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

Optimal. Leaf size=16 \[ -\frac{e^{-\coth ^{-1}(a x)}}{a c} \]

[Out]

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

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Rubi [A]  time = 0.0301235, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {6183} \[ -\frac{e^{-\coth ^{-1}(a x)}}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6183

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E^(n*ArcCoth[a*x])/(a*c*n), x] /; F
reeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2]

Rubi steps

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

Mathematica [A]  time = 0.0446563, size = 16, normalized size = 1. \[ -\frac{e^{-\coth ^{-1}(a x)}}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 24, normalized size = 1.5 \begin{align*} -{\frac{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)/(-a^2*c*x^2+c),x)

[Out]

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

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Maxima [A]  time = 1.04444, size = 31, normalized size = 1.94 \begin{align*} -\frac{\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)/(-a^2*c*x^2+c),x, algorithm="maxima")

[Out]

-sqrt((a*x - 1)/(a*x + 1))/(a*c)

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Fricas [A]  time = 1.82908, size = 46, normalized size = 2.88 \begin{align*} -\frac{\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)/(-a^2*c*x^2+c),x, algorithm="fricas")

[Out]

-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{\int \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a^{2} x^{2} - 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)/(-a**2*c*x**2+c),x)

[Out]

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

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

[Out]

undef