Integrand size = 24, antiderivative size = 88 \[ \int \frac {e^{\text {arctanh}(a x)} x^6}{\left (1-a^2 x^2\right )^{5/2}} \, dx=-\frac {x}{a^6}-\frac {x^2}{2 a^5}+\frac {1}{8 a^7 (1-a x)^2}-\frac {5}{4 a^7 (1-a x)}-\frac {1}{8 a^7 (1+a x)}-\frac {39 \log (1-a x)}{16 a^7}-\frac {9 \log (1+a x)}{16 a^7} \] Output:
-x/a^6-1/2*x^2/a^5+1/8/a^7/(-a*x+1)^2-5/4/a^7/(-a*x+1)-1/8/a^7/(a*x+1)-39/ 16*ln(-a*x+1)/a^7-9/16*ln(a*x+1)/a^7
Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.74 \[ \int \frac {e^{\text {arctanh}(a x)} x^6}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\frac {2 \left (-8 a x-4 a^2 x^2+\frac {1}{(-1+a x)^2}+\frac {10}{-1+a x}-\frac {1}{1+a x}\right )-39 \log (1-a x)-9 \log (1+a x)}{16 a^7} \] Input:
Integrate[(E^ArcTanh[a*x]*x^6)/(1 - a^2*x^2)^(5/2),x]
Output:
(2*(-8*a*x - 4*a^2*x^2 + (-1 + a*x)^(-2) + 10/(-1 + a*x) - (1 + a*x)^(-1)) - 39*Log[1 - a*x] - 9*Log[1 + a*x])/(16*a^7)
Time = 0.37 (sec) , antiderivative size = 88, 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.
\(\displaystyle \int \frac {x^6 e^{\text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \int \frac {x^6}{(1-a x)^3 (a x+1)^2}dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {39}{16 a^6 (a x-1)}-\frac {9}{16 a^6 (a x+1)}-\frac {5}{4 a^6 (a x-1)^2}+\frac {1}{8 a^6 (a x+1)^2}-\frac {1}{4 a^6 (a x-1)^3}-\frac {1}{a^6}-\frac {x}{a^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {5}{4 a^7 (1-a x)}-\frac {1}{8 a^7 (a x+1)}+\frac {1}{8 a^7 (1-a x)^2}-\frac {39 \log (1-a x)}{16 a^7}-\frac {9 \log (a x+1)}{16 a^7}-\frac {x}{a^6}-\frac {x^2}{2 a^5}\) |
Input:
Int[(E^ArcTanh[a*x]*x^6)/(1 - a^2*x^2)^(5/2),x]
Output:
-(x/a^6) - x^2/(2*a^5) + 1/(8*a^7*(1 - a*x)^2) - 5/(4*a^7*(1 - a*x)) - 1/( 8*a^7*(1 + a*x)) - (39*Log[1 - a*x])/(16*a^7) - (9*Log[1 + a*x])/(16*a^7)
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 = 0.18 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.83
method | result | size |
default | \(-\frac {\frac {1}{2} a \,x^{2}+x}{a^{6}}-\frac {9 \ln \left (a x +1\right )}{16 a^{7}}-\frac {1}{8 a^{7} \left (a x +1\right )}+\frac {1}{8 a^{7} \left (a x -1\right )^{2}}+\frac {5}{4 a^{7} \left (a x -1\right )}-\frac {39 \ln \left (a x -1\right )}{16 a^{7}}\) | \(73\) |
risch | \(-\frac {x^{2}}{2 a^{5}}-\frac {x}{a^{6}}+\frac {\frac {9 a \,x^{2}}{8}+\frac {3 x}{8}-\frac {5}{4 a}}{a^{6} \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}-\frac {39 \ln \left (a x -1\right )}{16 a^{7}}-\frac {9 \ln \left (-a x -1\right )}{16 a^{7}}\) | \(76\) |
norman | \(\frac {\frac {25 x^{3}}{8 a^{4}}-\frac {x^{5}}{a^{2}}-\frac {x^{6}}{2 a}-\frac {15 x}{8 a^{6}}-\frac {9}{4 a^{7}}+\frac {3 x^{2}}{a^{5}}}{\left (a^{2} x^{2}-1\right )^{2}}-\frac {39 \ln \left (a x -1\right )}{16 a^{7}}-\frac {9 \ln \left (a x +1\right )}{16 a^{7}}\) | \(80\) |
meijerg | \(-\frac {-\frac {x \left (-a^{2}\right )^{\frac {7}{2}} \left (56 a^{4} x^{4}-175 a^{2} x^{2}+105\right )}{14 a^{6} \left (-a^{2} x^{2}+1\right )^{2}}+\frac {15 \left (-a^{2}\right )^{\frac {7}{2}} \operatorname {arctanh}\left (a x \right )}{2 a^{7}}}{4 a^{6} \sqrt {-a^{2}}}+\frac {-\frac {a^{2} x^{2} \left (4 a^{4} x^{4}-18 a^{2} x^{2}+12\right )}{2 \left (-a^{2} x^{2}+1\right )^{2}}-6 \ln \left (-a^{2} x^{2}+1\right )}{4 a^{7}}\) | \(131\) |
parallelrisch | \(-\frac {44+8 a^{5} x^{5}+8 a^{4} x^{4}+39 a^{3} \ln \left (a x -1\right ) x^{3}+9 \ln \left (a x +1\right ) x^{3} a^{3}-39 a^{2} \ln \left (a x -1\right ) x^{2}-9 \ln \left (a x +1\right ) x^{2} a^{2}-50 a^{2} x^{2}-39 a \ln \left (a x -1\right ) x -9 \ln \left (a x +1\right ) x a -14 a x +39 \ln \left (a x -1\right )+9 \ln \left (a x +1\right )}{16 a^{7} \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}\) | \(146\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^3*x^6,x,method=_RETURNVERBOSE)
Output:
-1/a^6*(1/2*a*x^2+x)-9/16*ln(a*x+1)/a^7-1/8/a^7/(a*x+1)+1/8/a^7/(a*x-1)^2+ 5/4/a^7/(a*x-1)-39/16/a^7*ln(a*x-1)
Time = 0.08 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.42 \[ \int \frac {e^{\text {arctanh}(a x)} x^6}{\left (1-a^2 x^2\right )^{5/2}} \, dx=-\frac {8 \, a^{5} x^{5} + 8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 10 \, a x + 9 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right ) + 39 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x - 1\right ) + 20}{16 \, {\left (a^{10} x^{3} - a^{9} x^{2} - a^{8} x + a^{7}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^3*x^6,x, algorithm="fricas")
Output:
-1/16*(8*a^5*x^5 + 8*a^4*x^4 - 24*a^3*x^3 - 26*a^2*x^2 + 10*a*x + 9*(a^3*x ^3 - a^2*x^2 - a*x + 1)*log(a*x + 1) + 39*(a^3*x^3 - a^2*x^2 - a*x + 1)*lo g(a*x - 1) + 20)/(a^10*x^3 - a^9*x^2 - a^8*x + a^7)
Time = 0.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\text {arctanh}(a x)} x^6}{\left (1-a^2 x^2\right )^{5/2}} \, dx=- \frac {- 9 a^{2} x^{2} - 3 a x + 10}{8 a^{10} x^{3} - 8 a^{9} x^{2} - 8 a^{8} x + 8 a^{7}} - \frac {x^{2}}{2 a^{5}} - \frac {x}{a^{6}} - \frac {3 \cdot \left (\frac {13 \log {\left (x - \frac {1}{a} \right )}}{16} + \frac {3 \log {\left (x + \frac {1}{a} \right )}}{16}\right )}{a^{7}} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**3*x**6,x)
Output:
-(-9*a**2*x**2 - 3*a*x + 10)/(8*a**10*x**3 - 8*a**9*x**2 - 8*a**8*x + 8*a* *7) - x**2/(2*a**5) - x/a**6 - 3*(13*log(x - 1/a)/16 + 3*log(x + 1/a)/16)/ a**7
Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.91 \[ \int \frac {e^{\text {arctanh}(a x)} x^6}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\frac {9 \, a^{2} x^{2} + 3 \, a x - 10}{8 \, {\left (a^{10} x^{3} - a^{9} x^{2} - a^{8} x + a^{7}\right )}} - \frac {a x^{2} + 2 \, x}{2 \, a^{6}} - \frac {9 \, \log \left (a x + 1\right )}{16 \, a^{7}} - \frac {39 \, \log \left (a x - 1\right )}{16 \, a^{7}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^3*x^6,x, algorithm="maxima")
Output:
1/8*(9*a^2*x^2 + 3*a*x - 10)/(a^10*x^3 - a^9*x^2 - a^8*x + a^7) - 1/2*(a*x ^2 + 2*x)/a^6 - 9/16*log(a*x + 1)/a^7 - 39/16*log(a*x - 1)/a^7
Time = 0.13 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.88 \[ \int \frac {e^{\text {arctanh}(a x)} x^6}{\left (1-a^2 x^2\right )^{5/2}} \, dx=-\frac {9 \, \log \left ({\left | a x + 1 \right |}\right )}{16 \, a^{7}} - \frac {39 \, \log \left ({\left | a x - 1 \right |}\right )}{16 \, a^{7}} - \frac {a^{5} x^{2} + 2 \, a^{4} x}{2 \, a^{10}} + \frac {9 \, a^{2} x^{2} + 3 \, a x - 10}{8 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{2} a^{7}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^3*x^6,x, algorithm="giac")
Output:
-9/16*log(abs(a*x + 1))/a^7 - 39/16*log(abs(a*x - 1))/a^7 - 1/2*(a^5*x^2 + 2*a^4*x)/a^10 + 1/8*(9*a^2*x^2 + 3*a*x - 10)/((a*x + 1)*(a*x - 1)^2*a^7)
Time = 0.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\text {arctanh}(a x)} x^6}{\left (1-a^2 x^2\right )^{5/2}} \, dx=-\frac {39\,\ln \left (a\,x-1\right )}{16\,a^7}-\frac {9\,\ln \left (a\,x+1\right )}{16\,a^7}-\frac {x}{a^6}-\frac {\frac {3\,x}{8}+\frac {9\,a\,x^2}{8}-\frac {5}{4\,a}}{-a^9\,x^3+a^8\,x^2+a^7\,x-a^6}-\frac {x^2}{2\,a^5} \] Input:
int(-(x^6*(a*x + 1))/(a^2*x^2 - 1)^3,x)
Output:
- (39*log(a*x - 1))/(16*a^7) - (9*log(a*x + 1))/(16*a^7) - x/a^6 - ((3*x)/ 8 + (9*a*x^2)/8 - 5/(4*a))/(a^7*x - a^6 + a^8*x^2 - a^9*x^3) - x^2/(2*a^5)
Time = 0.16 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.70 \[ \int \frac {e^{\text {arctanh}(a x)} x^6}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\frac {-39 \,\mathrm {log}\left (a x -1\right ) a^{3} x^{3}+39 \,\mathrm {log}\left (a x -1\right ) a^{2} x^{2}+39 \,\mathrm {log}\left (a x -1\right ) a x -39 \,\mathrm {log}\left (a x -1\right )-9 \,\mathrm {log}\left (a x +1\right ) a^{3} x^{3}+9 \,\mathrm {log}\left (a x +1\right ) a^{2} x^{2}+9 \,\mathrm {log}\left (a x +1\right ) a x -9 \,\mathrm {log}\left (a x +1\right )-8 a^{5} x^{5}-8 a^{4} x^{4}+50 a^{3} x^{3}-36 a x +6}{16 a^{7} \left (a^{3} x^{3}-a^{2} x^{2}-a x +1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^3*x^6,x)
Output:
( - 39*log(a*x - 1)*a**3*x**3 + 39*log(a*x - 1)*a**2*x**2 + 39*log(a*x - 1 )*a*x - 39*log(a*x - 1) - 9*log(a*x + 1)*a**3*x**3 + 9*log(a*x + 1)*a**2*x **2 + 9*log(a*x + 1)*a*x - 9*log(a*x + 1) - 8*a**5*x**5 - 8*a**4*x**4 + 50 *a**3*x**3 - 36*a*x + 6)/(16*a**7*(a**3*x**3 - a**2*x**2 - a*x + 1))