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

Optimal. Leaf size=19 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c} \]

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x)/c

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Rubi [A]  time = 0.0397154, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6177, 264} \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x)/c

Rule 6177

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

Rule 264

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

Rubi steps

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

Mathematica [A]  time = 0.0669128, size = 19, normalized size = 1. \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x)/c

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Maple [A]  time = 0.04, size = 28, normalized size = 1.5 \begin{align*}{\frac{ax+1}{ac}\sqrt{{\frac{ax-1}{ax+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.02521, size = 59, normalized size = 3.11 \begin{align*} -\frac{2 \, a \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{{\left (a x - 1\right )} a^{2} c}{a x + 1} - a^{2} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.83666, size = 58, normalized size = 3.05 \begin{align*} \frac{{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x + 1)*sqrt((a*x - 1)/(a*x + 1))/(a*c)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17935, size = 32, normalized size = 1.68 \begin{align*} \frac{\sqrt{a^{2} x^{2} - 1} \mathrm{sgn}\left (a x + 1\right )}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a^2*x^2 - 1)*sgn(a*x + 1)/(a*c)