Integrand size = 22, antiderivative size = 76 \[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^6 \, dx=-\frac {c^6}{5 a^6 x^5}+\frac {c^6}{2 a^5 x^4}+\frac {c^6}{3 a^4 x^3}-\frac {2 c^6}{a^3 x^2}+\frac {c^6}{a^2 x}+c^6 x-\frac {2 c^6 \log (x)}{a} \] Output:
-1/5*c^6/a^6/x^5+1/2*c^6/a^5/x^4+1/3*c^6/a^4/x^3-2*c^6/a^3/x^2+c^6/a^2/x+c ^6*x-2*c^6*ln(x)/a
Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.79 \[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^6 \, dx=\frac {c^6 \left (-\frac {1}{5 x^5}+\frac {a}{2 x^4}+\frac {a^2}{3 x^3}-\frac {2 a^3}{x^2}+\frac {a^4}{x}+a^6 x-2 a^5 \log (x)\right )}{a^6} \] Input:
Integrate[E^(4*ArcTanh[a*x])*(c - c/(a*x))^6,x]
Output:
(c^6*(-1/5*1/x^5 + a/(2*x^4) + a^2/(3*x^3) - (2*a^3)/x^2 + a^4/x + a^6*x - 2*a^5*Log[x]))/a^6
Time = 0.39 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.79, 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.
\(\displaystyle \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^6 \, dx\) |
\(\Big \downarrow \) 6681 |
\(\displaystyle \frac {c^6 \int \frac {e^{4 \text {arctanh}(a x)} (1-a x)^6}{x^6}dx}{a^6}\) |
\(\Big \downarrow \) 6679 |
\(\displaystyle \frac {c^6 \int \frac {(1-a x)^4 (a x+1)^2}{x^6}dx}{a^6}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {c^6 \int \left (a^6-\frac {2 a^5}{x}-\frac {a^4}{x^2}+\frac {4 a^3}{x^3}-\frac {a^2}{x^4}-\frac {2 a}{x^5}+\frac {1}{x^6}\right )dx}{a^6}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c^6 \left (a^6 x-2 a^5 \log (x)+\frac {a^4}{x}-\frac {2 a^3}{x^2}+\frac {a^2}{3 x^3}+\frac {a}{2 x^4}-\frac {1}{5 x^5}\right )}{a^6}\) |
Input:
Int[E^(4*ArcTanh[a*x])*(c - c/(a*x))^6,x]
Output:
(c^6*(-1/5*1/x^5 + a/(2*x^4) + a^2/(3*x^3) - (2*a^3)/x^2 + a^4/x + a^6*x - 2*a^5*Log[x]))/a^6
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 = 0.23 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.72
method | result | size |
default | \(\frac {c^{6} \left (x \,a^{6}-\frac {2 a^{3}}{x^{2}}+\frac {a^{4}}{x}+\frac {a}{2 x^{4}}-2 a^{5} \ln \left (x \right )+\frac {a^{2}}{3 x^{3}}-\frac {1}{5 x^{5}}\right )}{a^{6}}\) | \(55\) |
risch | \(c^{6} x +\frac {a^{4} c^{6} x^{4}-2 a^{3} c^{6} x^{3}+\frac {1}{3} a^{2} c^{6} x^{2}+\frac {1}{2} a \,c^{6} x -\frac {1}{5} c^{6}}{a^{6} x^{5}}-\frac {2 c^{6} \ln \left (x \right )}{a}\) | \(69\) |
parallelrisch | \(-\frac {-30 a^{6} c^{6} x^{6}+60 c^{6} \ln \left (x \right ) a^{5} x^{5}-30 a^{4} c^{6} x^{4}+60 a^{3} c^{6} x^{3}-10 a^{2} c^{6} x^{2}-15 a \,c^{6} x +6 c^{6}}{30 a^{6} x^{5}}\) | \(79\) |
norman | \(\frac {-2 a^{6} c^{6} x^{7}+a^{7} c^{6} x^{8}+\frac {c^{6}}{5 a}-\frac {c^{6} x}{2}+\frac {5 a^{2} c^{6} x^{3}}{2}-\frac {8 c^{6} a \,x^{2}}{15}-\frac {2 c^{6} a^{3} x^{4}}{3}}{\left (a^{2} x^{2}-1\right ) a^{5} x^{5}}-\frac {2 c^{6} \ln \left (x \right )}{a}\) | \(97\) |
meijerg | \(-\frac {3 c^{6} \left (-\frac {2 \left (-15 a^{4} x^{4}+10 a^{2} x^{2}+2\right )}{3 x^{3} \left (-a^{2}\right )^{\frac {3}{2}} \left (-2 a^{2} x^{2}+2\right )}+\frac {5 a^{3} \operatorname {arctanh}\left (a x \right )}{\left (-a^{2}\right )^{\frac {3}{2}}}\right )}{2 \sqrt {-a^{2}}}-\frac {c^{6} \left (-\frac {1}{2 a^{4} x^{4}}-\frac {2}{a^{2} x^{2}}+1+6 \ln \left (x \right )+3 \ln \left (-a^{2}\right )+\frac {4 a^{2} x^{2}}{-4 a^{2} x^{2}+4}-3 \ln \left (-a^{2} x^{2}+1\right )\right )}{a}+\frac {4 a \,c^{6} x^{2}}{-a^{2} x^{2}+1}-\frac {4 c^{6} \left (\frac {1}{a^{2} x^{2}}-1-4 \ln \left (x \right )-2 \ln \left (-a^{2}\right )-\frac {3 a^{2} x^{2}}{-3 a^{2} x^{2}+3}+2 \ln \left (-a^{2} x^{2}+1\right )\right )}{a}+\frac {c^{6} \left (\frac {x \left (-a^{2}\right )^{\frac {5}{2}} \left (-10 a^{2} x^{2}+15\right )}{5 a^{4} \left (-a^{2} x^{2}+1\right )}-\frac {3 \left (-a^{2}\right )^{\frac {5}{2}} \operatorname {arctanh}\left (a x \right )}{a^{5}}\right )}{2 \sqrt {-a^{2}}}-\frac {c^{6} \left (\frac {a^{2} x^{2}}{-a^{2} x^{2}+1}+\ln \left (-a^{2} x^{2}+1\right )\right )}{a}-\frac {6 c^{6} \left (1+2 \ln \left (x \right )+\ln \left (-a^{2}\right )+\frac {2 a^{2} x^{2}}{-2 a^{2} x^{2}+2}-\ln \left (-a^{2} x^{2}+1\right )\right )}{a}+\frac {3 c^{6} \left (\frac {x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \left (-a^{2} x^{2}+1\right )}-\frac {\left (-a^{2}\right )^{\frac {3}{2}} \operatorname {arctanh}\left (a x \right )}{a^{3}}\right )}{2 \sqrt {-a^{2}}}+\frac {c^{6} \left (\frac {2 x \sqrt {-a^{2}}}{-2 a^{2} x^{2}+2}+\frac {\sqrt {-a^{2}}\, \operatorname {arctanh}\left (a x \right )}{a}\right )}{\sqrt {-a^{2}}}-\frac {c^{6} \left (-\frac {2 \left (-3 a^{2} x^{2}+2\right )}{x \sqrt {-a^{2}}\, \left (-2 a^{2} x^{2}+2\right )}+\frac {3 a \,\operatorname {arctanh}\left (a x \right )}{\sqrt {-a^{2}}}\right )}{\sqrt {-a^{2}}}-\frac {c^{6} \left (-\frac {2 \left (-105 x^{6} a^{6}+70 a^{4} x^{4}+14 a^{2} x^{2}+6\right )}{5 x^{5} \left (-a^{2}\right )^{\frac {5}{2}} \left (-6 a^{2} x^{2}+6\right )}+\frac {7 a^{5} \operatorname {arctanh}\left (a x \right )}{\left (-a^{2}\right )^{\frac {5}{2}}}\right )}{2 \sqrt {-a^{2}}}\) | \(625\) |
Input:
int((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a/x)^6,x,method=_RETURNVERBOSE)
Output:
c^6/a^6*(x*a^6-2*a^3/x^2+a^4/x+1/2*a/x^4-2*a^5*ln(x)+1/3*a^2/x^3-1/5/x^5)
Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.03 \[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^6 \, dx=\frac {30 \, a^{6} c^{6} x^{6} - 60 \, a^{5} c^{6} x^{5} \log \left (x\right ) + 30 \, a^{4} c^{6} x^{4} - 60 \, a^{3} c^{6} x^{3} + 10 \, a^{2} c^{6} x^{2} + 15 \, a c^{6} x - 6 \, c^{6}}{30 \, a^{6} x^{5}} \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a/x)^6,x, algorithm="fricas")
Output:
1/30*(30*a^6*c^6*x^6 - 60*a^5*c^6*x^5*log(x) + 30*a^4*c^6*x^4 - 60*a^3*c^6 *x^3 + 10*a^2*c^6*x^2 + 15*a*c^6*x - 6*c^6)/(a^6*x^5)
Time = 0.18 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00 \[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^6 \, dx=\frac {a^{6} c^{6} x - 2 a^{5} c^{6} \log {\left (x \right )} + \frac {30 a^{4} c^{6} x^{4} - 60 a^{3} c^{6} x^{3} + 10 a^{2} c^{6} x^{2} + 15 a c^{6} x - 6 c^{6}}{30 x^{5}}}{a^{6}} \] Input:
integrate((a*x+1)**4/(-a**2*x**2+1)**2*(c-c/a/x)**6,x)
Output:
(a**6*c**6*x - 2*a**5*c**6*log(x) + (30*a**4*c**6*x**4 - 60*a**3*c**6*x**3 + 10*a**2*c**6*x**2 + 15*a*c**6*x - 6*c**6)/(30*x**5))/a**6
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.92 \[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^6 \, dx=c^{6} x - \frac {2 \, c^{6} \log \left (x\right )}{a} + \frac {30 \, a^{4} c^{6} x^{4} - 60 \, a^{3} c^{6} x^{3} + 10 \, a^{2} c^{6} x^{2} + 15 \, a c^{6} x - 6 \, c^{6}}{30 \, a^{6} x^{5}} \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a/x)^6,x, algorithm="maxima")
Output:
c^6*x - 2*c^6*log(x)/a + 1/30*(30*a^4*c^6*x^4 - 60*a^3*c^6*x^3 + 10*a^2*c^ 6*x^2 + 15*a*c^6*x - 6*c^6)/(a^6*x^5)
Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.93 \[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^6 \, dx=c^{6} x - \frac {2 \, c^{6} \log \left ({\left | x \right |}\right )}{a} + \frac {30 \, a^{4} c^{6} x^{4} - 60 \, a^{3} c^{6} x^{3} + 10 \, a^{2} c^{6} x^{2} + 15 \, a c^{6} x - 6 \, c^{6}}{30 \, a^{6} x^{5}} \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a/x)^6,x, algorithm="giac")
Output:
c^6*x - 2*c^6*log(abs(x))/a + 1/30*(30*a^4*c^6*x^4 - 60*a^3*c^6*x^3 + 10*a ^2*c^6*x^2 + 15*a*c^6*x - 6*c^6)/(a^6*x^5)
Time = 15.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.74 \[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^6 \, dx=\frac {c^6\,\left (\frac {a\,x}{2}+\frac {a^2\,x^2}{3}-2\,a^3\,x^3+a^4\,x^4+a^6\,x^6-2\,a^5\,x^5\,\ln \left (x\right )-\frac {1}{5}\right )}{a^6\,x^5} \] Input:
int(((c - c/(a*x))^6*(a*x + 1)^4)/(a^2*x^2 - 1)^2,x)
Output:
(c^6*((a*x)/2 + (a^2*x^2)/3 - 2*a^3*x^3 + a^4*x^4 + a^6*x^6 - 2*a^5*x^5*lo g(x) - 1/5))/(a^6*x^5)
Time = 0.15 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.78 \[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^6 \, dx=\frac {c^{6} \left (-60 \,\mathrm {log}\left (x \right ) a^{5} x^{5}+30 a^{6} x^{6}+30 a^{4} x^{4}-60 a^{3} x^{3}+10 a^{2} x^{2}+15 a x -6\right )}{30 a^{6} x^{5}} \] Input:
int((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a/x)^6,x)
Output:
(c**6*( - 60*log(x)*a**5*x**5 + 30*a**6*x**6 + 30*a**4*x**4 - 60*a**3*x**3 + 10*a**2*x**2 + 15*a*x - 6))/(30*a**6*x**5)