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

Optimal. Leaf size=55 \[ \frac{a x+1}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{x}{3 a c^2 \sqrt{1-a^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0637577, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6148, 778, 191} \[ \frac{a x+1}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{x}{3 a c^2 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6148

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

Rule 778

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

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

Mathematica [A]  time = 0.0193405, size = 44, normalized size = 0.8 \[ \frac{-a^2 x^2+a x-1}{3 a^2 c^2 (a x-1) \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.03, size = 41, normalized size = 0.8 \begin{align*} -{\frac{{a}^{2}{x}^{2}-ax+1}{ \left ( 3\,ax-3 \right ){c}^{2}{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a \int \frac{x^{2}}{{\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}\,{d x} + \frac{1}{3 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.52715, size = 170, normalized size = 3.09 \begin{align*} \frac{a^{3} x^{3} - a^{2} x^{2} - a x +{\left (a^{2} x^{2} - a x + 1\right )} \sqrt{-a^{2} x^{2} + 1} + 1}{3 \,{\left (a^{5} c^{2} x^{3} - a^{4} c^{2} x^{2} - a^{3} c^{2} x + a^{2} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(a^3*x^3 - a^2*x^2 - a*x + (a^2*x^2 - a*x + 1)*sqrt(-a^2*x^2 + 1) + 1)/(a^5*c^2*x^3 - a^4*c^2*x^2 - a^3*c^
2*x + a^2*c^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{2}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \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((a*x+1)/(-a**2*x**2+1)**(1/2)*x/(-a**2*c*x**2+c)**2,x)

[Out]

(Integral(x/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + I
ntegral(a*x**2/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))
/c**2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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