3.221 \(\int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx\)

Optimal. Leaf size=28 \[ \frac{a-\frac{1}{x}}{a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}} \]

[Out]

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

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Rubi [A]  time = 0.103764, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6175, 6178, 637} \[ \frac{a-\frac{1}{x}}{a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^2),x]

[Out]

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

Rule 6175

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

Rule 6178

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> -Dist[c^n, Subst[Int[((c
+ d*x)^(p - n)*(1 - x^2/a^2)^(n/2))/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] &&
IntegerQ[(n - 1)/2] && IntegerQ[m] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1] || LtQ[-5, m, -1]) && In
tegerQ[2*p]

Rule 637

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-(a*e) + c*d*x)/(a*c*Sqrt[a + c*x^2]),
x] /; FreeQ[{a, c, d, e}, x]

Rubi steps

\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx &=\frac{\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^2 x^2} \, dx}{a^2 c^2}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1-\frac{x}{a}}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{a^2 c^2}\\ &=\frac{a-\frac{1}{x}}{a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}

Mathematica [A]  time = 0.0502169, size = 26, normalized size = 0.93 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2 (a x+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^2),x]

[Out]

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

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Maple [A]  time = 0.066, size = 35, normalized size = 1.3 \begin{align*}{\frac{ax+1}{ \left ( ax-1 \right ) a{c}^{2}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^2,x)

[Out]

((a*x-1)/(a*x+1))^(3/2)*(a*x+1)/(a*x-1)/a/c^2

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Maxima [A]  time = 1.09743, size = 30, normalized size = 1.07 \begin{align*} \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^2,x, algorithm="maxima")

[Out]

sqrt((a*x - 1)/(a*x + 1))/(a*c^2)

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Fricas [A]  time = 1.49414, size = 47, normalized size = 1.68 \begin{align*} \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^2,x, algorithm="fricas")

[Out]

sqrt((a*x - 1)/(a*x + 1))/(a*c^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}\, dx + \int \frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}\, dx}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))**(3/2)/(-a*c*x+c)**2,x)

[Out]

(Integral(-sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**3*x**3 - a**2*x**2 - a*x + 1), x) + Integral(a*x*sqrt(a*x/(a*
x + 1) - 1/(a*x + 1))/(a**3*x**3 - a**2*x**2 - a*x + 1), x))/c**2

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^2,x, algorithm="giac")

[Out]

undef