Integrand size = 22, antiderivative size = 76 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=-\frac {x}{c^4}+\frac {1}{4 a c^4 (1-a x)^2}-\frac {7}{4 a c^4 (1-a x)}-\frac {17 \log (1-a x)}{8 a c^4}+\frac {\log (1+a x)}{8 a c^4} \] Output:
-x/c^4+1/4/a/c^4/(-a*x+1)^2-7/4/a/c^4/(-a*x+1)-17/8*ln(-a*x+1)/a/c^4+1/8*l n(a*x+1)/a/c^4
Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {a^4 \left (-\frac {x}{a^4}+\frac {1}{4 a^5 (1-a x)^2}-\frac {7}{4 a^5 (1-a x)}-\frac {17 \log (1-a x)}{8 a^5}+\frac {\log (1+a x)}{8 a^5}\right )}{c^4} \] Input:
Integrate[1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))^4),x]
Output:
(a^4*(-(x/a^4) + 1/(4*a^5*(1 - a*x)^2) - 7/(4*a^5*(1 - a*x)) - (17*Log[1 - a*x])/(8*a^5) + Log[1 + a*x]/(8*a^5)))/c^4
Time = 0.40 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6681, 6679, 99, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx\) |
\(\Big \downarrow \) 6681 |
\(\displaystyle \frac {a^4 \int \frac {e^{-2 \text {arctanh}(a x)} x^4}{(1-a x)^4}dx}{c^4}\) |
\(\Big \downarrow \) 6679 |
\(\displaystyle \frac {a^4 \int \frac {x^4}{(1-a x)^3 (a x+1)}dx}{c^4}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {a^4 \int \left (\frac {1}{8 a^4 (a x+1)}-\frac {1}{a^4}-\frac {17}{8 a^4 (a x-1)}-\frac {7}{4 a^4 (a x-1)^2}-\frac {1}{2 a^4 (a x-1)^3}\right )dx}{c^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^4 \left (-\frac {7}{4 a^5 (1-a x)}+\frac {1}{4 a^5 (1-a x)^2}-\frac {17 \log (1-a x)}{8 a^5}+\frac {\log (a x+1)}{8 a^5}-\frac {x}{a^4}\right )}{c^4}\) |
Input:
Int[1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))^4),x]
Output:
(a^4*(-(x/a^4) + 1/(4*a^5*(1 - a*x)^2) - 7/(4*a^5*(1 - a*x)) - (17*Log[1 - a*x])/(8*a^5) + Log[1 + a*x]/(8*a^5)))/c^4
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(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]))
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Simp[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])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_.), x_Symbol ] :> Simp[d^p Int[u*(1 + c*(x/d))^p*(E^(n*ArcTanh[a*x])/x^p), x], x] /; F reeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]
Time = 0.16 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {a^{4} \left (-\frac {x}{a^{4}}+\frac {\ln \left (a x +1\right )}{8 a^{5}}+\frac {1}{4 a^{5} \left (a x -1\right )^{2}}+\frac {7}{4 a^{5} \left (a x -1\right )}-\frac {17 \ln \left (a x -1\right )}{8 a^{5}}\right )}{c^{4}}\) | \(61\) |
risch | \(-\frac {x}{c^{4}}+\frac {\frac {7 c^{4} x}{4}-\frac {3 c^{4}}{2 a}}{c^{8} \left (a x -1\right )^{2}}-\frac {17 \ln \left (a x -1\right )}{8 a \,c^{4}}+\frac {\ln \left (-a x -1\right )}{8 a \,c^{4}}\) | \(63\) |
norman | \(\frac {-\frac {7 a \,x^{2}}{2 c}-\frac {a^{4} x^{5}}{c}+\frac {9 x}{4 c}-\frac {5 a^{2} x^{3}}{4 c}+\frac {7 a^{3} x^{4}}{2 c}}{\left (a x +1\right ) c^{3} \left (a x -1\right )^{3}}-\frac {17 \ln \left (a x -1\right )}{8 a \,c^{4}}+\frac {\ln \left (a x +1\right )}{8 a \,c^{4}}\) | \(97\) |
parallelrisch | \(\frac {-8 a^{3} x^{3}-17 a^{2} \ln \left (a x -1\right ) x^{2}+\ln \left (a x +1\right ) x^{2} a^{2}+28 a^{2} x^{2}+34 a \ln \left (a x -1\right ) x -2 \ln \left (a x +1\right ) x a -18 a x -17 \ln \left (a x -1\right )+\ln \left (a x +1\right )}{8 c^{4} \left (a x -1\right )^{2} a}\) | \(98\) |
Input:
int(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^4,x,method=_RETURNVERBOSE)
Output:
a^4/c^4*(-x/a^4+1/8*ln(a*x+1)/a^5+1/4/a^5/(a*x-1)^2+7/4/a^5/(a*x-1)-17/8/a ^5*ln(a*x-1))
Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.22 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=-\frac {8 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 6 \, a x - {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x + 1\right ) + 17 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 12}{8 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \] Input:
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^4,x, algorithm="fricas")
Output:
-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)
Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.99 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=- a^{4} \left (\frac {- 7 a x + 6}{4 a^{7} c^{4} x^{2} - 8 a^{6} c^{4} x + 4 a^{5} c^{4}} + \frac {x}{a^{4} c^{4}} + \frac {\frac {17 \log {\left (x - \frac {1}{a} \right )}}{8} - \frac {\log {\left (x + \frac {1}{a} \right )}}{8}}{a^{5} c^{4}}\right ) \] Input:
integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a/x)**4,x)
Output:
-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 - log(x + 1/a)/8)/(a**5*c**4))
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {7 \, a x - 6}{4 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} - \frac {x}{c^{4}} + \frac {\log \left (a x + 1\right )}{8 \, a c^{4}} - \frac {17 \, \log \left (a x - 1\right )}{8 \, a c^{4}} \] Input:
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^4,x, algorithm="maxima")
Output:
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)
Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.25 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {2 \, \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a c^{4}} - \frac {17 \, \log \left ({\left | -\frac {2}{a x + 1} + 1 \right |}\right )}{8 \, a c^{4}} + \frac {{\left (a x + 1\right )} {\left (\frac {77}{a x + 1} - \frac {88}{{\left (a x + 1\right )}^{2}} - 16\right )}}{16 \, a c^{4} {\left (\frac {2}{a x + 1} - 1\right )}^{2}} \] Input:
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^4,x, algorithm="giac")
Output:
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)*(77/(a*x + 1) - 88/(a*x + 1)^2 - 16)/(a* c^4*(2/(a*x + 1) - 1)^2)
Time = 14.81 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.89 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {\frac {7\,x}{4}-\frac {3}{2\,a}}{a^2\,c^4\,x^2-2\,a\,c^4\,x+c^4}-\frac {x}{c^4}-\frac {17\,\ln \left (a\,x-1\right )}{8\,a\,c^4}+\frac {\ln \left (a\,x+1\right )}{8\,a\,c^4} \] Input:
int(-(a^2*x^2 - 1)/((c - c/(a*x))^4*(a*x + 1)^2),x)
Output:
((7*x)/4 - 3/(2*a))/(c^4 + a^2*c^4*x^2 - 2*a*c^4*x) - x/c^4 - (17*log(a*x - 1))/(8*a*c^4) + log(a*x + 1)/(8*a*c^4)
Time = 0.15 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.34 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {-17 \,\mathrm {log}\left (a x -1\right ) a^{2} x^{2}+34 \,\mathrm {log}\left (a x -1\right ) a x -17 \,\mathrm {log}\left (a x -1\right )+\mathrm {log}\left (a x +1\right ) a^{2} x^{2}-2 \,\mathrm {log}\left (a x +1\right ) a x +\mathrm {log}\left (a x +1\right )-8 a^{3} x^{3}+19 a^{2} x^{2}-9}{8 a \,c^{4} \left (a^{2} x^{2}-2 a x +1\right )} \] Input:
int(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^4,x)
Output:
( - 17*log(a*x - 1)*a**2*x**2 + 34*log(a*x - 1)*a*x - 17*log(a*x - 1) + lo g(a*x + 1)*a**2*x**2 - 2*log(a*x + 1)*a*x + log(a*x + 1) - 8*a**3*x**3 + 1 9*a**2*x**2 - 9)/(8*a*c**4*(a**2*x**2 - 2*a*x + 1))