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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0219697, 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 = {4339, 2178} \[ -\frac{\text{Ei}(n \cos (c (a+b x)))}{b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4339

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])

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

Mathematica [A]  time = 0.0607347, size = 19, normalized size = 1. \[ -\frac{\text{ExpIntegralEi}(n \cos (c (a+b x)))}{b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.02, size = 22, normalized size = 1.2 \begin{align*}{\frac{{\it Ei} \left ( 1,-n\cos \left ( bcx+ac \right ) \right ) }{cb}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.08073, size = 27, normalized size = 1.42 \begin{align*} -\frac{{\rm Ei}\left (n \cos \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*cos(b*c*x+a*c))*tan(c*(b*x+a)),x, algorithm="maxima")

[Out]

-Ei(n*cos(b*c*x + a*c))/(b*c)

________________________________________________________________________________________

Fricas [A]  time = 2.3288, size = 42, normalized size = 2.21 \begin{align*} -\frac{{\rm Ei}\left (n \cos \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*cos(b*c*x+a*c))*tan(c*(b*x+a)),x, algorithm="fricas")

[Out]

-Ei(n*cos(b*c*x + a*c))/(b*c)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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