3.1196 \(\int \frac{e^{-\tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0311933, 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 = {6137} \[ -\frac{e^{-\tanh ^{-1}(a x)}}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6137

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E^(n*ArcTanh[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^{-\tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=-\frac{e^{-\tanh ^{-1}(a x)}}{a c}\\ \end{align*}

Mathematica [A]  time = 0.0101139, size = 27, normalized size = 1.69 \[ -\frac{\sqrt{1-a x}}{a c \sqrt{a x+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(Sqrt[1 - a*x]/(a*c*Sqrt[1 + a*x]))

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Maple [A]  time = 0.03, size = 28, normalized size = 1.8 \begin{align*} -{\frac{1}{ \left ( ax+1 \right ) ac}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{\sqrt{-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )}{\left (a x + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.0768, size = 66, normalized size = 4.12 \begin{align*} -\frac{a x + \sqrt{-a^{2} x^{2} + 1} + 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^2*x^2+1)^(1/2)/(-a^2*c*x^2+c),x, algorithm="fricas")

[Out]

-(a*x + sqrt(-a^2*x^2 + 1) + 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 x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1947, size = 50, normalized size = 3.12 \begin{align*} \frac{2}{c{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} + 1\right )}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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