3.798 $$\int e^{4 \coth ^{-1}(a x)} (c-\frac{c}{a^2 x^2})^4 \, dx$$

Optimal. Leaf size=100 $\frac{2 c^4}{a^3 x^2}+\frac{10 c^4}{3 a^4 x^3}+\frac{c^4}{a^5 x^4}-\frac{4 c^4}{5 a^6 x^5}-\frac{2 c^4}{3 a^7 x^6}-\frac{c^4}{7 a^8 x^7}-\frac{4 c^4}{a^2 x}+\frac{4 c^4 \log (x)}{a}+c^4 x$

[Out]

-c^4/(7*a^8*x^7) - (2*c^4)/(3*a^7*x^6) - (4*c^4)/(5*a^6*x^5) + c^4/(a^5*x^4) + (10*c^4)/(3*a^4*x^3) + (2*c^4)/
(a^3*x^2) - (4*c^4)/(a^2*x) + c^4*x + (4*c^4*Log[x])/a

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Rubi [A]  time = 0.166298, antiderivative size = 100, 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} $\frac{2 c^4}{a^3 x^2}+\frac{10 c^4}{3 a^4 x^3}+\frac{c^4}{a^5 x^4}-\frac{4 c^4}{5 a^6 x^5}-\frac{2 c^4}{3 a^7 x^6}-\frac{c^4}{7 a^8 x^7}-\frac{4 c^4}{a^2 x}+\frac{4 c^4 \log (x)}{a}+c^4 x$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-c^4/(7*a^8*x^7) - (2*c^4)/(3*a^7*x^6) - (4*c^4)/(5*a^6*x^5) + c^4/(a^5*x^4) + (10*c^4)/(3*a^4*x^3) + (2*c^4)/
(a^3*x^2) - (4*c^4)/(a^2*x) + c^4*x + (4*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^{4 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^4 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^4 \, dx\\ &=\frac{c^4 \int \frac{e^{4 \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)^2 (1+a x)^6}{x^8} \, dx}{a^8}\\ &=\frac{c^4 \int \left (a^8+\frac{1}{x^8}+\frac{4 a}{x^7}+\frac{4 a^2}{x^6}-\frac{4 a^3}{x^5}-\frac{10 a^4}{x^4}-\frac{4 a^5}{x^3}+\frac{4 a^6}{x^2}+\frac{4 a^7}{x}\right ) \, dx}{a^8}\\ &=-\frac{c^4}{7 a^8 x^7}-\frac{2 c^4}{3 a^7 x^6}-\frac{4 c^4}{5 a^6 x^5}+\frac{c^4}{a^5 x^4}+\frac{10 c^4}{3 a^4 x^3}+\frac{2 c^4}{a^3 x^2}-\frac{4 c^4}{a^2 x}+c^4 x+\frac{4 c^4 \log (x)}{a}\\ \end{align*}

Mathematica [A]  time = 0.0329438, size = 100, normalized size = 1. $\frac{2 c^4}{a^3 x^2}+\frac{10 c^4}{3 a^4 x^3}+\frac{c^4}{a^5 x^4}-\frac{4 c^4}{5 a^6 x^5}-\frac{2 c^4}{3 a^7 x^6}-\frac{c^4}{7 a^8 x^7}-\frac{4 c^4}{a^2 x}+\frac{4 c^4 \log (x)}{a}+c^4 x$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-c^4/(7*a^8*x^7) - (2*c^4)/(3*a^7*x^6) - (4*c^4)/(5*a^6*x^5) + c^4/(a^5*x^4) + (10*c^4)/(3*a^4*x^3) + (2*c^4)/
(a^3*x^2) - (4*c^4)/(a^2*x) + c^4*x + (4*c^4*Log[x])/a

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Maple [A]  time = 0.053, size = 93, normalized size = 0.9 \begin{align*} -{\frac{{c}^{4}}{7\,{a}^{8}{x}^{7}}}-{\frac{2\,{c}^{4}}{3\,{a}^{7}{x}^{6}}}-{\frac{4\,{c}^{4}}{5\,{a}^{6}{x}^{5}}}+{\frac{{c}^{4}}{{a}^{5}{x}^{4}}}+{\frac{10\,{c}^{4}}{3\,{a}^{4}{x}^{3}}}+2\,{\frac{{c}^{4}}{{x}^{2}{a}^{3}}}-4\,{\frac{{c}^{4}}{{a}^{2}x}}+{c}^{4}x+4\,{\frac{{c}^{4}\ln \left ( x \right ) }{a}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*c^4/a^8/x^7-2/3*c^4/a^7/x^6-4/5*c^4/a^6/x^5+c^4/a^5/x^4+10/3*c^4/a^4/x^3+2*c^4/x^2/a^3-4*c^4/a^2/x+c^4*x+
4*c^4*ln(x)/a

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Maxima [A]  time = 1.02203, size = 124, normalized size = 1.24 \begin{align*} c^{4} x + \frac{4 \, c^{4} \log \left (x\right )}{a} - \frac{420 \, a^{6} c^{4} x^{6} - 210 \, a^{5} c^{4} x^{5} - 350 \, a^{4} c^{4} x^{4} - 105 \, a^{3} c^{4} x^{3} + 84 \, a^{2} c^{4} x^{2} + 70 \, a c^{4} x + 15 \, c^{4}}{105 \, a^{8} x^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*x + 4*c^4*log(x)/a - 1/105*(420*a^6*c^4*x^6 - 210*a^5*c^4*x^5 - 350*a^4*c^4*x^4 - 105*a^3*c^4*x^3 + 84*a^2
*c^4*x^2 + 70*a*c^4*x + 15*c^4)/(a^8*x^7)

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Fricas [A]  time = 1.4738, size = 231, normalized size = 2.31 \begin{align*} \frac{105 \, a^{8} c^{4} x^{8} + 420 \, a^{7} c^{4} x^{7} \log \left (x\right ) - 420 \, a^{6} c^{4} x^{6} + 210 \, a^{5} c^{4} x^{5} + 350 \, a^{4} c^{4} x^{4} + 105 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} - 70 \, a c^{4} x - 15 \, c^{4}}{105 \, a^{8} x^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(105*a^8*c^4*x^8 + 420*a^7*c^4*x^7*log(x) - 420*a^6*c^4*x^6 + 210*a^5*c^4*x^5 + 350*a^4*c^4*x^4 + 105*a^
3*c^4*x^3 - 84*a^2*c^4*x^2 - 70*a*c^4*x - 15*c^4)/(a^8*x^7)

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Sympy [A]  time = 0.721691, size = 100, normalized size = 1. \begin{align*} \frac{a^{8} c^{4} x + 4 a^{7} c^{4} \log{\left (x \right )} - \frac{420 a^{6} c^{4} x^{6} - 210 a^{5} c^{4} x^{5} - 350 a^{4} c^{4} x^{4} - 105 a^{3} c^{4} x^{3} + 84 a^{2} c^{4} x^{2} + 70 a c^{4} x + 15 c^{4}}{105 x^{7}}}{a^{8}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a**8*c**4*x + 4*a**7*c**4*log(x) - (420*a**6*c**4*x**6 - 210*a**5*c**4*x**5 - 350*a**4*c**4*x**4 - 105*a**3*c
**4*x**3 + 84*a**2*c**4*x**2 + 70*a*c**4*x + 15*c**4)/(105*x**7))/a**8

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Giac [A]  time = 1.12703, size = 216, normalized size = 2.16 \begin{align*} -\frac{4 \, c^{4} \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a} + \frac{4 \, c^{4} \log \left ({\left | -\frac{1}{a x - 1} - 1 \right |}\right )}{a} + \frac{{\left (105 \, c^{4} + \frac{659 \, c^{4}}{a x - 1} + \frac{1253 \, c^{4}}{{\left (a x - 1\right )}^{2}} - \frac{231 \, c^{4}}{{\left (a x - 1\right )}^{3}} - \frac{3885 \, c^{4}}{{\left (a x - 1\right )}^{4}} - \frac{5250 \, c^{4}}{{\left (a x - 1\right )}^{5}} - \frac{2730 \, c^{4}}{{\left (a x - 1\right )}^{6}} - \frac{420 \, c^{4}}{{\left (a x - 1\right )}^{7}}\right )}{\left (a x - 1\right )}}{105 \, a{\left (\frac{1}{a x - 1} + 1\right )}^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*c^4*log(abs(a*x - 1)/((a*x - 1)^2*abs(a)))/a + 4*c^4*log(abs(-1/(a*x - 1) - 1))/a + 1/105*(105*c^4 + 659*c^
4/(a*x - 1) + 1253*c^4/(a*x - 1)^2 - 231*c^4/(a*x - 1)^3 - 3885*c^4/(a*x - 1)^4 - 5250*c^4/(a*x - 1)^5 - 2730*
c^4/(a*x - 1)^6 - 420*c^4/(a*x - 1)^7)*(a*x - 1)/(a*(1/(a*x - 1) + 1)^7)