3.1258 \(\int \frac{e^{-3 \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx\)

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

[Out]

-1/(3*a*c*E^(3*ArcTanh[a*x]))

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Rubi [A]  time = 0.0332538, 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 = {6137} \[ -\frac{e^{-3 \tanh ^{-1}(a x)}}{3 a c} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)),x]

[Out]

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

Mathematica [A]  time = 0.0147526, size = 29, normalized size = 1.61 \[ -\frac{(1-a x)^{3/2}}{3 a c (a x+1)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)),x]

[Out]

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

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Maple [A]  time = 0.029, size = 28, normalized size = 1.6 \begin{align*} -{\frac{1}{3\, \left ( ax+1 \right ) ^{3}ca} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a^{2} c x^{2} - c\right )}{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.69853, size = 120, normalized size = 6.67 \begin{align*} -\frac{a^{2} x^{2} + 2 \, a x - \sqrt{-a^{2} x^{2} + 1}{\left (a x - 1\right )} + 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)^3*(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c),x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.1546, size = 89, normalized size = 4.94 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + 1\right )}}{3 \, c{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} + 1\right )}^{3}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) + 1)/(c*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 1)^3*abs
(a))