3.971 \(\int e^{n \cosh (a c+b c x)} \tanh (c (a+b x)) \, dx\)

Optimal. Leaf size=18 \[ \frac {\text {Ei}(n \cosh (c (a+b x)))}{b c} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 18, 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 = {4341, 2178} \[ \frac {\text {Ei}(n \cosh (c (a+b x)))}{b c} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*Cosh[a*c + b*c*x])*Tanh[c*(a + b*x)],x]

[Out]

ExpIntegralEi[n*Cosh[c*(a + b*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 4341

Int[(u_)*Tanh[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, Dist[1/(
b*c), Subst[Int[SubstFor[1/x, Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d], x] /; FunctionOfQ[Cosh[
c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x]

Rubi steps

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

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Mathematica [A]  time = 0.18, size = 18, normalized size = 1.00 \[ \frac {\text {Ei}(n \cosh (c (a+b x)))}{b c} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(n*Cosh[a*c + b*c*x])*Tanh[c*(a + b*x)],x]

[Out]

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

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fricas [A]  time = 0.51, size = 19, normalized size = 1.06 \[ \frac {{\rm Ei}\left (n \cosh \left (b c x + a c\right )\right )}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(n*cosh(b*c*x + a*c))/(b*c)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(n*cosh(b*c*x + a*c))*tanh((b*x + a)*c), x)

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maple [A]  time = 0.40, size = 31, normalized size = 1.72 \[ \frac {\Shi \left (n \cosh \left (c \left (b x +a \right )\right )\right )+\Chi \left (n \cosh \left (c \left (b x +a \right )\right )\right )}{c b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*cosh(b*c*x+a*c))*tanh(c*(b*x+a)),x)

[Out]

1/c/b*(Shi(n*cosh(c*(b*x+a)))+Chi(n*cosh(c*(b*x+a))))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(n*cosh(b*c*x + a*c))*tanh((b*x + a)*c), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \mathrm {tanh}\left (c\,\left (a+b\,x\right )\right )\,{\mathrm {e}}^{n\,\mathrm {cosh}\left (a\,c+b\,c\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(c*(a + b*x))*exp(n*cosh(a*c + b*c*x)),x)

[Out]

int(tanh(c*(a + b*x))*exp(n*cosh(a*c + b*c*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*cosh(b*c*x+a*c))*tanh(c*(b*x+a)),x)

[Out]

Integral(exp(n*cosh(a*c + b*c*x))*tanh(a*c + b*c*x), x)

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