### 3.976 $$\int e^{n \sinh (a+b x)} \coth (a+b x) \, dx$$

Optimal. Leaf size=13 $\frac{\text{Ei}(n \sinh (a+b x))}{b}$

[Out]

ExpIntegralEi[n*Sinh[a + b*x]]/b

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Rubi [A]  time = 0.0228231, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {4340, 2178} $\frac{\text{Ei}(n \sinh (a+b x))}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(n*Sinh[a + b*x])*Coth[a + b*x],x]

[Out]

ExpIntegralEi[n*Sinh[a + b*x]]/b

Rule 4340

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

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

Mathematica [A]  time = 0.0334604, size = 13, normalized size = 1. $\frac{\text{Ei}(n \sinh (a+b x))}{b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^(n*Sinh[a + b*x])*Coth[a + b*x],x]

[Out]

ExpIntegralEi[n*Sinh[a + b*x]]/b

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Maple [A]  time = 0.01, size = 17, normalized size = 1.3 \begin{align*} -{\frac{{\it Ei} \left ( 1,-n\sinh \left ( bx+a \right ) \right ) }{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*sinh(b*x+a))*coth(b*x+a),x)

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*sinh(b*x+a))*coth(b*x+a),x, algorithm="maxima")

[Out]

integrate(coth(b*x + a)*e^(n*sinh(b*x + a)), x)

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Fricas [A]  time = 1.97297, size = 31, normalized size = 2.38 \begin{align*} \frac{{\rm Ei}\left (n \sinh \left (b x + a\right )\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*sinh(b*x+a))*coth(b*x+a),x, algorithm="fricas")

[Out]

Ei(n*sinh(b*x + a))/b

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{n \sinh{\left (a + b x \right )}} \coth{\left (a + b x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*sinh(b*x+a))*coth(b*x+a),x)

[Out]

Integral(exp(n*sinh(a + b*x))*coth(a + b*x), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*sinh(b*x+a))*coth(b*x+a),x, algorithm="giac")

[Out]

integrate(coth(b*x + a)*e^(n*sinh(b*x + a)), x)