3.1045 \(\int \frac{e^{2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^3}{x^2} \, dx\)

Optimal. Leaf size=76 \[ \frac{1}{5} a^6 c^3 x^5+\frac{1}{2} a^5 c^3 x^4-\frac{1}{3} a^4 c^3 x^3-2 a^3 c^3 x^2-a^2 c^3 x+2 a c^3 \log (x)-\frac{c^3}{x} \]

[Out]

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

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Rubi [A]  time = 0.0908948, antiderivative size = 76, 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} \[ \frac{1}{5} a^6 c^3 x^5+\frac{1}{2} a^5 c^3 x^4-\frac{1}{3} a^4 c^3 x^3-2 a^3 c^3 x^2-a^2 c^3 x+2 a c^3 \log (x)-\frac{c^3}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.025573, size = 59, normalized size = 0.78 \[ c^3 \left (\frac{a^6 x^5}{5}+\frac{a^5 x^4}{2}-\frac{a^4 x^3}{3}-2 a^3 x^2-a^2 x+2 a \log (x)-\frac{1}{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.031, size = 71, normalized size = 0.9 \begin{align*} -{\frac{{c}^{3}}{x}}-{a}^{2}{c}^{3}x-2\,{a}^{3}{c}^{3}{x}^{2}-{\frac{{a}^{4}{c}^{3}{x}^{3}}{3}}+{\frac{{a}^{5}{c}^{3}{x}^{4}}{2}}+{\frac{{a}^{6}{c}^{3}{x}^{5}}{5}}+2\,a{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^2,x)

[Out]

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

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Maxima [A]  time = 0.956938, size = 95, normalized size = 1.25 \begin{align*} \frac{1}{5} \, a^{6} c^{3} x^{5} + \frac{1}{2} \, a^{5} c^{3} x^{4} - \frac{1}{3} \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} - a^{2} c^{3} x + 2 \, a c^{3} \log \left (x\right ) - \frac{c^{3}}{x} \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^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.93189, size = 163, normalized size = 2.14 \begin{align*} \frac{6 \, a^{6} c^{3} x^{6} + 15 \, a^{5} c^{3} x^{5} - 10 \, a^{4} c^{3} x^{4} - 60 \, a^{3} c^{3} x^{3} - 30 \, a^{2} c^{3} x^{2} + 60 \, a c^{3} x \log \left (x\right ) - 30 \, c^{3}}{30 \, x} \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^2,x, algorithm="fricas")

[Out]

1/30*(6*a^6*c^3*x^6 + 15*a^5*c^3*x^5 - 10*a^4*c^3*x^4 - 60*a^3*c^3*x^3 - 30*a^2*c^3*x^2 + 60*a*c^3*x*log(x) -
30*c^3)/x

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Sympy [A]  time = 0.343045, size = 70, normalized size = 0.92 \begin{align*} \frac{a^{6} c^{3} x^{5}}{5} + \frac{a^{5} c^{3} x^{4}}{2} - \frac{a^{4} c^{3} x^{3}}{3} - 2 a^{3} c^{3} x^{2} - a^{2} c^{3} x + 2 a c^{3} \log{\left (x \right )} - \frac{c^{3}}{x} \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**2,x)

[Out]

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

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Giac [A]  time = 1.14991, size = 96, normalized size = 1.26 \begin{align*} \frac{1}{5} \, a^{6} c^{3} x^{5} + \frac{1}{2} \, a^{5} c^{3} x^{4} - \frac{1}{3} \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} - a^{2} c^{3} x + 2 \, a c^{3} \log \left ({\left | x \right |}\right ) - \frac{c^{3}}{x} \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^2,x, algorithm="giac")

[Out]

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