### 3.788 $$\int \frac{e^{2 \coth ^{-1}(a x)}}{(c-\frac{c}{a^2 x^2})^4} \, dx$$

Optimal. Leaf size=145 $\frac{99}{32 a c^4 (1-a x)}-\frac{11}{64 a c^4 (a x+1)}-\frac{35}{32 a c^4 (1-a x)^2}+\frac{1}{64 a c^4 (a x+1)^2}+\frac{13}{48 a c^4 (1-a x)^3}-\frac{1}{32 a c^4 (1-a x)^4}+\frac{303 \log (1-a x)}{128 a c^4}-\frac{47 \log (a x+1)}{128 a c^4}+\frac{x}{c^4}$

[Out]

x/c^4 - 1/(32*a*c^4*(1 - a*x)^4) + 13/(48*a*c^4*(1 - a*x)^3) - 35/(32*a*c^4*(1 - a*x)^2) + 99/(32*a*c^4*(1 - a
*x)) + 1/(64*a*c^4*(1 + a*x)^2) - 11/(64*a*c^4*(1 + a*x)) + (303*Log[1 - a*x])/(128*a*c^4) - (47*Log[1 + a*x])
/(128*a*c^4)

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Rubi [A]  time = 0.228441, antiderivative size = 145, 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} $\frac{99}{32 a c^4 (1-a x)}-\frac{11}{64 a c^4 (a x+1)}-\frac{35}{32 a c^4 (1-a x)^2}+\frac{1}{64 a c^4 (a x+1)^2}+\frac{13}{48 a c^4 (1-a x)^3}-\frac{1}{32 a c^4 (1-a x)^4}+\frac{303 \log (1-a x)}{128 a c^4}-\frac{47 \log (a x+1)}{128 a c^4}+\frac{x}{c^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

x/c^4 - 1/(32*a*c^4*(1 - a*x)^4) + 13/(48*a*c^4*(1 - a*x)^3) - 35/(32*a*c^4*(1 - a*x)^2) + 99/(32*a*c^4*(1 - a
*x)) + 1/(64*a*c^4*(1 + a*x)^2) - 11/(64*a*c^4*(1 + a*x)) + (303*Log[1 - a*x])/(128*a*c^4) - (47*Log[1 + a*x])
/(128*a*c^4)

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 \frac{e^{2 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^4} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^4} \, dx\\ &=-\frac{a^8 \int \frac{e^{2 \tanh ^{-1}(a x)} x^8}{\left (1-a^2 x^2\right )^4} \, dx}{c^4}\\ &=-\frac{a^8 \int \frac{x^8}{(1-a x)^5 (1+a x)^3} \, dx}{c^4}\\ &=-\frac{a^8 \int \left (-\frac{1}{a^8}-\frac{1}{8 a^8 (-1+a x)^5}-\frac{13}{16 a^8 (-1+a x)^4}-\frac{35}{16 a^8 (-1+a x)^3}-\frac{99}{32 a^8 (-1+a x)^2}-\frac{303}{128 a^8 (-1+a x)}+\frac{1}{32 a^8 (1+a x)^3}-\frac{11}{64 a^8 (1+a x)^2}+\frac{47}{128 a^8 (1+a x)}\right ) \, dx}{c^4}\\ &=\frac{x}{c^4}-\frac{1}{32 a c^4 (1-a x)^4}+\frac{13}{48 a c^4 (1-a x)^3}-\frac{35}{32 a c^4 (1-a x)^2}+\frac{99}{32 a c^4 (1-a x)}+\frac{1}{64 a c^4 (1+a x)^2}-\frac{11}{64 a c^4 (1+a x)}+\frac{303 \log (1-a x)}{128 a c^4}-\frac{47 \log (1+a x)}{128 a c^4}\\ \end{align*}

Mathematica [A]  time = 0.103408, size = 98, normalized size = 0.68 $\frac{\frac{2 \left (192 a^7 x^7-384 a^6 x^6-819 a^5 x^5+1254 a^4 x^4+866 a^3 x^3-1258 a^2 x^2-275 a x+400\right )}{(a x-1)^4 (a x+1)^2}+909 \log (1-a x)-141 \log (a x+1)}{384 a c^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((2*(400 - 275*a*x - 1258*a^2*x^2 + 866*a^3*x^3 + 1254*a^4*x^4 - 819*a^5*x^5 - 384*a^6*x^6 + 192*a^7*x^7))/((-
1 + a*x)^4*(1 + a*x)^2) + 909*Log[1 - a*x] - 141*Log[1 + a*x])/(384*a*c^4)

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Maple [A]  time = 0.053, size = 125, normalized size = 0.9 \begin{align*}{\frac{x}{{c}^{4}}}+{\frac{1}{64\,a{c}^{4} \left ( ax+1 \right ) ^{2}}}-{\frac{11}{64\,a{c}^{4} \left ( ax+1 \right ) }}-{\frac{47\,\ln \left ( ax+1 \right ) }{128\,a{c}^{4}}}-{\frac{1}{32\,a{c}^{4} \left ( ax-1 \right ) ^{4}}}-{\frac{13}{48\,a{c}^{4} \left ( ax-1 \right ) ^{3}}}-{\frac{35}{32\,a{c}^{4} \left ( ax-1 \right ) ^{2}}}-{\frac{99}{32\,a{c}^{4} \left ( ax-1 \right ) }}+{\frac{303\,\ln \left ( ax-1 \right ) }{128\,a{c}^{4}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/c^4+1/64/a/c^4/(a*x+1)^2-11/64/a/c^4/(a*x+1)-47/128*ln(a*x+1)/a/c^4-1/32/a/c^4/(a*x-1)^4-13/48/a/c^4/(a*x-1)
^3-35/32/a/c^4/(a*x-1)^2-99/32/a/c^4/(a*x-1)+303/128/a/c^4*ln(a*x-1)

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Maxima [A]  time = 1.04468, size = 196, normalized size = 1.35 \begin{align*} -\frac{627 \, a^{5} x^{5} - 486 \, a^{4} x^{4} - 1058 \, a^{3} x^{3} + 874 \, a^{2} x^{2} + 467 \, a x - 400}{192 \,{\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} + \frac{x}{c^{4}} - \frac{47 \, \log \left (a x + 1\right )}{128 \, a c^{4}} + \frac{303 \, \log \left (a x - 1\right )}{128 \, a c^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*(627*a^5*x^5 - 486*a^4*x^4 - 1058*a^3*x^3 + 874*a^2*x^2 + 467*a*x - 400)/(a^7*c^4*x^6 - 2*a^6*c^4*x^5 -
a^5*c^4*x^4 + 4*a^4*c^4*x^3 - a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4) + x/c^4 - 47/128*log(a*x + 1)/(a*c^4) + 303/
128*log(a*x - 1)/(a*c^4)

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Fricas [A]  time = 1.59104, size = 509, normalized size = 3.51 \begin{align*} \frac{384 \, a^{7} x^{7} - 768 \, a^{6} x^{6} - 1638 \, a^{5} x^{5} + 2508 \, a^{4} x^{4} + 1732 \, a^{3} x^{3} - 2516 \, a^{2} x^{2} - 550 \, a x - 141 \,{\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x + 1\right ) + 909 \,{\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 800}{384 \,{\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(384*a^7*x^7 - 768*a^6*x^6 - 1638*a^5*x^5 + 2508*a^4*x^4 + 1732*a^3*x^3 - 2516*a^2*x^2 - 550*a*x - 141*(
a^6*x^6 - 2*a^5*x^5 - a^4*x^4 + 4*a^3*x^3 - a^2*x^2 - 2*a*x + 1)*log(a*x + 1) + 909*(a^6*x^6 - 2*a^5*x^5 - a^4
*x^4 + 4*a^3*x^3 - a^2*x^2 - 2*a*x + 1)*log(a*x - 1) + 800)/(a^7*c^4*x^6 - 2*a^6*c^4*x^5 - a^5*c^4*x^4 + 4*a^4
*c^4*x^3 - a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4)

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Sympy [A]  time = 1.30888, size = 156, normalized size = 1.08 \begin{align*} a^{8} \left (- \frac{627 a^{5} x^{5} - 486 a^{4} x^{4} - 1058 a^{3} x^{3} + 874 a^{2} x^{2} + 467 a x - 400}{192 a^{15} c^{4} x^{6} - 384 a^{14} c^{4} x^{5} - 192 a^{13} c^{4} x^{4} + 768 a^{12} c^{4} x^{3} - 192 a^{11} c^{4} x^{2} - 384 a^{10} c^{4} x + 192 a^{9} c^{4}} + \frac{x}{a^{8} c^{4}} + \frac{\frac{303 \log{\left (x - \frac{1}{a} \right )}}{128} - \frac{47 \log{\left (x + \frac{1}{a} \right )}}{128}}{a^{9} c^{4}}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**8*(-(627*a**5*x**5 - 486*a**4*x**4 - 1058*a**3*x**3 + 874*a**2*x**2 + 467*a*x - 400)/(192*a**15*c**4*x**6 -
384*a**14*c**4*x**5 - 192*a**13*c**4*x**4 + 768*a**12*c**4*x**3 - 192*a**11*c**4*x**2 - 384*a**10*c**4*x + 19
2*a**9*c**4) + x/(a**8*c**4) + (303*log(x - 1/a)/128 - 47*log(x + 1/a)/128)/(a**9*c**4))

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Giac [A]  time = 1.13694, size = 130, normalized size = 0.9 \begin{align*} \frac{x}{c^{4}} - \frac{47 \, \log \left ({\left | a x + 1 \right |}\right )}{128 \, a c^{4}} + \frac{303 \, \log \left ({\left | a x - 1 \right |}\right )}{128 \, a c^{4}} - \frac{627 \, a^{5} x^{5} - 486 \, a^{4} x^{4} - 1058 \, a^{3} x^{3} + 874 \, a^{2} x^{2} + 467 \, a x - 400}{192 \,{\left (a x + 1\right )}^{2}{\left (a x - 1\right )}^{4} a c^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/c^4 - 47/128*log(abs(a*x + 1))/(a*c^4) + 303/128*log(abs(a*x - 1))/(a*c^4) - 1/192*(627*a^5*x^5 - 486*a^4*x^
4 - 1058*a^3*x^3 + 874*a^2*x^2 + 467*a*x - 400)/((a*x + 1)^2*(a*x - 1)^4*a*c^4)