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

Optimal. Leaf size=90 \[ -\frac{3 c^4}{a^3 x^2}+\frac{3 c^4}{2 a^5 x^4}-\frac{2 c^4}{5 a^6 x^5}-\frac{c^4}{3 a^7 x^6}+\frac{c^4}{7 a^8 x^7}+\frac{2 c^4}{a^2 x}-\frac{2 c^4 \log (x)}{a}+c^4 x \]

[Out]

c^4/(7*a^8*x^7) - c^4/(3*a^7*x^6) - (2*c^4)/(5*a^6*x^5) + (3*c^4)/(2*a^5*x^4) - (3*c^4)/(a^3*x^2) + (2*c^4)/(a
^2*x) + c^4*x - (2*c^4*Log[x])/a

________________________________________________________________________________________

Rubi [A]  time = 0.153422, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6167, 6157, 6150, 88} \[ -\frac{3 c^4}{a^3 x^2}+\frac{3 c^4}{2 a^5 x^4}-\frac{2 c^4}{5 a^6 x^5}-\frac{c^4}{3 a^7 x^6}+\frac{c^4}{7 a^8 x^7}+\frac{2 c^4}{a^2 x}-\frac{2 c^4 \log (x)}{a}+c^4 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

c^4/(7*a^8*x^7) - c^4/(3*a^7*x^6) - (2*c^4)/(5*a^6*x^5) + (3*c^4)/(2*a^5*x^4) - (3*c^4)/(a^3*x^2) + (2*c^4)/(a
^2*x) + c^4*x - (2*c^4*Log[x])/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 6157

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

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

Mathematica [A]  time = 0.0329902, size = 90, normalized size = 1. \[ -\frac{3 c^4}{a^3 x^2}+\frac{3 c^4}{2 a^5 x^4}-\frac{2 c^4}{5 a^6 x^5}-\frac{c^4}{3 a^7 x^6}+\frac{c^4}{7 a^8 x^7}+\frac{2 c^4}{a^2 x}-\frac{2 c^4 \log (x)}{a}+c^4 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

c^4/(7*a^8*x^7) - c^4/(3*a^7*x^6) - (2*c^4)/(5*a^6*x^5) + (3*c^4)/(2*a^5*x^4) - (3*c^4)/(a^3*x^2) + (2*c^4)/(a
^2*x) + c^4*x - (2*c^4*Log[x])/a

________________________________________________________________________________________

Maple [A]  time = 0.049, size = 83, normalized size = 0.9 \begin{align*}{\frac{{c}^{4}}{7\,{a}^{8}{x}^{7}}}-{\frac{{c}^{4}}{3\,{a}^{7}{x}^{6}}}-{\frac{2\,{c}^{4}}{5\,{a}^{6}{x}^{5}}}+{\frac{3\,{c}^{4}}{2\,{a}^{5}{x}^{4}}}-3\,{\frac{{c}^{4}}{{x}^{2}{a}^{3}}}+2\,{\frac{{c}^{4}}{{a}^{2}x}}+{c}^{4}x-2\,{\frac{{c}^{4}\ln \left ( x \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.03559, size = 109, normalized size = 1.21 \begin{align*} c^{4} x - \frac{2 \, c^{4} \log \left (x\right )}{a} + \frac{420 \, a^{6} c^{4} x^{6} - 630 \, a^{5} c^{4} x^{5} + 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} - 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*x - 2*c^4*log(x)/a + 1/210*(420*a^6*c^4*x^6 - 630*a^5*c^4*x^5 + 315*a^3*c^4*x^3 - 84*a^2*c^4*x^2 - 70*a*c^
4*x + 30*c^4)/(a^8*x^7)

________________________________________________________________________________________

Fricas [A]  time = 1.34407, size = 207, normalized size = 2.3 \begin{align*} \frac{210 \, a^{8} c^{4} x^{8} - 420 \, a^{7} c^{4} x^{7} \log \left (x\right ) + 420 \, a^{6} c^{4} x^{6} - 630 \, a^{5} c^{4} x^{5} + 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} - 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/210*(210*a^8*c^4*x^8 - 420*a^7*c^4*x^7*log(x) + 420*a^6*c^4*x^6 - 630*a^5*c^4*x^5 + 315*a^3*c^4*x^3 - 84*a^2
*c^4*x^2 - 70*a*c^4*x + 30*c^4)/(a^8*x^7)

________________________________________________________________________________________

Sympy [A]  time = 0.653665, size = 88, normalized size = 0.98 \begin{align*} \frac{a^{8} c^{4} x - 2 a^{7} c^{4} \log{\left (x \right )} + \frac{420 a^{6} c^{4} x^{6} - 630 a^{5} c^{4} x^{5} + 315 a^{3} c^{4} x^{3} - 84 a^{2} c^{4} x^{2} - 70 a c^{4} x + 30 c^{4}}{210 x^{7}}}{a^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a**8*c**4*x - 2*a**7*c**4*log(x) + (420*a**6*c**4*x**6 - 630*a**5*c**4*x**5 + 315*a**3*c**4*x**3 - 84*a**2*c*
*4*x**2 - 70*a*c**4*x + 30*c**4)/(210*x**7))/a**8

________________________________________________________________________________________

Giac [A]  time = 1.12806, size = 111, normalized size = 1.23 \begin{align*} c^{4} x - \frac{2 \, c^{4} \log \left ({\left | x \right |}\right )}{a} + \frac{420 \, a^{6} c^{4} x^{6} - 630 \, a^{5} c^{4} x^{5} + 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} - 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*x - 2*c^4*log(abs(x))/a + 1/210*(420*a^6*c^4*x^6 - 630*a^5*c^4*x^5 + 315*a^3*c^4*x^3 - 84*a^2*c^4*x^2 - 70
*a*c^4*x + 30*c^4)/(a^8*x^7)