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

Optimal. Leaf size=13 \[ \frac{x}{c^2 (1-a x)^2} \]

[Out]

x/(c^2*(1 - a*x)^2)

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Rubi [A]  time = 0.0261538, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6129, 34} \[ \frac{x}{c^2 (1-a x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(c^2*(1 - a*x)^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 34

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_)), x_Symbol] :> Simp[(d*x*(a + b*x)^(m + 1))/(b*(m + 2)), x] /
; FreeQ[{a, b, c, d, m}, x] && EqQ[a*d - b*c*(m + 2), 0]

Rubi steps

\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)}}{(c-a c x)^2} \, dx &=\frac{\int \frac{1+a x}{(1-a x)^3} \, dx}{c^2}\\ &=\frac{x}{c^2 (1-a x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0069741, size = 25, normalized size = 1.92 \[ \frac{(a x+1)^2}{4 a c^2 (1-a x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + a*x)^2/(4*a*c^2*(1 - a*x)^2)

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Maple [B]  time = 0.039, size = 28, normalized size = 2.2 \begin{align*}{\frac{1}{{c}^{2}} \left ({\frac{1}{a \left ( ax-1 \right ) }}+{\frac{1}{a \left ( ax-1 \right ) ^{2}}} \right ) } \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)^2,x)

[Out]

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

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Maxima [B]  time = 0.938602, size = 34, normalized size = 2.62 \begin{align*} \frac{x}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}} \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)^2,x, algorithm="maxima")

[Out]

x/(a^2*c^2*x^2 - 2*a*c^2*x + c^2)

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Fricas [B]  time = 1.68245, size = 47, normalized size = 3.62 \begin{align*} \frac{x}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}} \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)^2,x, algorithm="fricas")

[Out]

x/(a^2*c^2*x^2 - 2*a*c^2*x + c^2)

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Sympy [B]  time = 0.377952, size = 22, normalized size = 1.69 \begin{align*} \frac{x}{a^{2} c^{2} x^{2} - 2 a c^{2} x + c^{2}} \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)**2,x)

[Out]

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

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Giac [B]  time = 1.2299, size = 43, normalized size = 3.31 \begin{align*} \frac{1}{{\left (a c x - c\right )}^{2} a} + \frac{1}{{\left (a c x - c\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)^2,x, algorithm="giac")

[Out]

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