3.1228 \(\int e^{-2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^4 \, dx\)

Optimal. Leaf size=73 \[ \frac{c^4 (1-a x)^9}{9 a}-\frac{3 c^4 (1-a x)^8}{4 a}+\frac{12 c^4 (1-a x)^7}{7 a}-\frac{4 c^4 (1-a x)^6}{3 a} \]

[Out]

(-4*c^4*(1 - a*x)^6)/(3*a) + (12*c^4*(1 - a*x)^7)/(7*a) - (3*c^4*(1 - a*x)^8)/(4*a) + (c^4*(1 - a*x)^9)/(9*a)

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Rubi [A]  time = 0.0553365, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6140, 43} \[ \frac{c^4 (1-a x)^9}{9 a}-\frac{3 c^4 (1-a x)^8}{4 a}+\frac{12 c^4 (1-a x)^7}{7 a}-\frac{4 c^4 (1-a x)^6}{3 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*c^4*(1 - a*x)^6)/(3*a) + (12*c^4*(1 - a*x)^7)/(7*a) - (3*c^4*(1 - a*x)^8)/(4*a) + (c^4*(1 - a*x)^9)/(9*a)

Rule 6140

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a*x)^(p - n/2)*
(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx &=c^4 \int (1-a x)^5 (1+a x)^3 \, dx\\ &=c^4 \int \left (8 (1-a x)^5-12 (1-a x)^6+6 (1-a x)^7-(1-a x)^8\right ) \, dx\\ &=-\frac{4 c^4 (1-a x)^6}{3 a}+\frac{12 c^4 (1-a x)^7}{7 a}-\frac{3 c^4 (1-a x)^8}{4 a}+\frac{c^4 (1-a x)^9}{9 a}\\ \end{align*}

Mathematica [A]  time = 0.026989, size = 39, normalized size = 0.53 \[ -\frac{c^4 (a x-1)^6 \left (28 a^3 x^3+105 a^2 x^2+138 a x+65\right )}{252 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^4*(-1 + a*x)^6*(65 + 138*a*x + 105*a^2*x^2 + 28*a^3*x^3))/(252*a)

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Maple [A]  time = 0.024, size = 61, normalized size = 0.8 \begin{align*}{c}^{4} \left ( -{\frac{{x}^{9}{a}^{8}}{9}}+{\frac{{a}^{7}{x}^{8}}{4}}+{\frac{2\,{x}^{7}{a}^{6}}{7}}-{x}^{6}{a}^{5}+{\frac{3\,{x}^{4}{a}^{3}}{2}}-{\frac{2\,{x}^{3}{a}^{2}}{3}}-a{x}^{2}+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*(-1/9*x^9*a^8+1/4*a^7*x^8+2/7*x^7*a^6-x^6*a^5+3/2*x^4*a^3-2/3*x^3*a^2-a*x^2+x)

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Maxima [A]  time = 0.963072, size = 109, normalized size = 1.49 \begin{align*} -\frac{1}{9} \, a^{8} c^{4} x^{9} + \frac{1}{4} \, a^{7} c^{4} x^{8} + \frac{2}{7} \, a^{6} c^{4} x^{7} - a^{5} c^{4} x^{6} + \frac{3}{2} \, a^{3} c^{4} x^{4} - \frac{2}{3} \, a^{2} c^{4} x^{3} - a c^{4} x^{2} + c^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*a^8*c^4*x^9 + 1/4*a^7*c^4*x^8 + 2/7*a^6*c^4*x^7 - a^5*c^4*x^6 + 3/2*a^3*c^4*x^4 - 2/3*a^2*c^4*x^3 - a*c^4
*x^2 + c^4*x

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Fricas [A]  time = 2.07214, size = 167, normalized size = 2.29 \begin{align*} -\frac{1}{9} \, a^{8} c^{4} x^{9} + \frac{1}{4} \, a^{7} c^{4} x^{8} + \frac{2}{7} \, a^{6} c^{4} x^{7} - a^{5} c^{4} x^{6} + \frac{3}{2} \, a^{3} c^{4} x^{4} - \frac{2}{3} \, a^{2} c^{4} x^{3} - a c^{4} x^{2} + c^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*a^8*c^4*x^9 + 1/4*a^7*c^4*x^8 + 2/7*a^6*c^4*x^7 - a^5*c^4*x^6 + 3/2*a^3*c^4*x^4 - 2/3*a^2*c^4*x^3 - a*c^4
*x^2 + c^4*x

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Sympy [A]  time = 0.112576, size = 87, normalized size = 1.19 \begin{align*} - \frac{a^{8} c^{4} x^{9}}{9} + \frac{a^{7} c^{4} x^{8}}{4} + \frac{2 a^{6} c^{4} x^{7}}{7} - a^{5} c^{4} x^{6} + \frac{3 a^{3} c^{4} x^{4}}{2} - \frac{2 a^{2} c^{4} x^{3}}{3} - a c^{4} x^{2} + c^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**8*c**4*x**9/9 + a**7*c**4*x**8/4 + 2*a**6*c**4*x**7/7 - a**5*c**4*x**6 + 3*a**3*c**4*x**4/2 - 2*a**2*c**4*
x**3/3 - a*c**4*x**2 + c**4*x

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Giac [A]  time = 1.13551, size = 105, normalized size = 1.44 \begin{align*} -\frac{{\left (28 \, c^{4} - \frac{315 \, c^{4}}{a x + 1} + \frac{1440 \, c^{4}}{{\left (a x + 1\right )}^{2}} - \frac{3360 \, c^{4}}{{\left (a x + 1\right )}^{3}} + \frac{4032 \, c^{4}}{{\left (a x + 1\right )}^{4}} - \frac{2016 \, c^{4}}{{\left (a x + 1\right )}^{5}}\right )}{\left (a x + 1\right )}^{9}}{252 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/252*(28*c^4 - 315*c^4/(a*x + 1) + 1440*c^4/(a*x + 1)^2 - 3360*c^4/(a*x + 1)^3 + 4032*c^4/(a*x + 1)^4 - 2016
*c^4/(a*x + 1)^5)*(a*x + 1)^9/a