Integrand size = 24, antiderivative size = 76 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (1-a^2 x^2\right )^{5/2}} \, dx=-\frac {x}{a^5}+\frac {1}{8 a^6 (1-a x)^2}-\frac {1}{a^6 (1-a x)}+\frac {1}{8 a^6 (1+a x)}-\frac {23 \log (1-a x)}{16 a^6}+\frac {7 \log (1+a x)}{16 a^6} \] Output:
-x/a^5+1/8/a^6/(-a*x+1)^2-1/a^6/(-a*x+1)+1/8/a^6/(a*x+1)-23/16*ln(-a*x+1)/ a^6+7/16*ln(a*x+1)/a^6
Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.72 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\frac {2 \left (-8 a x+\frac {1}{(-1+a x)^2}+\frac {8}{-1+a x}+\frac {1}{1+a x}\right )-23 \log (1-a x)+7 \log (1+a x)}{16 a^6} \] Input:
Integrate[(E^ArcTanh[a*x]*x^5)/(1 - a^2*x^2)^(5/2),x]
Output:
(2*(-8*a*x + (-1 + a*x)^(-2) + 8/(-1 + a*x) + (1 + a*x)^(-1)) - 23*Log[1 - a*x] + 7*Log[1 + a*x])/(16*a^6)
Time = 0.56 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6700, 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.
\(\displaystyle \int \frac {x^5 e^{\text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \int \frac {x^5}{(1-a x)^3 (a x+1)^2}dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {7}{16 a^5 (a x+1)}-\frac {1}{8 a^5 (a x+1)^2}-\frac {23}{16 a^5 (a x-1)}-\frac {1}{a^5 (a x-1)^2}-\frac {1}{4 a^5 (a x-1)^3}-\frac {1}{a^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{a^6 (1-a x)}+\frac {1}{8 a^6 (a x+1)}+\frac {1}{8 a^6 (1-a x)^2}-\frac {23 \log (1-a x)}{16 a^6}+\frac {7 \log (a x+1)}{16 a^6}-\frac {x}{a^5}\) |
Input:
Int[(E^ArcTanh[a*x]*x^5)/(1 - a^2*x^2)^(5/2),x]
Output:
-(x/a^5) + 1/(8*a^6*(1 - a*x)^2) - 1/(a^6*(1 - a*x)) + 1/(8*a^6*(1 + a*x)) - (23*Log[1 - a*x])/(16*a^6) + (7*Log[1 + a*x])/(16*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_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[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])
Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.86
method | result | size |
default | \(-\frac {x}{a^{5}}+\frac {7 \ln \left (a x +1\right )}{16 a^{6}}+\frac {1}{8 a^{6} \left (a x +1\right )}+\frac {1}{8 \left (a x -1\right )^{2} a^{6}}+\frac {1}{\left (a x -1\right ) a^{6}}-\frac {23 \ln \left (a x -1\right )}{16 a^{6}}\) | \(65\) |
risch | \(-\frac {x}{a^{5}}+\frac {\frac {9 a \,x^{2}}{8}-\frac {x}{8}-\frac {3}{4 a}}{a^{5} \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}-\frac {23 \ln \left (a x -1\right )}{16 a^{6}}+\frac {7 \ln \left (-a x -1\right )}{16 a^{6}}\) | \(68\) |
norman | \(\frac {-\frac {x^{5}}{a}-\frac {15 x}{8 a^{5}}+\frac {25 x^{3}}{8 a^{3}}-\frac {3}{4 a^{6}}+\frac {x^{2}}{a^{4}}}{\left (a^{2} x^{2}-1\right )^{2}}-\frac {23 \ln \left (a x -1\right )}{16 a^{6}}+\frac {7 \ln \left (a x +1\right )}{16 a^{6}}\) | \(71\) |
meijerg | \(-\frac {\frac {x^{2} a^{2} \left (-9 a^{2} x^{2}+6\right )}{3 \left (-a^{2} x^{2}+1\right )^{2}}+2 \ln \left (-a^{2} x^{2}+1\right )}{4 a^{6}}-\frac {-\frac {x \left (-a^{2}\right )^{\frac {7}{2}} \left (56 a^{4} x^{4}-175 a^{2} x^{2}+105\right )}{14 a^{6} \left (-a^{2} x^{2}+1\right )^{2}}+\frac {15 \left (-a^{2}\right )^{\frac {7}{2}} \operatorname {arctanh}\left (a x \right )}{2 a^{7}}}{4 a^{5} \sqrt {-a^{2}}}\) | \(123\) |
parallelrisch | \(-\frac {28+16 a^{4} x^{4}+23 a^{3} \ln \left (a x -1\right ) x^{3}-7 \ln \left (a x +1\right ) x^{3} a^{3}-23 a^{2} \ln \left (a x -1\right ) x^{2}+7 \ln \left (a x +1\right ) x^{2} a^{2}-50 a^{2} x^{2}-23 a \ln \left (a x -1\right ) x +7 \ln \left (a x +1\right ) x a +2 a x +23 \ln \left (a x -1\right )-7 \ln \left (a x +1\right )}{16 a^{6} \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}\) | \(138\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^3*x^5,x,method=_RETURNVERBOSE)
Output:
-x/a^5+7/16*ln(a*x+1)/a^6+1/8/a^6/(a*x+1)+1/8/(a*x-1)^2/a^6+1/(a*x-1)/a^6- 23/16/a^6*ln(a*x-1)
Time = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.54 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (1-a^2 x^2\right )^{5/2}} \, dx=-\frac {16 \, a^{4} x^{4} - 16 \, a^{3} x^{3} - 34 \, a^{2} x^{2} + 18 \, a x - 7 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right ) + 23 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x - 1\right ) + 12}{16 \, {\left (a^{9} x^{3} - a^{8} x^{2} - a^{7} x + a^{6}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^3*x^5,x, algorithm="fricas")
Output:
-1/16*(16*a^4*x^4 - 16*a^3*x^3 - 34*a^2*x^2 + 18*a*x - 7*(a^3*x^3 - a^2*x^ 2 - a*x + 1)*log(a*x + 1) + 23*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x - 1) + 12)/(a^9*x^3 - a^8*x^2 - a^7*x + a^6)
Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (1-a^2 x^2\right )^{5/2}} \, dx=- \frac {- 9 a^{2} x^{2} + a x + 6}{8 a^{9} x^{3} - 8 a^{8} x^{2} - 8 a^{7} x + 8 a^{6}} - \frac {x}{a^{5}} - \frac {\frac {23 \log {\left (x - \frac {1}{a} \right )}}{16} - \frac {7 \log {\left (x + \frac {1}{a} \right )}}{16}}{a^{6}} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**3*x**5,x)
Output:
-(-9*a**2*x**2 + a*x + 6)/(8*a**9*x**3 - 8*a**8*x**2 - 8*a**7*x + 8*a**6) - x/a**5 - (23*log(x - 1/a)/16 - 7*log(x + 1/a)/16)/a**6
Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.95 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\frac {9 \, a^{2} x^{2} - a x - 6}{8 \, {\left (a^{9} x^{3} - a^{8} x^{2} - a^{7} x + a^{6}\right )}} - \frac {x}{a^{5}} + \frac {7 \, \log \left (a x + 1\right )}{16 \, a^{6}} - \frac {23 \, \log \left (a x - 1\right )}{16 \, a^{6}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^3*x^5,x, algorithm="maxima")
Output:
1/8*(9*a^2*x^2 - a*x - 6)/(a^9*x^3 - a^8*x^2 - a^7*x + a^6) - x/a^5 + 7/16 *log(a*x + 1)/a^6 - 23/16*log(a*x - 1)/a^6
Time = 0.12 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.84 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (1-a^2 x^2\right )^{5/2}} \, dx=-\frac {x}{a^{5}} + \frac {7 \, \log \left ({\left | a x + 1 \right |}\right )}{16 \, a^{6}} - \frac {23 \, \log \left ({\left | a x - 1 \right |}\right )}{16 \, a^{6}} + \frac {9 \, a^{2} x^{2} - a x - 6}{8 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{2} a^{6}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^3*x^5,x, algorithm="giac")
Output:
-x/a^5 + 7/16*log(abs(a*x + 1))/a^6 - 23/16*log(abs(a*x - 1))/a^6 + 1/8*(9 *a^2*x^2 - a*x - 6)/((a*x + 1)*(a*x - 1)^2*a^6)
Time = 23.76 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\frac {7\,\ln \left (a\,x+1\right )}{16\,a^6}-\frac {23\,\ln \left (a\,x-1\right )}{16\,a^6}-\frac {x}{a^5}+\frac {\frac {x}{8}-\frac {9\,a\,x^2}{8}+\frac {3}{4\,a}}{-a^8\,x^3+a^7\,x^2+a^6\,x-a^5} \] Input:
int(-(x^5*(a*x + 1))/(a^2*x^2 - 1)^3,x)
Output:
(7*log(a*x + 1))/(16*a^6) - (23*log(a*x - 1))/(16*a^6) - x/a^5 + (x/8 - (9 *a*x^2)/8 + 3/(4*a))/(a^6*x - a^5 + a^7*x^2 - a^8*x^3)
Time = 0.15 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.87 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\frac {-23 \,\mathrm {log}\left (a x -1\right ) a^{3} x^{3}+23 \,\mathrm {log}\left (a x -1\right ) a^{2} x^{2}+23 \,\mathrm {log}\left (a x -1\right ) a x -23 \,\mathrm {log}\left (a x -1\right )+7 \,\mathrm {log}\left (a x +1\right ) a^{3} x^{3}-7 \,\mathrm {log}\left (a x +1\right ) a^{2} x^{2}-7 \,\mathrm {log}\left (a x +1\right ) a x +7 \,\mathrm {log}\left (a x +1\right )-16 a^{4} x^{4}+50 a^{3} x^{3}-52 a x +22}{16 a^{6} \left (a^{3} x^{3}-a^{2} x^{2}-a x +1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^3*x^5,x)
Output:
( - 23*log(a*x - 1)*a**3*x**3 + 23*log(a*x - 1)*a**2*x**2 + 23*log(a*x - 1 )*a*x - 23*log(a*x - 1) + 7*log(a*x + 1)*a**3*x**3 - 7*log(a*x + 1)*a**2*x **2 - 7*log(a*x + 1)*a*x + 7*log(a*x + 1) - 16*a**4*x**4 + 50*a**3*x**3 - 52*a*x + 22)/(16*a**6*(a**3*x**3 - a**2*x**2 - a*x + 1))