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

Optimal. Leaf size=31 \[ \frac{2}{a c (1-a x)}+\frac{\log (1-a x)}{a c} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0381136, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6129, 43} \[ \frac{2}{a c (1-a x)}+\frac{\log (1-a x)}{a c} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])/(c - a*c*x),x]

[Out]

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

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 \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 = 0.0162467, size = 25, normalized size = 0.81 \[ \frac{\frac{2}{1-a x}+\log (1-a x)}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2/(1 - a*x) + Log[1 - a*x])/(a*c)

________________________________________________________________________________________

Maple [A]  time = 0.036, size = 30, 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)^2/(-a^2*x^2+1)/(-a*c*x+c),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.934844, size = 39, normalized size = 1.26 \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((a*x+1)^2/(-a^2*x^2+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 = 1.68026, size = 62, normalized size = 2. \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((a*x+1)^2/(-a^2*x^2+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 = 0.333816, size = 20, normalized size = 0.65 \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((a*x+1)**2/(-a**2*x**2+1)/(-a*c*x+c),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.2026, size = 41, normalized size = 1.32 \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((a*x+1)^2/(-a^2*x^2+1)/(-a*c*x+c),x, algorithm="giac")

[Out]

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