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]

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

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Rubi [A]  time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.05, size = 18, normalized size = 1.00 \[ \frac {e^{3 \coth ^{-1}(a x)}}{3 a c} \]

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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fricas [B]  time = 0.59, size = 51, normalized size = 2.83 \[ \frac {{\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \]

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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giac [A]  time = 0.14, size = 35, normalized size = 1.94 \[ \frac {a x + 1}{3 \, {\left (a x - 1\right )} a c \sqrt {\frac {a x - 1}{a x + 1}}} \]

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)))

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maple [A]  time = 0.04, size = 24, normalized size = 1.33 \[ \frac {1}{3 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a c} \]

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 = 0.30, size = 23, normalized size = 1.28 \[ \frac {1}{3 \, a c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \]

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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mupad [B]  time = 0.03, size = 23, normalized size = 1.28 \[ \frac {1}{3\,a\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(3*a*c*((a*x - 1)/(a*x + 1))^(3/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {1}{\frac {a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c} \]

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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