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

Optimal. Leaf size=18 \[ \frac{e^{3 \coth ^{-1}(a x)}}{3 a c} \]

[Out]

E^(3*ArcCoth[a*x])/(3*a*c)

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Rubi [A]  time = 0.0313877, antiderivative size = 18, 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} \[ \frac{e^{3 \coth ^{-1}(a x)}}{3 a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^(3*ArcCoth[a*x])/(3*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^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=\frac{e^{3 \coth ^{-1}(a x)}}{3 a c}\\ \end{align*}

Mathematica [A]  time = 0.0464494, size = 18, normalized size = 1. \[ \frac{e^{3 \coth ^{-1}(a x)}}{3 a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^(3*ArcCoth[a*x])/(3*a*c)

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Maple [A]  time = 0.123, size = 24, normalized size = 1.3 \begin{align*}{\frac{1}{3\,ac} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0772, size = 31, normalized size = 1.72 \begin{align*} \frac{1}{3 \, a c \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3/(a*c*((a*x - 1)/(a*x + 1))^(3/2))

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Fricas [B]  time = 1.58278, size = 112, normalized size = 6.22 \begin{align*} \frac{{\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(a^2*x^2 + 2*a*x + 1)*sqrt((a*x - 1)/(a*x + 1))/(a^3*c*x^2 - 2*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}{\frac{a^{3} x^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{a^{2} x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} + \frac{\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(1/((a*x-1)/(a*x+1))**(3/2)/(-a**2*c*x**2+c),x)

[Out]

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

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Giac [A]  time = 1.17372, size = 47, normalized size = 2.61 \begin{align*} \frac{a x + 1}{3 \,{\left (a x - 1\right )} 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))^(3/2)/(-a^2*c*x^2+c),x, algorithm="giac")

[Out]

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