3.583 \(\int e^{4 \coth ^{-1}(a x)} (c-a^2 c x^2)^3 \, dx\)

Optimal. Leaf size=35 \[ \frac{c^3 (a x+1)^6}{3 a}-\frac{c^3 (a x+1)^7}{7 a} \]

[Out]

(c^3*(1 + a*x)^6)/(3*a) - (c^3*(1 + a*x)^7)/(7*a)

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Rubi [A]  time = 0.0628281, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6167, 6140, 43} \[ \frac{c^3 (a x+1)^6}{3 a}-\frac{c^3 (a x+1)^7}{7 a} \]

Antiderivative was successfully verified.

[In]

Int[E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^3,x]

[Out]

(c^3*(1 + a*x)^6)/(3*a) - (c^3*(1 + a*x)^7)/(7*a)

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

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

Mathematica [A]  time = 0.0206693, size = 23, normalized size = 0.66 \[ -\frac{c^3 (a x+1)^6 (3 a x-4)}{21 a} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^3,x]

[Out]

-(c^3*(1 + a*x)^6*(-4 + 3*a*x))/(21*a)

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Maple [A]  time = 0.041, size = 45, normalized size = 1.3 \begin{align*}{c}^{3} \left ( -{\frac{{x}^{7}{a}^{6}}{7}}-{\frac{2\,{x}^{6}{a}^{5}}{3}}-{x}^{5}{a}^{4}+{\frac{5\,{x}^{3}{a}^{2}}{3}}+2\,a{x}^{2}+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x-1)^2*(a*x+1)^2*(-a^2*c*x^2+c)^3,x)

[Out]

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

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Maxima [A]  time = 0.980798, size = 80, normalized size = 2.29 \begin{align*} -\frac{1}{7} \, a^{6} c^{3} x^{7} - \frac{2}{3} \, a^{5} c^{3} x^{6} - a^{4} c^{3} x^{5} + \frac{5}{3} \, a^{2} c^{3} x^{3} + 2 \, a c^{3} x^{2} + c^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(-a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.51495, size = 122, normalized size = 3.49 \begin{align*} -\frac{1}{7} \, a^{6} c^{3} x^{7} - \frac{2}{3} \, a^{5} c^{3} x^{6} - a^{4} c^{3} x^{5} + \frac{5}{3} \, a^{2} c^{3} x^{3} + 2 \, a c^{3} x^{2} + c^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.096159, size = 63, normalized size = 1.8 \begin{align*} - \frac{a^{6} c^{3} x^{7}}{7} - \frac{2 a^{5} c^{3} x^{6}}{3} - a^{4} c^{3} x^{5} + \frac{5 a^{2} c^{3} x^{3}}{3} + 2 a c^{3} x^{2} + c^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)**2*(a*x+1)**2*(-a**2*c*x**2+c)**3,x)

[Out]

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

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Giac [B]  time = 1.15696, size = 105, normalized size = 3. \begin{align*} -\frac{{\left (3 \, c^{3} + \frac{35 \, c^{3}}{a x - 1} + \frac{168 \, c^{3}}{{\left (a x - 1\right )}^{2}} + \frac{420 \, c^{3}}{{\left (a x - 1\right )}^{3}} + \frac{560 \, c^{3}}{{\left (a x - 1\right )}^{4}} + \frac{336 \, c^{3}}{{\left (a x - 1\right )}^{5}}\right )}{\left (a x - 1\right )}^{7}}{21 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(-a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

-1/21*(3*c^3 + 35*c^3/(a*x - 1) + 168*c^3/(a*x - 1)^2 + 420*c^3/(a*x - 1)^3 + 560*c^3/(a*x - 1)^4 + 336*c^3/(a
*x - 1)^5)*(a*x - 1)^7/a