3.1044 \(\int \frac{e^{2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^3}{x} \, dx\)

Optimal. Leaf size=79 \[ \frac{1}{6} a^6 c^3 x^6+\frac{2}{5} a^5 c^3 x^5-\frac{1}{4} a^4 c^3 x^4-\frac{4}{3} a^3 c^3 x^3-\frac{1}{2} a^2 c^3 x^2+2 a c^3 x+c^3 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0856803, antiderivative size = 79, 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} \[ \frac{1}{6} a^6 c^3 x^6+\frac{2}{5} a^5 c^3 x^5-\frac{1}{4} a^4 c^3 x^4-\frac{4}{3} a^3 c^3 x^3-\frac{1}{2} a^2 c^3 x^2+2 a c^3 x+c^3 \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0290283, size = 51, normalized size = 0.65 \[ c^3 \left (\frac{1}{60} a x \left (10 a^5 x^5+24 a^4 x^4-15 a^3 x^3-80 a^2 x^2-30 a x+120\right )+\log (x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

c^3*((a*x*(120 - 30*a*x - 80*a^2*x^2 - 15*a^3*x^3 + 24*a^4*x^4 + 10*a^5*x^5))/60 + Log[x])

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Maple [A]  time = 0.026, size = 70, normalized size = 0.9 \begin{align*} 2\,a{c}^{3}x-{\frac{{a}^{2}{c}^{3}{x}^{2}}{2}}-{\frac{4\,{a}^{3}{c}^{3}{x}^{3}}{3}}-{\frac{{a}^{4}{c}^{3}{x}^{4}}{4}}+{\frac{2\,{a}^{5}{c}^{3}{x}^{5}}{5}}+{\frac{{a}^{6}{c}^{3}{x}^{6}}{6}}+{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,x)

[Out]

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

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Maxima [A]  time = 0.961181, size = 93, normalized size = 1.18 \begin{align*} \frac{1}{6} \, a^{6} c^{3} x^{6} + \frac{2}{5} \, a^{5} c^{3} x^{5} - \frac{1}{4} \, a^{4} c^{3} x^{4} - \frac{4}{3} \, a^{3} c^{3} x^{3} - \frac{1}{2} \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x + c^{3} \log \left (x\right ) \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,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.9625, size = 154, normalized size = 1.95 \begin{align*} \frac{1}{6} \, a^{6} c^{3} x^{6} + \frac{2}{5} \, a^{5} c^{3} x^{5} - \frac{1}{4} \, a^{4} c^{3} x^{4} - \frac{4}{3} \, a^{3} c^{3} x^{3} - \frac{1}{2} \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x + c^{3} \log \left (x\right ) \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,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.32037, size = 76, normalized size = 0.96 \begin{align*} \frac{a^{6} c^{3} x^{6}}{6} + \frac{2 a^{5} c^{3} x^{5}}{5} - \frac{a^{4} c^{3} x^{4}}{4} - \frac{4 a^{3} c^{3} x^{3}}{3} - \frac{a^{2} c^{3} x^{2}}{2} + 2 a c^{3} x + c^{3} \log{\left (x \right )} \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,x)

[Out]

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

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Giac [A]  time = 1.18015, size = 95, normalized size = 1.2 \begin{align*} \frac{1}{6} \, a^{6} c^{3} x^{6} + \frac{2}{5} \, a^{5} c^{3} x^{5} - \frac{1}{4} \, a^{4} c^{3} x^{4} - \frac{4}{3} \, a^{3} c^{3} x^{3} - \frac{1}{2} \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x + c^{3} \log \left ({\left | x \right |}\right ) \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,x, algorithm="giac")

[Out]

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