### 3.173 $$\int \frac{e^{2 \coth ^{-1}(a x)}}{c-a c x} \, dx$$

Optimal. Leaf size=32 $-\frac{2}{a c (1-a x)}-\frac{\log (1-a x)}{a c}$

[Out]

-2/(a*c*(1 - a*x)) - Log[1 - a*x]/(a*c)

________________________________________________________________________________________

Rubi [A]  time = 0.0637809, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {6167, 6129, 43} $-\frac{2}{a c (1-a x)}-\frac{\log (1-a x)}{a c}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(2*ArcCoth[a*x])/(c - a*c*x),x]

[Out]

-2/(a*c*(1 - a*x)) - Log[1 - a*x]/(a*c)

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rule 6129

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^p, Int[(u*(1 + (d*x)/c)
^p*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ
[p] || GtQ[c, 0])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{c-a c x} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{c-a c x} \, dx\\ &=-\frac{\int \frac{1+a x}{(1-a x)^2} \, dx}{c}\\ &=-\frac{\int \left (\frac{2}{(-1+a x)^2}+\frac{1}{-1+a x}\right ) \, dx}{c}\\ &=-\frac{2}{a c (1-a x)}-\frac{\log (1-a x)}{a c}\\ \end{align*}

Mathematica [A]  time = 0.0149316, size = 30, normalized size = 0.94 $-\frac{\frac{2}{a (1-a x)}+\frac{\log (1-a x)}{a}}{c}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^(2*ArcCoth[a*x])/(c - a*c*x),x]

[Out]

-((2/(a*(1 - a*x)) + Log[1 - a*x]/a)/c)

________________________________________________________________________________________

Maple [A]  time = 0.056, size = 31, normalized size = 1. \begin{align*} 2\,{\frac{1}{ac \left ( ax-1 \right ) }}-{\frac{\ln \left ( ax-1 \right ) }{ac}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(a*x-1)/(-a*c*x+c),x)

[Out]

2/c/a/(a*x-1)-1/c/a*ln(a*x-1)

________________________________________________________________________________________

Maxima [A]  time = 1.02256, size = 41, normalized size = 1.28 \begin{align*} \frac{2}{a^{2} c x - a c} - \frac{\log \left (a x - 1\right )}{a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)/(-a*c*x+c),x, algorithm="maxima")

[Out]

2/(a^2*c*x - a*c) - log(a*x - 1)/(a*c)

________________________________________________________________________________________

Fricas [A]  time = 1.45197, size = 63, normalized size = 1.97 \begin{align*} -\frac{{\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{a^{2} c x - a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)/(-a*c*x+c),x, algorithm="fricas")

[Out]

-((a*x - 1)*log(a*x - 1) - 2)/(a^2*c*x - a*c)

________________________________________________________________________________________

Sympy [A]  time = 0.352277, size = 20, normalized size = 0.62 \begin{align*} \frac{2}{a^{2} c x - a c} - \frac{\log{\left (a x - 1 \right )}}{a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)/(-a*c*x+c),x)

[Out]

2/(a**2*c*x - a*c) - log(a*x - 1)/(a*c)

________________________________________________________________________________________

Giac [A]  time = 1.14623, size = 42, normalized size = 1.31 \begin{align*} -\frac{\log \left ({\left | a x - 1 \right |}\right )}{a c} + \frac{2}{{\left (a x - 1\right )} a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)/(-a*c*x+c),x, algorithm="giac")

[Out]

-log(abs(a*x - 1))/(a*c) + 2/((a*x - 1)*a*c)