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]

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

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Rubi [A]  time = 0.09, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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 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]

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])

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*}

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Mathematica [A]  time = 0.04, 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]

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

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fricas [A]  time = 0.61, size = 49, normalized size = 1.58 \[ -\frac {2 \, a x + 1}{6 \, {\left (a^{7} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{3} - 2 \, a^{4} c^{3} x - a^{3} c^{3}\right )}} \]

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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giac [B]  time = 0.17, size = 76, normalized size = 2.45 \[ \frac {1}{32 \, a^{3} c^{3} {\left (\frac {2}{a x + 1} - 1\right )}} + \frac {\frac {3 \, a^{3} c^{6}}{a x + 1} + \frac {6 \, a^{3} c^{6}}{{\left (a x + 1\right )}^{2}} - \frac {4 \, a^{3} c^{6}}{{\left (a x + 1\right )}^{3}}}{48 \, a^{6} c^{9}} \]

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)

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maple [A]  time = 0.04, size = 54, normalized size = 1.74 \[ \frac {-\frac {1}{16 a^{3} \left (a x -1\right )}-\frac {1}{12 a^{3} \left (a x +1\right )^{3}}+\frac {1}{8 a^{3} \left (a x +1\right )^{2}}+\frac {1}{16 a^{3} \left (a x +1\right )}}{c^{3}} \]

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

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maxima [A]  time = 0.33, size = 49, normalized size = 1.58 \[ -\frac {2 \, a x + 1}{6 \, {\left (a^{7} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{3} - 2 \, a^{4} c^{3} x - a^{3} c^{3}\right )}} \]

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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mupad [B]  time = 0.11, size = 28, normalized size = 0.90 \[ -\frac {2\,a\,x+1}{6\,a^3\,c^3\,\left (a\,x-1\right )\,{\left (a\,x+1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.36, size = 49, normalized size = 1.58 \[ \frac {- 2 a x - 1}{6 a^{7} c^{3} x^{4} + 12 a^{6} c^{3} x^{3} - 12 a^{4} c^{3} x - 6 a^{3} c^{3}} \]

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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