3.1046 \(\int \frac{e^{2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^3}{x^3} \, dx\)

Optimal. Leaf size=78 \[ \frac{1}{4} a^6 c^3 x^4+\frac{2}{3} a^5 c^3 x^3-\frac{1}{2} a^4 c^3 x^2-4 a^3 c^3 x-a^2 c^3 \log (x)-\frac{2 a c^3}{x}-\frac{c^3}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.095062, antiderivative size = 78, 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} \[ \frac{1}{4} a^6 c^3 x^4+\frac{2}{3} a^5 c^3 x^3-\frac{1}{2} a^4 c^3 x^2-4 a^3 c^3 x-a^2 c^3 \log (x)-\frac{2 a c^3}{x}-\frac{c^3}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0239966, size = 58, normalized size = 0.74 \[ \frac{c^3 \left (3 a^6 x^6+8 a^5 x^5-6 a^4 x^4-48 a^3 x^3-12 a^2 x^2 \log (x)-24 a x-6\right )}{12 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*(-6 - 24*a*x - 48*a^3*x^3 - 6*a^4*x^4 + 8*a^5*x^5 + 3*a^6*x^6 - 12*a^2*x^2*Log[x]))/(12*x^2)

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Maple [A]  time = 0.033, size = 71, normalized size = 0.9 \begin{align*} -{\frac{{c}^{3}}{2\,{x}^{2}}}-2\,{\frac{a{c}^{3}}{x}}-4\,{a}^{3}{c}^{3}x-{\frac{{a}^{4}{c}^{3}{x}^{2}}{2}}+{\frac{2\,{a}^{5}{c}^{3}{x}^{3}}{3}}+{\frac{{a}^{6}{c}^{3}{x}^{4}}{4}}-{a}^{2}{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^3,x)

[Out]

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

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Maxima [A]  time = 0.965557, size = 93, normalized size = 1.19 \begin{align*} \frac{1}{4} \, a^{6} c^{3} x^{4} + \frac{2}{3} \, a^{5} c^{3} x^{3} - \frac{1}{2} \, a^{4} c^{3} x^{2} - 4 \, a^{3} c^{3} x - a^{2} c^{3} \log \left (x\right ) - \frac{4 \, a c^{3} x + c^{3}}{2 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.01215, size = 162, normalized size = 2.08 \begin{align*} \frac{3 \, a^{6} c^{3} x^{6} + 8 \, a^{5} c^{3} x^{5} - 6 \, a^{4} c^{3} x^{4} - 48 \, a^{3} c^{3} x^{3} - 12 \, a^{2} c^{3} x^{2} \log \left (x\right ) - 24 \, a c^{3} x - 6 \, c^{3}}{12 \, x^{2}} \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^3,x, algorithm="fricas")

[Out]

1/12*(3*a^6*c^3*x^6 + 8*a^5*c^3*x^5 - 6*a^4*c^3*x^4 - 48*a^3*c^3*x^3 - 12*a^2*c^3*x^2*log(x) - 24*a*c^3*x - 6*
c^3)/x^2

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Sympy [A]  time = 0.370847, size = 73, normalized size = 0.94 \begin{align*} \frac{a^{6} c^{3} x^{4}}{4} + \frac{2 a^{5} c^{3} x^{3}}{3} - \frac{a^{4} c^{3} x^{2}}{2} - 4 a^{3} c^{3} x - a^{2} c^{3} \log{\left (x \right )} - \frac{4 a c^{3} x + c^{3}}{2 x^{2}} \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**3,x)

[Out]

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

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Giac [A]  time = 1.16721, size = 95, normalized size = 1.22 \begin{align*} \frac{1}{4} \, a^{6} c^{3} x^{4} + \frac{2}{3} \, a^{5} c^{3} x^{3} - \frac{1}{2} \, a^{4} c^{3} x^{2} - 4 \, a^{3} c^{3} x - a^{2} c^{3} \log \left ({\left | x \right |}\right ) - \frac{4 \, a c^{3} x + c^{3}}{2 \, x^{2}} \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^3,x, algorithm="giac")

[Out]

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