3.1047 \(\int \frac{e^{2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^3}{x^4} \, dx\)

Optimal. Leaf size=72 \[ \frac{1}{3} a^6 c^3 x^3+a^5 c^3 x^2-a^4 c^3 x+\frac{a^2 c^3}{x}-4 a^3 c^3 \log (x)-\frac{a c^3}{x^2}-\frac{c^3}{3 x^3} \]

[Out]

-c^3/(3*x^3) - (a*c^3)/x^2 + (a^2*c^3)/x - a^4*c^3*x + a^5*c^3*x^2 + (a^6*c^3*x^3)/3 - 4*a^3*c^3*Log[x]

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Rubi [A]  time = 0.0929093, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6150, 88} \[ \frac{1}{3} a^6 c^3 x^3+a^5 c^3 x^2-a^4 c^3 x+\frac{a^2 c^3}{x}-4 a^3 c^3 \log (x)-\frac{a c^3}{x^2}-\frac{c^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^3)/x^4,x]

[Out]

-c^3/(3*x^3) - (a*c^3)/x^2 + (a^2*c^3)/x - a^4*c^3*x + a^5*c^3*x^2 + (a^6*c^3*x^3)/3 - 4*a^3*c^3*Log[x]

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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^3}{x^4} \, dx &=c^3 \int \frac{(1-a x)^2 (1+a x)^4}{x^4} \, dx\\ &=c^3 \int \left (-a^4+\frac{1}{x^4}+\frac{2 a}{x^3}-\frac{a^2}{x^2}-\frac{4 a^3}{x}+2 a^5 x+a^6 x^2\right ) \, dx\\ &=-\frac{c^3}{3 x^3}-\frac{a c^3}{x^2}+\frac{a^2 c^3}{x}-a^4 c^3 x+a^5 c^3 x^2+\frac{1}{3} a^6 c^3 x^3-4 a^3 c^3 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0293488, size = 55, normalized size = 0.76 \[ c^3 \left (\frac{a^6 x^3}{3}+a^5 x^2-a^4 x+\frac{a^2}{x}-4 a^3 \log (x)-\frac{a}{x^2}-\frac{1}{3 x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^3)/x^4,x]

[Out]

c^3*(-1/(3*x^3) - a/x^2 + a^2/x - a^4*x + a^5*x^2 + (a^6*x^3)/3 - 4*a^3*Log[x])

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Maple [A]  time = 0.031, size = 69, normalized size = 1. \begin{align*} -{\frac{{c}^{3}}{3\,{x}^{3}}}-{\frac{a{c}^{3}}{{x}^{2}}}+{\frac{{a}^{2}{c}^{3}}{x}}-{a}^{4}{c}^{3}x+{a}^{5}{c}^{3}{x}^{2}+{\frac{{a}^{6}{c}^{3}{x}^{3}}{3}}-4\,{a}^{3}{c}^{3}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*c^3/x^3-a*c^3/x^2+a^2*c^3/x-a^4*c^3*x+a^5*c^3*x^2+1/3*a^6*c^3*x^3-4*a^3*c^3*ln(x)

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Maxima [A]  time = 0.942724, size = 95, normalized size = 1.32 \begin{align*} \frac{1}{3} \, a^{6} c^{3} x^{3} + a^{5} c^{3} x^{2} - a^{4} c^{3} x - 4 \, a^{3} c^{3} \log \left (x\right ) + \frac{3 \, a^{2} c^{3} x^{2} - 3 \, a c^{3} x - c^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^3/x^4,x, algorithm="maxima")

[Out]

1/3*a^6*c^3*x^3 + a^5*c^3*x^2 - a^4*c^3*x - 4*a^3*c^3*log(x) + 1/3*(3*a^2*c^3*x^2 - 3*a*c^3*x - c^3)/x^3

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Fricas [A]  time = 2.03493, size = 153, normalized size = 2.12 \begin{align*} \frac{a^{6} c^{3} x^{6} + 3 \, a^{5} c^{3} x^{5} - 3 \, a^{4} c^{3} x^{4} - 12 \, a^{3} c^{3} x^{3} \log \left (x\right ) + 3 \, a^{2} c^{3} x^{2} - 3 \, a c^{3} x - c^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^3/x^4,x, algorithm="fricas")

[Out]

1/3*(a^6*c^3*x^6 + 3*a^5*c^3*x^5 - 3*a^4*c^3*x^4 - 12*a^3*c^3*x^3*log(x) + 3*a^2*c^3*x^2 - 3*a*c^3*x - c^3)/x^
3

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Sympy [A]  time = 0.409236, size = 70, normalized size = 0.97 \begin{align*} \frac{a^{6} c^{3} x^{3}}{3} + a^{5} c^{3} x^{2} - a^{4} c^{3} x - 4 a^{3} c^{3} \log{\left (x \right )} + \frac{3 a^{2} c^{3} x^{2} - 3 a c^{3} x - c^{3}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**6*c**3*x**3/3 + a**5*c**3*x**2 - a**4*c**3*x - 4*a**3*c**3*log(x) + (3*a**2*c**3*x**2 - 3*a*c**3*x - c**3)/
(3*x**3)

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Giac [A]  time = 1.15008, size = 96, normalized size = 1.33 \begin{align*} \frac{1}{3} \, a^{6} c^{3} x^{3} + a^{5} c^{3} x^{2} - a^{4} c^{3} x - 4 \, a^{3} c^{3} \log \left ({\left | x \right |}\right ) + \frac{3 \, a^{2} c^{3} x^{2} - 3 \, a c^{3} x - c^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^3/x^4,x, algorithm="giac")

[Out]

1/3*a^6*c^3*x^3 + a^5*c^3*x^2 - a^4*c^3*x - 4*a^3*c^3*log(abs(x)) + 1/3*(3*a^2*c^3*x^2 - 3*a*c^3*x - c^3)/x^3