b c x + a c\right )\right )}{b c} \]

[In]

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

[Out]

-Ei(n*cos(b*c*x + a*c))/(b*c)

Giac [F]

\[ \int e^{n \cos (c (a+b x))} \tan (a c+b c x) \, dx=\int { e^{\left (n \cos \left ({\left (b x + a\right )} c\right )\right )} \tan \left (b c x + a c\right ) \,d x } \]

[In]

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

[Out]

integrate(e^(n*cos((b*x + a)*c))*tan(b*c*x + a*c), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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