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

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

[Out]

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

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Rubi [A]  time = 0.0288307, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {6183} \[ \frac{e^{\coth ^{-1}(a x)}}{a c} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCoth[a*x]/(c - a^2*c*x^2),x]

[Out]

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

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 = 0.0438475, size = 13, normalized size = 1. \[ \frac{e^{\coth ^{-1}(a x)}}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 23, normalized size = 1.8 \begin{align*}{\frac{1}{ac}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.08499, size = 30, normalized size = 2.31 \begin{align*} \frac{1}{a c \sqrt{\frac{a x - 1}{a x + 1}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a*c*sqrt((a*x - 1)/(a*x + 1)))

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Fricas [A]  time = 1.54037, size = 72, normalized size = 5.54 \begin{align*} \frac{{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c x - a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12888, size = 30, normalized size = 2.31 \begin{align*} \frac{1}{a c \sqrt{\frac{a x - 1}{a x + 1}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a*c*sqrt((a*x - 1)/(a*x + 1)))