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

Optimal. Leaf size=19 \[ \frac{x}{c^3 \sqrt{1-a^2 x^2}} \]

[Out]

x/(c^3*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.0278557, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6127, 191} \[ \frac{x}{c^3 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(c^3*Sqrt[1 - a^2*x^2])

Rule 6127

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^n, Int[(c + d*x)^(p - n)*(1 -
a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

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

Mathematica [A]  time = 0.0111319, size = 19, normalized size = 1. \[ \frac{x}{c^3 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(c^3*Sqrt[1 - a^2*x^2])

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Maple [A]  time = 0.03, size = 32, normalized size = 1.7 \begin{align*}{\frac{x}{ \left ( ax-1 \right ) ^{2}{c}^{3} \left ( ax+1 \right ) ^{2}} \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*c*x+c)^3,x)

[Out]

(-a^2*x^2+1)^(3/2)*x/(a*x-1)^2/c^3/(a*x+1)^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 c x - c\right )}^{3}{\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*c*x+c)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.65895, size = 58, normalized size = 3.05 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} x}{a^{2} c^{3} x^{2} - c^{3}} \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*c*x+c)^3,x, algorithm="fricas")

[Out]

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

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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^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx + \int - \frac{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx}{c^{3}} \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*c*x+c)**3,x)

[Out]

-(Integral(sqrt(-a**2*x**2 + 1)/(a**6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 1), x) + Integral(-a**2*x**2*sqrt(-a*
*2*x**2 + 1)/(a**6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 1), x))/c**3

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Giac [A]  time = 1.21393, size = 39, normalized size = 2.05 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} x}{{\left (a^{2} x^{2} - 1\right )} c^{3}} \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*c*x+c)^3,x, algorithm="giac")

[Out]

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