Integrand size = 22, antiderivative size = 170 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^9} \, dx=-\frac {a^2}{168 x^6}+\frac {a^4}{84 x^4}+\frac {5 a^6}{504 x^2}-\frac {a \text {arctanh}(a x)}{28 x^7}+\frac {a^3 \text {arctanh}(a x)}{12 x^5}-\frac {a^5 \text {arctanh}(a x)}{36 x^3}-\frac {a^7 \text {arctanh}(a x)}{12 x}+\frac {1}{24} a^8 \text {arctanh}(a x)^2-\frac {\text {arctanh}(a x)^2}{8 x^8}+\frac {a^2 \text {arctanh}(a x)^2}{3 x^6}-\frac {a^4 \text {arctanh}(a x)^2}{4 x^4}+\frac {4}{63} a^8 \log (x)-\frac {2}{63} a^8 \log \left (1-a^2 x^2\right ) \] Output:
-1/168*a^2/x^6+1/84*a^4/x^4+5/504*a^6/x^2-1/28*a*arctanh(a*x)/x^7+1/12*a^3 *arctanh(a*x)/x^5-1/36*a^5*arctanh(a*x)/x^3-1/12*a^7*arctanh(a*x)/x+1/24*a ^8*arctanh(a*x)^2-1/8*arctanh(a*x)^2/x^8+1/3*a^2*arctanh(a*x)^2/x^6-1/4*a^ 4*arctanh(a*x)^2/x^4+4/63*a^8*ln(x)-2/63*a^8*ln(-a^2*x^2+1)
Time = 0.05 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.73 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^9} \, dx=\frac {-2 a x \left (9-21 a^2 x^2+7 a^4 x^4+21 a^6 x^6\right ) \text {arctanh}(a x)+21 \left (-1+a^2 x^2\right )^3 \left (3+a^2 x^2\right ) \text {arctanh}(a x)^2+a^2 x^2 \left (-3+6 a^2 x^2+5 a^4 x^4+32 a^6 x^6 \log (x)-16 a^6 x^6 \log \left (1-a^2 x^2\right )\right )}{504 x^8} \] Input:
Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^9,x]
Output:
(-2*a*x*(9 - 21*a^2*x^2 + 7*a^4*x^4 + 21*a^6*x^6)*ArcTanh[a*x] + 21*(-1 + a^2*x^2)^3*(3 + a^2*x^2)*ArcTanh[a*x]^2 + a^2*x^2*(-3 + 6*a^2*x^2 + 5*a^4* x^4 + 32*a^6*x^6*Log[x] - 16*a^6*x^6*Log[1 - a^2*x^2]))/(504*x^8)
Time = 1.13 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6574, 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.
| \(\displaystyle \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^9} \, dx\) |
| \(\Big \downarrow \) 6574 |
| \(\displaystyle \int \left (\frac {a^4 \text {arctanh}(a x)^2}{x^5}-\frac {2 a^2 \text {arctanh}(a x)^2}{x^7}+\frac {\text {arctanh}(a x)^2}{x^9}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{24} a^8 \text {arctanh}(a x)^2+\frac {4}{63} a^8 \log (x)-\frac {a^7 \text {arctanh}(a x)}{12 x}+\frac {5 a^6}{504 x^2}-\frac {a^5 \text {arctanh}(a x)}{36 x^3}-\frac {a^4 \text {arctanh}(a x)^2}{4 x^4}+\frac {a^4}{84 x^4}+\frac {a^3 \text {arctanh}(a x)}{12 x^5}+\frac {a^2 \text {arctanh}(a x)^2}{3 x^6}-\frac {a^2}{168 x^6}-\frac {2}{63} a^8 \log \left (1-a^2 x^2\right )-\frac {\text {arctanh}(a x)^2}{8 x^8}-\frac {a \text {arctanh}(a x)}{28 x^7}\) |
Input:
Int[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^9,x]
Output:
-1/168*a^2/x^6 + a^4/(84*x^4) + (5*a^6)/(504*x^2) - (a*ArcTanh[a*x])/(28*x ^7) + (a^3*ArcTanh[a*x])/(12*x^5) - (a^5*ArcTanh[a*x])/(36*x^3) - (a^7*Arc Tanh[a*x])/(12*x) + (a^8*ArcTanh[a*x]^2)/24 - ArcTanh[a*x]^2/(8*x^8) + (a^ 2*ArcTanh[a*x]^2)/(3*x^6) - (a^4*ArcTanh[a*x]^2)/(4*x^4) + (4*a^8*Log[x])/ 63 - (2*a^8*Log[1 - a^2*x^2])/63
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(q_), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[q, 1]
Time = 0.46 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.99
| method | result | size |
| parallelrisch | \(\frac {21 \operatorname {arctanh}\left (a x \right )^{2} a^{8} x^{8}+32 \ln \left (x \right ) a^{8} x^{8}-32 \ln \left (a x -1\right ) x^{8} a^{8}-32 \,\operatorname {arctanh}\left (a x \right ) a^{8} x^{8}+5 a^{8} x^{8}-42 \,\operatorname {arctanh}\left (a x \right ) a^{7} x^{7}+5 a^{6} x^{6}-14 \,\operatorname {arctanh}\left (a x \right ) a^{5} x^{5}-126 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}+6 a^{4} x^{4}+42 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )+168 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}-3 a^{2} x^{2}-18 a x \,\operatorname {arctanh}\left (a x \right )-63 \operatorname {arctanh}\left (a x \right )^{2}}{504 x^{8}}\) | \(169\) |
| derivativedivides | \(a^{8} \left (\frac {\operatorname {arctanh}\left (a x \right )^{2}}{3 a^{6} x^{6}}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{8 a^{8} x^{8}}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 a^{4} x^{4}}-\frac {\operatorname {arctanh}\left (a x \right )}{28 a^{7} x^{7}}+\frac {\operatorname {arctanh}\left (a x \right )}{12 a^{5} x^{5}}-\frac {\operatorname {arctanh}\left (a x \right )}{36 a^{3} x^{3}}-\frac {\operatorname {arctanh}\left (a x \right )}{12 a x}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{24}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{24}-\frac {\ln \left (a x -1\right )^{2}}{96}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{48}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{48}-\frac {\ln \left (a x +1\right )^{2}}{96}-\frac {1}{168 a^{6} x^{6}}+\frac {1}{84 a^{4} x^{4}}+\frac {5}{504 a^{2} x^{2}}+\frac {4 \ln \left (a x \right )}{63}-\frac {2 \ln \left (a x -1\right )}{63}-\frac {2 \ln \left (a x +1\right )}{63}\right )\) | \(226\) |
| default | \(a^{8} \left (\frac {\operatorname {arctanh}\left (a x \right )^{2}}{3 a^{6} x^{6}}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{8 a^{8} x^{8}}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{4 a^{4} x^{4}}-\frac {\operatorname {arctanh}\left (a x \right )}{28 a^{7} x^{7}}+\frac {\operatorname {arctanh}\left (a x \right )}{12 a^{5} x^{5}}-\frac {\operatorname {arctanh}\left (a x \right )}{36 a^{3} x^{3}}-\frac {\operatorname {arctanh}\left (a x \right )}{12 a x}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{24}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{24}-\frac {\ln \left (a x -1\right )^{2}}{96}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{48}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{48}-\frac {\ln \left (a x +1\right )^{2}}{96}-\frac {1}{168 a^{6} x^{6}}+\frac {1}{84 a^{4} x^{4}}+\frac {5}{504 a^{2} x^{2}}+\frac {4 \ln \left (a x \right )}{63}-\frac {2 \ln \left (a x -1\right )}{63}-\frac {2 \ln \left (a x +1\right )}{63}\right )\) | \(226\) |
| parts | \(-\frac {a^{4} \operatorname {arctanh}\left (a x \right )^{2}}{4 x^{4}}+\frac {a^{2} \operatorname {arctanh}\left (a x \right )^{2}}{3 x^{6}}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{8 x^{8}}-\frac {a \,\operatorname {arctanh}\left (a x \right )}{28 x^{7}}+\frac {a^{3} \operatorname {arctanh}\left (a x \right )}{12 x^{5}}-\frac {a^{5} \operatorname {arctanh}\left (a x \right )}{36 x^{3}}-\frac {a^{7} \operatorname {arctanh}\left (a x \right )}{12 x}-\frac {a^{8} \operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{24}+\frac {a^{8} \operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{24}+\frac {a^{8} \left (-\frac {21 \ln \left (a x -1\right )^{2}}{4}+\frac {21 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {21 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {21 \ln \left (a x +1\right )^{2}}{4}-\frac {3}{a^{6} x^{6}}+\frac {6}{a^{4} x^{4}}+\frac {5}{a^{2} x^{2}}+32 \ln \left (a x \right )-16 \ln \left (a x -1\right )-16 \ln \left (a x +1\right )\right )}{504}\) | \(229\) |
| risch | \(\frac {\left (a^{8} x^{8}-6 a^{4} x^{4}+8 a^{2} x^{2}-3\right ) \ln \left (a x +1\right )^{2}}{96 x^{8}}-\frac {\left (21 a^{8} x^{8} \ln \left (-a x +1\right )+42 a^{7} x^{7}+14 a^{5} x^{5}-126 x^{4} \ln \left (-a x +1\right ) a^{4}-42 a^{3} x^{3}+168 x^{2} \ln \left (-a x +1\right ) a^{2}+18 a x -63 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{1008 x^{8}}+\frac {21 a^{8} x^{8} \ln \left (-a x +1\right )^{2}+128 \ln \left (x \right ) a^{8} x^{8}-64 \ln \left (a^{2} x^{2}-1\right ) a^{8} x^{8}+84 a^{7} x^{7} \ln \left (-a x +1\right )+20 a^{6} x^{6}+28 x^{5} \ln \left (-a x +1\right ) a^{5}-126 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+24 a^{4} x^{4}-84 a^{3} x^{3} \ln \left (-a x +1\right )+168 a^{2} x^{2} \ln \left (-a x +1\right )^{2}-12 a^{2} x^{2}+36 a x \ln \left (-a x +1\right )-63 \ln \left (-a x +1\right )^{2}}{2016 x^{8}}\) | \(310\) |
Input:
int((-a^2*x^2+1)^2*arctanh(a*x)^2/x^9,x,method=_RETURNVERBOSE)
Output:
1/504*(21*arctanh(a*x)^2*a^8*x^8+32*ln(x)*a^8*x^8-32*ln(a*x-1)*x^8*a^8-32* arctanh(a*x)*a^8*x^8+5*a^8*x^8-42*arctanh(a*x)*a^7*x^7+5*a^6*x^6-14*arctan h(a*x)*a^5*x^5-126*a^4*x^4*arctanh(a*x)^2+6*a^4*x^4+42*a^3*x^3*arctanh(a*x )+168*a^2*x^2*arctanh(a*x)^2-3*a^2*x^2-18*a*x*arctanh(a*x)-63*arctanh(a*x) ^2)/x^8
Time = 0.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.87 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^9} \, dx=-\frac {64 \, a^{8} x^{8} \log \left (a^{2} x^{2} - 1\right ) - 128 \, a^{8} x^{8} \log \left (x\right ) - 20 \, a^{6} x^{6} - 24 \, a^{4} x^{4} + 12 \, a^{2} x^{2} - 21 \, {\left (a^{8} x^{8} - 6 \, a^{4} x^{4} + 8 \, a^{2} x^{2} - 3\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, {\left (21 \, a^{7} x^{7} + 7 \, a^{5} x^{5} - 21 \, a^{3} x^{3} + 9 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{2016 \, x^{8}} \] Input:
integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^9,x, algorithm="fricas")
Output:
-1/2016*(64*a^8*x^8*log(a^2*x^2 - 1) - 128*a^8*x^8*log(x) - 20*a^6*x^6 - 2 4*a^4*x^4 + 12*a^2*x^2 - 21*(a^8*x^8 - 6*a^4*x^4 + 8*a^2*x^2 - 3)*log(-(a* x + 1)/(a*x - 1))^2 + 4*(21*a^7*x^7 + 7*a^5*x^5 - 21*a^3*x^3 + 9*a*x)*log( -(a*x + 1)/(a*x - 1)))/x^8
Time = 0.99 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.99 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^9} \, dx=\begin {cases} \frac {4 a^{8} \log {\left (x \right )}}{63} - \frac {4 a^{8} \log {\left (x - \frac {1}{a} \right )}}{63} + \frac {a^{8} \operatorname {atanh}^{2}{\left (a x \right )}}{24} - \frac {4 a^{8} \operatorname {atanh}{\left (a x \right )}}{63} - \frac {a^{7} \operatorname {atanh}{\left (a x \right )}}{12 x} + \frac {5 a^{6}}{504 x^{2}} - \frac {a^{5} \operatorname {atanh}{\left (a x \right )}}{36 x^{3}} - \frac {a^{4} \operatorname {atanh}^{2}{\left (a x \right )}}{4 x^{4}} + \frac {a^{4}}{84 x^{4}} + \frac {a^{3} \operatorname {atanh}{\left (a x \right )}}{12 x^{5}} + \frac {a^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{3 x^{6}} - \frac {a^{2}}{168 x^{6}} - \frac {a \operatorname {atanh}{\left (a x \right )}}{28 x^{7}} - \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{8 x^{8}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate((-a**2*x**2+1)**2*atanh(a*x)**2/x**9,x)
Output:
Piecewise((4*a**8*log(x)/63 - 4*a**8*log(x - 1/a)/63 + a**8*atanh(a*x)**2/ 24 - 4*a**8*atanh(a*x)/63 - a**7*atanh(a*x)/(12*x) + 5*a**6/(504*x**2) - a **5*atanh(a*x)/(36*x**3) - a**4*atanh(a*x)**2/(4*x**4) + a**4/(84*x**4) + a**3*atanh(a*x)/(12*x**5) + a**2*atanh(a*x)**2/(3*x**6) - a**2/(168*x**6) - a*atanh(a*x)/(28*x**7) - atanh(a*x)**2/(8*x**8), Ne(a, 0)), (0, True))
Time = 0.04 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.20 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^9} \, dx=\frac {1}{2016} \, {\left (128 \, a^{6} \log \left (x\right ) - \frac {21 \, a^{6} x^{6} \log \left (a x + 1\right )^{2} + 21 \, a^{6} x^{6} \log \left (a x - 1\right )^{2} + 64 \, a^{6} x^{6} \log \left (a x - 1\right ) - 20 \, a^{4} x^{4} - 24 \, a^{2} x^{2} - 2 \, {\left (21 \, a^{6} x^{6} \log \left (a x - 1\right ) - 32 \, a^{6} x^{6}\right )} \log \left (a x + 1\right ) + 12}{x^{6}}\right )} a^{2} + \frac {1}{504} \, {\left (21 \, a^{7} \log \left (a x + 1\right ) - 21 \, a^{7} \log \left (a x - 1\right ) - \frac {2 \, {\left (21 \, a^{6} x^{6} + 7 \, a^{4} x^{4} - 21 \, a^{2} x^{2} + 9\right )}}{x^{7}}\right )} a \operatorname {artanh}\left (a x\right ) - \frac {{\left (6 \, a^{4} x^{4} - 8 \, a^{2} x^{2} + 3\right )} \operatorname {artanh}\left (a x\right )^{2}}{24 \, x^{8}} \] Input:
integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^9,x, algorithm="maxima")
Output:
1/2016*(128*a^6*log(x) - (21*a^6*x^6*log(a*x + 1)^2 + 21*a^6*x^6*log(a*x - 1)^2 + 64*a^6*x^6*log(a*x - 1) - 20*a^4*x^4 - 24*a^2*x^2 - 2*(21*a^6*x^6* log(a*x - 1) - 32*a^6*x^6)*log(a*x + 1) + 12)/x^6)*a^2 + 1/504*(21*a^7*log (a*x + 1) - 21*a^7*log(a*x - 1) - 2*(21*a^6*x^6 + 7*a^4*x^4 - 21*a^2*x^2 + 9)/x^7)*a*arctanh(a*x) - 1/24*(6*a^4*x^4 - 8*a^2*x^2 + 3)*arctanh(a*x)^2/ x^8
Leaf count of result is larger than twice the leaf count of optimal. 651 vs. \(2 (144) = 288\).
Time = 0.13 (sec) , antiderivative size = 651, normalized size of antiderivative = 3.83 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^9} \, dx =\text {Too large to display} \] Input:
integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^9,x, algorithm="giac")
Output:
2/63*(2*a^7*log(-(a*x + 1)/(a*x - 1) - 1) - 2*a^7*log(-(a*x + 1)/(a*x - 1) ) + 84*((a*x + 1)^5*a^7/(a*x - 1)^5 - (a*x + 1)^4*a^7/(a*x - 1)^4 + (a*x + 1)^3*a^7/(a*x - 1)^3)*log(-(a*x + 1)/(a*x - 1))^2/((a*x + 1)^8/(a*x - 1)^ 8 + 8*(a*x + 1)^7/(a*x - 1)^7 + 28*(a*x + 1)^6/(a*x - 1)^6 + 56*(a*x + 1)^ 5/(a*x - 1)^5 + 70*(a*x + 1)^4/(a*x - 1)^4 + 56*(a*x + 1)^3/(a*x - 1)^3 + 28*(a*x + 1)^2/(a*x - 1)^2 + 8*(a*x + 1)/(a*x - 1) + 1) + 2*(28*(a*x + 1)^ 4*a^7/(a*x - 1)^4 + 7*(a*x + 1)^3*a^7/(a*x - 1)^3 + 21*(a*x + 1)^2*a^7/(a* x - 1)^2 + 7*(a*x + 1)*a^7/(a*x - 1) + a^7)*log(-(a*x + 1)/(a*x - 1))/((a* x + 1)^7/(a*x - 1)^7 + 7*(a*x + 1)^6/(a*x - 1)^6 + 21*(a*x + 1)^5/(a*x - 1 )^5 + 35*(a*x + 1)^4/(a*x - 1)^4 + 35*(a*x + 1)^3/(a*x - 1)^3 + 21*(a*x + 1)^2/(a*x - 1)^2 + 7*(a*x + 1)/(a*x - 1) + 1) - (2*(a*x + 1)^5*a^7/(a*x - 1)^5 + 11*(a*x + 1)^4*a^7/(a*x - 1)^4 + 6*(a*x + 1)^3*a^7/(a*x - 1)^3 + 11 *(a*x + 1)^2*a^7/(a*x - 1)^2 + 2*(a*x + 1)*a^7/(a*x - 1))/((a*x + 1)^6/(a* x - 1)^6 + 6*(a*x + 1)^5/(a*x - 1)^5 + 15*(a*x + 1)^4/(a*x - 1)^4 + 20*(a* x + 1)^3/(a*x - 1)^3 + 15*(a*x + 1)^2/(a*x - 1)^2 + 6*(a*x + 1)/(a*x - 1) + 1))*a
Time = 5.12 (sec) , antiderivative size = 357, normalized size of antiderivative = 2.10 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^9} \, dx=\frac {4\,a^8\,\ln \left (x\right )}{63}+\frac {a^8\,{\ln \left (a\,x+1\right )}^2}{96}+\frac {a^8\,{\ln \left (1-a\,x\right )}^2}{96}-\frac {{\ln \left (a\,x+1\right )}^2}{32\,x^8}-\frac {{\ln \left (1-a\,x\right )}^2}{32\,x^8}-\frac {2\,a^8\,\ln \left (a^2\,x^2-1\right )}{63}-\frac {a^2}{168\,x^6}+\frac {a^4}{84\,x^4}+\frac {5\,a^6}{504\,x^2}-\frac {a^8\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{48}+\frac {\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{16\,x^8}+\frac {a^2\,{\ln \left (a\,x+1\right )}^2}{12\,x^6}-\frac {a^4\,{\ln \left (a\,x+1\right )}^2}{16\,x^4}+\frac {a^2\,{\ln \left (1-a\,x\right )}^2}{12\,x^6}-\frac {a^4\,{\ln \left (1-a\,x\right )}^2}{16\,x^4}-\frac {a\,\ln \left (a\,x+1\right )}{56\,x^7}+\frac {a\,\ln \left (1-a\,x\right )}{56\,x^7}+\frac {a^3\,\ln \left (a\,x+1\right )}{24\,x^5}-\frac {a^5\,\ln \left (a\,x+1\right )}{72\,x^3}-\frac {a^7\,\ln \left (a\,x+1\right )}{24\,x}-\frac {a^3\,\ln \left (1-a\,x\right )}{24\,x^5}+\frac {a^5\,\ln \left (1-a\,x\right )}{72\,x^3}+\frac {a^7\,\ln \left (1-a\,x\right )}{24\,x}-\frac {a^2\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{6\,x^6}+\frac {a^4\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{8\,x^4} \] Input:
int((atanh(a*x)^2*(a^2*x^2 - 1)^2)/x^9,x)
Output:
(4*a^8*log(x))/63 + (a^8*log(a*x + 1)^2)/96 + (a^8*log(1 - a*x)^2)/96 - lo g(a*x + 1)^2/(32*x^8) - log(1 - a*x)^2/(32*x^8) - (2*a^8*log(a^2*x^2 - 1)) /63 - a^2/(168*x^6) + a^4/(84*x^4) + (5*a^6)/(504*x^2) - (a^8*log(a*x + 1) *log(1 - a*x))/48 + (log(a*x + 1)*log(1 - a*x))/(16*x^8) + (a^2*log(a*x + 1)^2)/(12*x^6) - (a^4*log(a*x + 1)^2)/(16*x^4) + (a^2*log(1 - a*x)^2)/(12* x^6) - (a^4*log(1 - a*x)^2)/(16*x^4) - (a*log(a*x + 1))/(56*x^7) + (a*log( 1 - a*x))/(56*x^7) + (a^3*log(a*x + 1))/(24*x^5) - (a^5*log(a*x + 1))/(72* x^3) - (a^7*log(a*x + 1))/(24*x) - (a^3*log(1 - a*x))/(24*x^5) + (a^5*log( 1 - a*x))/(72*x^3) + (a^7*log(1 - a*x))/(24*x) - (a^2*log(a*x + 1)*log(1 - a*x))/(6*x^6) + (a^4*log(a*x + 1)*log(1 - a*x))/(8*x^4)
Time = 0.17 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.96 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^9} \, dx=\frac {21 \mathit {atanh} \left (a x \right )^{2} a^{8} x^{8}-126 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}+168 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-63 \mathit {atanh} \left (a x \right )^{2}-32 \mathit {atanh} \left (a x \right ) a^{8} x^{8}-42 \mathit {atanh} \left (a x \right ) a^{7} x^{7}-14 \mathit {atanh} \left (a x \right ) a^{5} x^{5}+42 \mathit {atanh} \left (a x \right ) a^{3} x^{3}-18 \mathit {atanh} \left (a x \right ) a x -32 \,\mathrm {log}\left (a^{2} x -a \right ) a^{8} x^{8}+32 \,\mathrm {log}\left (x \right ) a^{8} x^{8}+5 a^{6} x^{6}+6 a^{4} x^{4}-3 a^{2} x^{2}}{504 x^{8}} \] Input:
int((-a^2*x^2+1)^2*atanh(a*x)^2/x^9,x)
Output:
(21*atanh(a*x)**2*a**8*x**8 - 126*atanh(a*x)**2*a**4*x**4 + 168*atanh(a*x) **2*a**2*x**2 - 63*atanh(a*x)**2 - 32*atanh(a*x)*a**8*x**8 - 42*atanh(a*x) *a**7*x**7 - 14*atanh(a*x)*a**5*x**5 + 42*atanh(a*x)*a**3*x**3 - 18*atanh( a*x)*a*x - 32*log(a**2*x - a)*a**8*x**8 + 32*log(x)*a**8*x**8 + 5*a**6*x** 6 + 6*a**4*x**4 - 3*a**2*x**2)/(504*x**8)