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

Optimal. Leaf size=76 \[ -\frac{7}{4 a c^4 (1-a x)}+\frac{1}{4 a c^4 (1-a x)^2}-\frac{17 \log (1-a x)}{8 a c^4}+\frac{\log (a x+1)}{8 a c^4}-\frac{x}{c^4} \]

[Out]

-(x/c^4) + 1/(4*a*c^4*(1 - a*x)^2) - 7/(4*a*c^4*(1 - a*x)) - (17*Log[1 - a*x])/(8*a*c^4) + Log[1 + a*x]/(8*a*c
^4)

________________________________________________________________________________________

Rubi [A]  time = 0.132721, antiderivative size = 76, 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 = {6131, 6129, 88} \[ -\frac{7}{4 a c^4 (1-a x)}+\frac{1}{4 a c^4 (1-a x)^2}-\frac{17 \log (1-a x)}{8 a c^4}+\frac{\log (a x+1)}{8 a c^4}-\frac{x}{c^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))^4),x]

[Out]

-(x/c^4) + 1/(4*a*c^4*(1 - a*x)^2) - 7/(4*a*c^4*(1 - a*x)) - (17*Log[1 - a*x])/(8*a*c^4) + Log[1 + a*x]/(8*a*c
^4)

Rule 6131

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

Rule 6129

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^p, Int[(u*(1 + (d*x)/c)
^p*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 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 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^4} \, dx &=\frac{a^4 \int \frac{e^{-2 \tanh ^{-1}(a x)} x^4}{(1-a x)^4} \, dx}{c^4}\\ &=\frac{a^4 \int \frac{x^4}{(1-a x)^3 (1+a x)} \, dx}{c^4}\\ &=\frac{a^4 \int \left (-\frac{1}{a^4}-\frac{1}{2 a^4 (-1+a x)^3}-\frac{7}{4 a^4 (-1+a x)^2}-\frac{17}{8 a^4 (-1+a x)}+\frac{1}{8 a^4 (1+a x)}\right ) \, dx}{c^4}\\ &=-\frac{x}{c^4}+\frac{1}{4 a c^4 (1-a x)^2}-\frac{7}{4 a c^4 (1-a x)}-\frac{17 \log (1-a x)}{8 a c^4}+\frac{\log (1+a x)}{8 a c^4}\\ \end{align*}

Mathematica [A]  time = 0.138221, size = 69, normalized size = 0.91 \[ \frac{-8 a^3 x^3+16 a^2 x^2+6 a x-17 (a x-1)^2 \log (1-a x)+(a x-1)^2 \log (a x+1)-12}{8 a c^4 (a x-1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))^4),x]

[Out]

(-12 + 6*a*x + 16*a^2*x^2 - 8*a^3*x^3 - 17*(-1 + a*x)^2*Log[1 - a*x] + (-1 + a*x)^2*Log[1 + a*x])/(8*a*c^4*(-1
+ a*x)^2)

________________________________________________________________________________________

Maple [A]  time = 0.037, size = 66, normalized size = 0.9 \begin{align*} -{\frac{x}{{c}^{4}}}+{\frac{\ln \left ( ax+1 \right ) }{8\,a{c}^{4}}}+{\frac{1}{4\,a{c}^{4} \left ( ax-1 \right ) ^{2}}}+{\frac{7}{4\,a{c}^{4} \left ( ax-1 \right ) }}-{\frac{17\,\ln \left ( ax-1 \right ) }{8\,a{c}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/c^4+1/8*ln(a*x+1)/a/c^4+1/4/c^4/a/(a*x-1)^2+7/4/c^4/a/(a*x-1)-17/8/c^4/a*ln(a*x-1)

________________________________________________________________________________________

Maxima [A]  time = 0.948014, size = 95, normalized size = 1.25 \begin{align*} \frac{7 \, a x - 6}{4 \,{\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} - \frac{x}{c^{4}} + \frac{\log \left (a x + 1\right )}{8 \, a c^{4}} - \frac{17 \, \log \left (a x - 1\right )}{8 \, a c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(7*a*x - 6)/(a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4) - x/c^4 + 1/8*log(a*x + 1)/(a*c^4) - 17/8*log(a*x - 1)/(a*
c^4)

________________________________________________________________________________________

Fricas [A]  time = 2.28046, size = 212, normalized size = 2.79 \begin{align*} -\frac{8 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 6 \, a x -{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x + 1\right ) + 17 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 12}{8 \,{\left (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)^2*(-a^2*x^2+1)/(c-c/a/x)^4,x, algorithm="fricas")

[Out]

-1/8*(8*a^3*x^3 - 16*a^2*x^2 - 6*a*x - (a^2*x^2 - 2*a*x + 1)*log(a*x + 1) + 17*(a^2*x^2 - 2*a*x + 1)*log(a*x -
1) + 12)/(a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4)

________________________________________________________________________________________

Sympy [A]  time = 0.675496, size = 75, normalized size = 0.99 \begin{align*} - a^{4} \left (- \frac{7 a x - 6}{4 a^{7} c^{4} x^{2} - 8 a^{6} c^{4} x + 4 a^{5} c^{4}} + \frac{x}{a^{4} c^{4}} + \frac{\frac{17 \log{\left (x - \frac{1}{a} \right )}}{8} - \frac{\log{\left (x + \frac{1}{a} \right )}}{8}}{a^{5} c^{4}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**4*(-(7*a*x - 6)/(4*a**7*c**4*x**2 - 8*a**6*c**4*x + 4*a**5*c**4) + x/(a**4*c**4) + (17*log(x - 1/a)/8 - lo
g(x + 1/a)/8)/(a**5*c**4))

________________________________________________________________________________________

Giac [A]  time = 1.16696, size = 128, normalized size = 1.68 \begin{align*} \frac{2 \, \log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a c^{4}} - \frac{17 \, \log \left ({\left | -\frac{2}{a x + 1} + 1 \right |}\right )}{8 \, a c^{4}} + \frac{{\left (a x + 1\right )}{\left (\frac{77}{a x + 1} - \frac{88}{{\left (a x + 1\right )}^{2}} - 16\right )}}{16 \, a c^{4}{\left (\frac{2}{a x + 1} - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(abs(a*x + 1)/((a*x + 1)^2*abs(a)))/(a*c^4) - 17/8*log(abs(-2/(a*x + 1) + 1))/(a*c^4) + 1/16*(a*x + 1)*(7
7/(a*x + 1) - 88/(a*x + 1)^2 - 16)/(a*c^4*(2/(a*x + 1) - 1)^2)