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]

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

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Rubi [A]  time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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*}

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Mathematica [A]  time = 0.05, size = 13, normalized size = 1.00 \[ \frac {e^{\coth ^{-1}(a x)}}{a c} \]

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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fricas [A]  time = 0.79, size = 34, normalized size = 2.62 \[ \frac {{\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c x - a c} \]

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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giac [A]  time = 0.15, size = 22, normalized size = 1.69 \[ \frac {1}{a c \sqrt {\frac {a x - 1}{a x + 1}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a*c*sqrt((a*x - 1)/(a*x + 1)))

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maple [A]  time = 0.04, size = 23, normalized size = 1.77 \[ \frac {1}{\sqrt {\frac {a x -1}{a x +1}}\, a c} \]

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 = 0.30, size = 22, normalized size = 1.69 \[ \frac {1}{a c \sqrt {\frac {a x - 1}{a x + 1}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(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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mupad [B]  time = 1.22, size = 22, normalized size = 1.69 \[ \frac {1}{a\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \]

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))^(1/2)),x)

[Out]

1/(a*c*((a*x - 1)/(a*x + 1))^(1/2))

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

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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