3.1371 \(\int \frac{e^{-2 \tanh ^{-1}(a x)} x^2}{(c-a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=31 \[ \frac{2 a x+1}{6 a^3 c^3 (1-a x) (a x+1)^3} \]

[Out]

(1 + 2*a*x)/(6*a^3*c^3*(1 - a*x)*(1 + a*x)^3)

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Rubi [A]  time = 0.0893645, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6150, 81} \[ \frac{2 a x+1}{6 a^3 c^3 (1-a x) (a x+1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + 2*a*x)/(6*a^3*c^3*(1 - a*x)*(1 + a*x)^3)

Rule 6150

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

Rule 81

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*
x)^(n + 1)*(e + f*x)^(p + 1)*(2*a*d*f*(n + p + 3) - b*(d*e*(n + 2) + c*f*(p + 2)) + b*d*f*(n + p + 2)*x))/(d^2
*f^2*(n + p + 2)*(n + p + 3)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && NeQ[n + p + 3,
0] && EqQ[d*f*(n + p + 2)*(a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1)))) - b*(d*e*(n + 1)
+ c*f*(p + 1))*(a*d*f*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2))), 0]

Rubi steps

\begin{align*} \int \frac{e^{-2 \tanh ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac{\int \frac{x^2}{(1-a x)^2 (1+a x)^4} \, dx}{c^3}\\ &=\frac{1+2 a x}{6 a^3 c^3 (1-a x) (1+a x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0304598, size = 30, normalized size = 0.97 \[ -\frac{2 a x+1}{6 a^3 c^3 (a x-1) (a x+1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1 + 2*a*x)/(6*a^3*c^3*(-1 + a*x)*(1 + a*x)^3)

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Maple [A]  time = 0.033, size = 54, normalized size = 1.7 \begin{align*}{\frac{1}{{c}^{3}} \left ( -{\frac{1}{12\,{a}^{3} \left ( ax+1 \right ) ^{3}}}+{\frac{1}{8\,{a}^{3} \left ( ax+1 \right ) ^{2}}}+{\frac{1}{16\,{a}^{3} \left ( ax+1 \right ) }}-{\frac{1}{16\,{a}^{3} \left ( ax-1 \right ) }} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(-1/12/a^3/(a*x+1)^3+1/8/a^3/(a*x+1)^2+1/16/a^3/(a*x+1)-1/16/a^3/(a*x-1))

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Maxima [A]  time = 0.959972, size = 66, normalized size = 2.13 \begin{align*} -\frac{2 \, a x + 1}{6 \,{\left (a^{7} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{3} - 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(x^2/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.90108, size = 97, normalized size = 3.13 \begin{align*} -\frac{2 \, a x + 1}{6 \,{\left (a^{7} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{3} - 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(x^2/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.580487, size = 49, normalized size = 1.58 \begin{align*} - \frac{2 a x + 1}{6 a^{7} c^{3} x^{4} + 12 a^{6} c^{3} x^{3} - 12 a^{4} c^{3} x - 6 a^{3} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a*x + 1)/(6*a**7*c**3*x**4 + 12*a**6*c**3*x**3 - 12*a**4*c**3*x - 6*a**3*c**3)

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Giac [B]  time = 1.136, size = 103, normalized size = 3.32 \begin{align*} \frac{1}{32 \, a^{3} c^{3}{\left (\frac{2}{a x + 1} - 1\right )}} + \frac{\frac{3 \, a^{3} c^{6}}{a x + 1} + \frac{6 \, a^{3} c^{6}}{{\left (a x + 1\right )}^{2}} - \frac{4 \, a^{3} c^{6}}{{\left (a x + 1\right )}^{3}}}{48 \, a^{6} c^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32/(a^3*c^3*(2/(a*x + 1) - 1)) + 1/48*(3*a^3*c^6/(a*x + 1) + 6*a^3*c^6/(a*x + 1)^2 - 4*a^3*c^6/(a*x + 1)^3)/
(a^6*c^9)