3.1197 \(\int \frac{e^{-\tanh ^{-1}(a x)}}{(c-a^2 c x^2)^2} \, dx\)

Optimal. Leaf size=53 \[ \frac{2 x}{3 c^2 \sqrt{1-a^2 x^2}}-\frac{1-a x}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0431805, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6139, 639, 191} \[ \frac{2 x}{3 c^2 \sqrt{1-a^2 x^2}}-\frac{1-a x}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6139

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

Rule 639

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a
*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

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^{-\tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac{\int \frac{1-a x}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=-\frac{1-a x}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{2 \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^2}\\ &=-\frac{1-a x}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x}{3 c^2 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0173273, size = 43, normalized size = 0.81 \[ \frac{2 a^2 x^2+2 a x-1}{3 a c^2 \sqrt{1-a x} (a x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.03, size = 42, normalized size = 0.8 \begin{align*}{\frac{2\,{a}^{2}{x}^{2}+2\,ax-1}{ \left ( 3\,ax+3 \right ) a{c}^{2}}{\frac{1}{\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)^2,x)

[Out]

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

________________________________________________________________________________________

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 )}^{2}{\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)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.33526, size = 174, normalized size = 3.28 \begin{align*} -\frac{a^{3} x^{3} + a^{2} x^{2} - a x +{\left (2 \, a^{2} x^{2} + 2 \, a x - 1\right )} \sqrt{-a^{2} x^{2} + 1} - 1}{3 \,{\left (a^{4} c^{2} x^{3} + a^{3} c^{2} x^{2} - a^{2} c^{2} x - a c^{2}\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)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{- a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}} \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)**2,x)

[Out]

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

________________________________________________________________________________________

Giac [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 )}^{2}{\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)^2,x, algorithm="giac")

[Out]

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