3.1031 \(\int e^{2 \tanh ^{-1}(a x)} x (c-a^2 c x^2)^2 \, dx\)

Optimal. Leaf size=48 \[ -\frac{1}{6} a^4 c^2 x^6-\frac{2}{5} a^3 c^2 x^5+\frac{2}{3} a c^2 x^3+\frac{c^2 x^2}{2} \]

[Out]

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

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Rubi [A]  time = 0.066772, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {6150, 75} \[ -\frac{1}{6} a^4 c^2 x^6-\frac{2}{5} a^3 c^2 x^5+\frac{2}{3} a c^2 x^3+\frac{c^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

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

Mathematica [A]  time = 0.0182033, size = 32, normalized size = 0.67 \[ -\frac{1}{30} c^2 x^2 \left (5 a^4 x^4+12 a^3 x^3-20 a x-15\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^2*x^2*(-15 - 20*a*x + 12*a^3*x^3 + 5*a^4*x^4))/30

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Maple [A]  time = 0.024, size = 33, normalized size = 0.7 \begin{align*}{c}^{2} \left ( -{\frac{{x}^{6}{a}^{4}}{6}}-{\frac{2\,{x}^{5}{a}^{3}}{5}}+{\frac{2\,{x}^{3}a}{3}}+{\frac{{x}^{2}}{2}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.948752, size = 54, normalized size = 1.12 \begin{align*} -\frac{1}{6} \, a^{4} c^{2} x^{6} - \frac{2}{5} \, a^{3} c^{2} x^{5} + \frac{2}{3} \, a c^{2} x^{3} + \frac{1}{2} \, c^{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)*x*(-a^2*c*x^2+c)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.69476, size = 89, normalized size = 1.85 \begin{align*} -\frac{1}{6} \, a^{4} c^{2} x^{6} - \frac{2}{5} \, a^{3} c^{2} x^{5} + \frac{2}{3} \, a c^{2} x^{3} + \frac{1}{2} \, c^{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)*x*(-a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.087026, size = 44, normalized size = 0.92 \begin{align*} - \frac{a^{4} c^{2} x^{6}}{6} - \frac{2 a^{3} c^{2} x^{5}}{5} + \frac{2 a c^{2} x^{3}}{3} + \frac{c^{2} x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14133, size = 54, normalized size = 1.12 \begin{align*} -\frac{1}{6} \, a^{4} c^{2} x^{6} - \frac{2}{5} \, a^{3} c^{2} x^{5} + \frac{2}{3} \, a c^{2} x^{3} + \frac{1}{2} \, c^{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)*x*(-a^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

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