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

Optimal. Leaf size=88 \[ \frac{x^2 (a x+1)}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{2 x}{15 a^2 c^3 \sqrt{1-a^2 x^2}}-\frac{2 (1-a x)}{15 a^3 c^3 \left (1-a^2 x^2\right )^{3/2}} \]

[Out]

(x^2*(1 + a*x))/(5*a*c^3*(1 - a^2*x^2)^(5/2)) - (2*(1 - a*x))/(15*a^3*c^3*(1 - a^2*x^2)^(3/2)) - (2*x)/(15*a^2
*c^3*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.107533, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {6148, 796, 778, 191} \[ \frac{x^2 (a x+1)}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{2 x}{15 a^2 c^3 \sqrt{1-a^2 x^2}}-\frac{2 (1-a x)}{15 a^3 c^3 \left (1-a^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(1 + a*x))/(5*a*c^3*(1 - a^2*x^2)^(5/2)) - (2*(1 - a*x))/(15*a^3*c^3*(1 - a^2*x^2)^(3/2)) - (2*x)/(15*a^2
*c^3*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 796

Int[(x_)^2*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(x^2*(a*g - c*f*x)*(a + c*x^2)^(p
+ 1))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)), Int[x*Simp[2*a*g - c*f*(2*p + 5)*x, x]*(a + c*x^2)^(p + 1
), x], x] /; FreeQ[{a, c, f, g}, x] && EqQ[a*g^2 + f^2*c, 0] && LtQ[p, -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^2}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac{\int \frac{x^2 (1+a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=\frac{x^2 (1+a x)}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{\int \frac{x \left (2 a-2 a^2 x\right )}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 a^2 c^3}\\ &=\frac{x^2 (1+a x)}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{2 (1-a x)}{15 a^3 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{2 \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 a^2 c^3}\\ &=\frac{x^2 (1+a x)}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{2 (1-a x)}{15 a^3 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{2 x}{15 a^2 c^3 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0263293, size = 68, normalized size = 0.77 \[ \frac{-2 a^4 x^4+2 a^3 x^3+3 a^2 x^2+2 a x-2}{15 a^3 c^3 (a x-1)^2 (a x+1) \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2 + 2*a*x + 3*a^2*x^2 + 2*a^3*x^3 - 2*a^4*x^4)/(15*a^3*c^3*(-1 + a*x)^2*(1 + a*x)*Sqrt[1 - a^2*x^2])

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Maple [A]  time = 0.033, size = 58, normalized size = 0.7 \begin{align*}{\frac{2\,{x}^{4}{a}^{4}-2\,{x}^{3}{a}^{3}-3\,{a}^{2}{x}^{2}-2\,ax+2}{ \left ( 15\,ax-15 \right ){c}^{3}{a}^{3}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \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^2/(-a^2*c*x^2+c)^3,x)

[Out]

1/15*(2*a^4*x^4-2*a^3*x^3-3*a^2*x^2-2*a*x+2)/(a*x-1)/c^3/(-a^2*x^2+1)^(3/2)/a^3

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

[Out]

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

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Fricas [A]  time = 1.57908, size = 294, normalized size = 3.34 \begin{align*} -\frac{2 \, a^{5} x^{5} - 2 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + 2 \, a x -{\left (2 \, a^{4} x^{4} - 2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 2 \, a x + 2\right )} \sqrt{-a^{2} x^{2} + 1} - 2}{15 \,{\left (a^{8} c^{3} x^{5} - a^{7} c^{3} x^{4} - 2 \, a^{6} c^{3} x^{3} + 2 \, a^{5} c^{3} x^{2} + a^{4} c^{3} x - a^{3} c^{3}\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^2/(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

-1/15*(2*a^5*x^5 - 2*a^4*x^4 - 4*a^3*x^3 + 4*a^2*x^2 + 2*a*x - (2*a^4*x^4 - 2*a^3*x^3 - 3*a^2*x^2 - 2*a*x + 2)
*sqrt(-a^2*x^2 + 1) - 2)/(a^8*c^3*x^5 - a^7*c^3*x^4 - 2*a^6*c^3*x^3 + 2*a^5*c^3*x^2 + a^4*c^3*x - a^3*c^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{2}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{3}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}} \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**2/(-a**2*c*x**2+c)**3,x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (a x + 1\right )} x^{2}}{{\left (a^{2} c x^{2} - c\right )}^{3} \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^2/(-a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

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