Integrand size = 22, antiderivative size = 113 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^7} \, dx=-\frac {a^2}{60 x^4}+\frac {7 a^4}{90 x^2}-\frac {a \text {arctanh}(a x)}{15 x^5}+\frac {2 a^3 \text {arctanh}(a x)}{9 x^3}-\frac {a^5 \text {arctanh}(a x)}{3 x}-\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{6 x^6}+\frac {8}{45} a^6 \log (x)-\frac {4}{45} a^6 \log \left (1-a^2 x^2\right ) \] Output:
-1/60*a^2/x^4+7/90*a^4/x^2-1/15*a*arctanh(a*x)/x^5+2/9*a^3*arctanh(a*x)/x^ 3-1/3*a^5*arctanh(a*x)/x-1/6*(-a^2*x^2+1)^3*arctanh(a*x)^2/x^6+8/45*a^6*ln (x)-4/45*a^6*ln(-a^2*x^2+1)
Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.88 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^7} \, dx=\frac {-4 a x \left (3-10 a^2 x^2+15 a^4 x^4\right ) \text {arctanh}(a x)+30 \left (-1+a^2 x^2\right )^3 \text {arctanh}(a x)^2+a^2 x^2 \left (-3+14 a^2 x^2+32 a^4 x^4 \log (x)-16 a^4 x^4 \log \left (1-a^2 x^2\right )\right )}{180 x^6} \] Input:
Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^7,x]
Output:
(-4*a*x*(3 - 10*a^2*x^2 + 15*a^4*x^4)*ArcTanh[a*x] + 30*(-1 + a^2*x^2)^3*A rcTanh[a*x]^2 + a^2*x^2*(-3 + 14*a^2*x^2 + 32*a^4*x^4*Log[x] - 16*a^4*x^4* Log[1 - a^2*x^2]))/(180*x^6)
Time = 0.51 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6570, 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.
| \(\displaystyle \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^7} \, dx\) |
| \(\Big \downarrow \) 6570 |
| \(\displaystyle \frac {1}{3} a \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{x^6}dx-\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{6 x^6}\) |
| \(\Big \downarrow \) 6574 |
| \(\displaystyle \frac {1}{3} a \int \left (\frac {\text {arctanh}(a x) a^4}{x^2}-\frac {2 \text {arctanh}(a x) a^2}{x^4}+\frac {\text {arctanh}(a x)}{x^6}\right )dx-\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{6 x^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} a \left (\frac {8}{15} a^5 \log (x)-\frac {a^4 \text {arctanh}(a x)}{x}+\frac {7 a^3}{30 x^2}+\frac {2 a^2 \text {arctanh}(a x)}{3 x^3}-\frac {4}{15} a^5 \log \left (1-a^2 x^2\right )-\frac {\text {arctanh}(a x)}{5 x^5}-\frac {a}{20 x^4}\right )-\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{6 x^6}\) |
Input:
Int[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^7,x]
Output:
-1/6*((1 - a^2*x^2)^3*ArcTanh[a*x]^2)/x^6 + (a*(-1/20*a/x^4 + (7*a^3)/(30* x^2) - ArcTanh[a*x]/(5*x^5) + (2*a^2*ArcTanh[a*x])/(3*x^3) - (a^4*ArcTanh[ a*x])/x + (8*a^5*Log[x])/15 - (4*a^5*Log[1 - a^2*x^2])/15))/3
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e _.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p/(d*(m + 1))), x] - Simp[b*c*(p/(m + 1)) Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
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 = 0.44 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.32
| method | result | size |
| parallelrisch | \(\frac {30 \operatorname {arctanh}\left (a x \right )^{2} a^{6} x^{6}+32 \ln \left (x \right ) a^{6} x^{6}-32 \ln \left (a x -1\right ) x^{6} a^{6}-32 \,\operatorname {arctanh}\left (a x \right ) a^{6} x^{6}+14 a^{6} x^{6}-60 \,\operatorname {arctanh}\left (a x \right ) a^{5} x^{5}-90 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}+14 a^{4} x^{4}+40 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )+90 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}-3 a^{2} x^{2}-12 a x \,\operatorname {arctanh}\left (a x \right )-30 \operatorname {arctanh}\left (a x \right )^{2}}{180 x^{6}}\) | \(149\) |
| derivativedivides | \(a^{6} \left (-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{6 a^{6} x^{6}}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{2 a^{2} x^{2}}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{2 a^{4} x^{4}}-\frac {\operatorname {arctanh}\left (a x \right )}{15 a^{5} x^{5}}+\frac {2 \,\operatorname {arctanh}\left (a x \right )}{9 a^{3} x^{3}}-\frac {\operatorname {arctanh}\left (a x \right )}{3 a x}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{6}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{6}-\frac {\ln \left (a x -1\right )^{2}}{24}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{12}-\frac {\ln \left (a x +1\right )^{2}}{24}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{12}-\frac {1}{60 a^{4} x^{4}}+\frac {7}{90 a^{2} x^{2}}+\frac {8 \ln \left (a x \right )}{45}-\frac {4 \ln \left (a x -1\right )}{45}-\frac {4 \ln \left (a x +1\right )}{45}\right )\) | \(206\) |
| default | \(a^{6} \left (-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{6 a^{6} x^{6}}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{2 a^{2} x^{2}}+\frac {\operatorname {arctanh}\left (a x \right )^{2}}{2 a^{4} x^{4}}-\frac {\operatorname {arctanh}\left (a x \right )}{15 a^{5} x^{5}}+\frac {2 \,\operatorname {arctanh}\left (a x \right )}{9 a^{3} x^{3}}-\frac {\operatorname {arctanh}\left (a x \right )}{3 a x}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{6}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{6}-\frac {\ln \left (a x -1\right )^{2}}{24}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{12}-\frac {\ln \left (a x +1\right )^{2}}{24}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{12}-\frac {1}{60 a^{4} x^{4}}+\frac {7}{90 a^{2} x^{2}}+\frac {8 \ln \left (a x \right )}{45}-\frac {4 \ln \left (a x -1\right )}{45}-\frac {4 \ln \left (a x +1\right )}{45}\right )\) | \(206\) |
| parts | \(\frac {a^{2} \operatorname {arctanh}\left (a x \right )^{2}}{2 x^{4}}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{6 x^{6}}-\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{4}}{2 x^{2}}-\frac {a \,\operatorname {arctanh}\left (a x \right )}{15 x^{5}}+\frac {2 a^{3} \operatorname {arctanh}\left (a x \right )}{9 x^{3}}-\frac {a^{5} \operatorname {arctanh}\left (a x \right )}{3 x}-\frac {a^{6} \operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{6}+\frac {a^{6} \operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{6}+\frac {a^{6} \left (-\frac {15 \ln \left (a x -1\right )^{2}}{4}+\frac {15 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {15 \ln \left (a x +1\right )^{2}}{4}+\frac {15 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {3}{2 a^{4} x^{4}}+\frac {7}{a^{2} x^{2}}+16 \ln \left (a x \right )-8 \ln \left (a x -1\right )-8 \ln \left (a x +1\right )\right )}{90}\) | \(209\) |
| risch | \(\frac {\left (a^{6} x^{6}-3 a^{4} x^{4}+3 a^{2} x^{2}-1\right ) \ln \left (a x +1\right )^{2}}{24 x^{6}}-\frac {\left (15 a^{6} x^{6} \ln \left (-a x +1\right )+30 a^{5} x^{5}-45 x^{4} \ln \left (-a x +1\right ) a^{4}-20 a^{3} x^{3}+45 x^{2} \ln \left (-a x +1\right ) a^{2}+6 a x -15 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{180 x^{6}}+\frac {15 a^{6} x^{6} \ln \left (-a x +1\right )^{2}+64 \ln \left (x \right ) a^{6} x^{6}-32 \ln \left (a^{2} x^{2}-1\right ) a^{6} x^{6}+60 x^{5} \ln \left (-a x +1\right ) a^{5}-45 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+28 a^{4} x^{4}-40 a^{3} x^{3} \ln \left (-a x +1\right )+45 a^{2} x^{2} \ln \left (-a x +1\right )^{2}-6 a^{2} x^{2}+12 a x \ln \left (-a x +1\right )-15 \ln \left (-a x +1\right )^{2}}{360 x^{6}}\) | \(279\) |
Input:
int((-a^2*x^2+1)^2*arctanh(a*x)^2/x^7,x,method=_RETURNVERBOSE)
Output:
1/180*(30*arctanh(a*x)^2*a^6*x^6+32*ln(x)*a^6*x^6-32*ln(a*x-1)*x^6*a^6-32* arctanh(a*x)*a^6*x^6+14*a^6*x^6-60*arctanh(a*x)*a^5*x^5-90*a^4*x^4*arctanh (a*x)^2+14*a^4*x^4+40*a^3*x^3*arctanh(a*x)+90*a^2*x^2*arctanh(a*x)^2-3*a^2 *x^2-12*a*x*arctanh(a*x)-30*arctanh(a*x)^2)/x^6
Time = 0.08 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.17 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^7} \, dx=-\frac {32 \, a^{6} x^{6} \log \left (a^{2} x^{2} - 1\right ) - 64 \, a^{6} x^{6} \log \left (x\right ) - 28 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 15 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, {\left (15 \, a^{5} x^{5} - 10 \, a^{3} x^{3} + 3 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{360 \, x^{6}} \] Input:
integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^7,x, algorithm="fricas")
Output:
-1/360*(32*a^6*x^6*log(a^2*x^2 - 1) - 64*a^6*x^6*log(x) - 28*a^4*x^4 + 6*a ^2*x^2 - 15*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(-(a*x + 1)/(a*x - 1) )^2 + 4*(15*a^5*x^5 - 10*a^3*x^3 + 3*a*x)*log(-(a*x + 1)/(a*x - 1)))/x^6
Time = 0.69 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.31 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^7} \, dx=\begin {cases} \frac {8 a^{6} \log {\left (x \right )}}{45} - \frac {8 a^{6} \log {\left (x - \frac {1}{a} \right )}}{45} + \frac {a^{6} \operatorname {atanh}^{2}{\left (a x \right )}}{6} - \frac {8 a^{6} \operatorname {atanh}{\left (a x \right )}}{45} - \frac {a^{5} \operatorname {atanh}{\left (a x \right )}}{3 x} - \frac {a^{4} \operatorname {atanh}^{2}{\left (a x \right )}}{2 x^{2}} + \frac {7 a^{4}}{90 x^{2}} + \frac {2 a^{3} \operatorname {atanh}{\left (a x \right )}}{9 x^{3}} + \frac {a^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{2 x^{4}} - \frac {a^{2}}{60 x^{4}} - \frac {a \operatorname {atanh}{\left (a x \right )}}{15 x^{5}} - \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{6 x^{6}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate((-a**2*x**2+1)**2*atanh(a*x)**2/x**7,x)
Output:
Piecewise((8*a**6*log(x)/45 - 8*a**6*log(x - 1/a)/45 + a**6*atanh(a*x)**2/ 6 - 8*a**6*atanh(a*x)/45 - a**5*atanh(a*x)/(3*x) - a**4*atanh(a*x)**2/(2*x **2) + 7*a**4/(90*x**2) + 2*a**3*atanh(a*x)/(9*x**3) + a**2*atanh(a*x)**2/ (2*x**4) - a**2/(60*x**4) - a*atanh(a*x)/(15*x**5) - atanh(a*x)**2/(6*x**6 ), Ne(a, 0)), (0, True))
Time = 0.03 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.66 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^7} \, dx=\frac {1}{360} \, {\left (64 \, a^{4} \log \left (x\right ) - \frac {15 \, a^{4} x^{4} \log \left (a x + 1\right )^{2} + 15 \, a^{4} x^{4} \log \left (a x - 1\right )^{2} + 32 \, a^{4} x^{4} \log \left (a x - 1\right ) - 28 \, a^{2} x^{2} - 2 \, {\left (15 \, a^{4} x^{4} \log \left (a x - 1\right ) - 16 \, a^{4} x^{4}\right )} \log \left (a x + 1\right ) + 6}{x^{4}}\right )} a^{2} + \frac {1}{90} \, {\left (15 \, a^{5} \log \left (a x + 1\right ) - 15 \, a^{5} \log \left (a x - 1\right ) - \frac {2 \, {\left (15 \, a^{4} x^{4} - 10 \, a^{2} x^{2} + 3\right )}}{x^{5}}\right )} a \operatorname {artanh}\left (a x\right ) - \frac {{\left (3 \, a^{4} x^{4} - 3 \, a^{2} x^{2} + 1\right )} \operatorname {artanh}\left (a x\right )^{2}}{6 \, x^{6}} \] Input:
integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^7,x, algorithm="maxima")
Output:
1/360*(64*a^4*log(x) - (15*a^4*x^4*log(a*x + 1)^2 + 15*a^4*x^4*log(a*x - 1 )^2 + 32*a^4*x^4*log(a*x - 1) - 28*a^2*x^2 - 2*(15*a^4*x^4*log(a*x - 1) - 16*a^4*x^4)*log(a*x + 1) + 6)/x^4)*a^2 + 1/90*(15*a^5*log(a*x + 1) - 15*a^ 5*log(a*x - 1) - 2*(15*a^4*x^4 - 10*a^2*x^2 + 3)/x^5)*a*arctanh(a*x) - 1/6 *(3*a^4*x^4 - 3*a^2*x^2 + 1)*arctanh(a*x)^2/x^6
Leaf count of result is larger than twice the leaf count of optimal. 440 vs. \(2 (96) = 192\).
Time = 0.13 (sec) , antiderivative size = 440, normalized size of antiderivative = 3.89 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^7} \, dx=\frac {4}{45} \, {\left (2 \, a^{5} \log \left (-\frac {a x + 1}{a x - 1} - 1\right ) - 2 \, a^{5} \log \left (-\frac {a x + 1}{a x - 1}\right ) + \frac {30 \, {\left (a x + 1\right )}^{3} a^{5} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}}{{\left (a x - 1\right )}^{3} {\left (\frac {{\left (a x + 1\right )}^{6}}{{\left (a x - 1\right )}^{6}} + \frac {6 \, {\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} + \frac {15 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {20 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {15 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {6 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}} + \frac {2 \, {\left (\frac {10 \, {\left (a x + 1\right )}^{2} a^{5}}{{\left (a x - 1\right )}^{2}} + \frac {5 \, {\left (a x + 1\right )} a^{5}}{a x - 1} + a^{5}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{\frac {{\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} + \frac {5 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {10 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {10 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {5 \, {\left (a x + 1\right )}}{a x - 1} + 1} - \frac {\frac {2 \, {\left (a x + 1\right )}^{3} a^{5}}{{\left (a x - 1\right )}^{3}} + \frac {7 \, {\left (a x + 1\right )}^{2} a^{5}}{{\left (a x - 1\right )}^{2}} + \frac {2 \, {\left (a x + 1\right )} a^{5}}{a x - 1}}{\frac {{\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {4 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {6 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {4 \, {\left (a x + 1\right )}}{a x - 1} + 1}\right )} a \] Input:
integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^7,x, algorithm="giac")
Output:
4/45*(2*a^5*log(-(a*x + 1)/(a*x - 1) - 1) - 2*a^5*log(-(a*x + 1)/(a*x - 1) ) + 30*(a*x + 1)^3*a^5*log(-(a*x + 1)/(a*x - 1))^2/((a*x - 1)^3*((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)) + 2*(10*(a*x + 1)^2*a^5/(a*x - 1)^2 + 5*(a*x + 1)*a^5/(a*x - 1 ) + a^5)*log(-(a*x + 1)/(a*x - 1))/((a*x + 1)^5/(a*x - 1)^5 + 5*(a*x + 1)^ 4/(a*x - 1)^4 + 10*(a*x + 1)^3/(a*x - 1)^3 + 10*(a*x + 1)^2/(a*x - 1)^2 + 5*(a*x + 1)/(a*x - 1) + 1) - (2*(a*x + 1)^3*a^5/(a*x - 1)^3 + 7*(a*x + 1)^ 2*a^5/(a*x - 1)^2 + 2*(a*x + 1)*a^5/(a*x - 1))/((a*x + 1)^4/(a*x - 1)^4 + 4*(a*x + 1)^3/(a*x - 1)^3 + 6*(a*x + 1)^2/(a*x - 1)^2 + 4*(a*x + 1)/(a*x - 1) + 1))*a
Time = 4.32 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.96 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^7} \, dx=\frac {8\,a^6\,\ln \left (x\right )}{45}-\frac {\frac {3\,a^2}{4}-\frac {7\,a^4\,x^2}{2}}{45\,x^4}-{\ln \left (1-a\,x\right )}^2\,\left (\frac {\frac {a^4\,x^4}{2}-\frac {a^2\,x^2}{2}+\frac {1}{6}}{4\,x^6}-\frac {a^6}{24}\right )-{\ln \left (a\,x+1\right )}^2\,\left (\frac {\frac {a^4\,x^4}{8}-\frac {a^2\,x^2}{8}+\frac {1}{24}}{x^6}-\frac {a^6}{24}\right )-\ln \left (1-a\,x\right )\,\left (\frac {a\,\left (\frac {137\,a^5\,x^5}{2}-30\,a^4\,x^4+15\,a^3\,x^3-10\,a^2\,x^2+\frac {15\,a\,x}{2}-6\right )}{360\,x^5}-\ln \left (a\,x+1\right )\,\left (\frac {\frac {a^4\,x^4}{2}-\frac {a^2\,x^2}{2}+\frac {1}{6}}{2\,x^6}-\frac {a^6}{12}\right )-\frac {a\,\left (137\,a^5\,x^5+60\,a^4\,x^4+30\,a^3\,x^3+20\,a^2\,x^2+15\,a\,x+12\right )}{720\,x^5}+\frac {5\,a^8\,x^2-\frac {15\,a^9\,x^3}{2}}{60\,a^5\,x^5}+\frac {\frac {15\,a^9\,x^3}{2}+5\,a^8\,x^2}{60\,a^5\,x^5}\right )-\frac {4\,a^6\,\ln \left (a^2\,x^2-1\right )}{45}-\frac {a\,\ln \left (a\,x+1\right )\,\left (\frac {a^4\,x^4}{6}-\frac {a^2\,x^2}{9}+\frac {1}{30}\right )}{x^5} \] Input:
int((atanh(a*x)^2*(a^2*x^2 - 1)^2)/x^7,x)
Output:
(8*a^6*log(x))/45 - ((3*a^2)/4 - (7*a^4*x^2)/2)/(45*x^4) - log(1 - a*x)^2* (((a^4*x^4)/2 - (a^2*x^2)/2 + 1/6)/(4*x^6) - a^6/24) - log(a*x + 1)^2*(((a ^4*x^4)/8 - (a^2*x^2)/8 + 1/24)/x^6 - a^6/24) - log(1 - a*x)*((a*((15*a*x) /2 - 10*a^2*x^2 + 15*a^3*x^3 - 30*a^4*x^4 + (137*a^5*x^5)/2 - 6))/(360*x^5 ) - log(a*x + 1)*(((a^4*x^4)/2 - (a^2*x^2)/2 + 1/6)/(2*x^6) - a^6/12) - (a *(15*a*x + 20*a^2*x^2 + 30*a^3*x^3 + 60*a^4*x^4 + 137*a^5*x^5 + 12))/(720* x^5) + (5*a^8*x^2 - (15*a^9*x^3)/2)/(60*a^5*x^5) + (5*a^8*x^2 + (15*a^9*x^ 3)/2)/(60*a^5*x^5)) - (4*a^6*log(a^2*x^2 - 1))/45 - (a*log(a*x + 1)*((a^4* x^4)/6 - (a^2*x^2)/9 + 1/30))/x^5
Time = 0.18 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.27 \[ \int \frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{x^7} \, dx=\frac {30 \mathit {atanh} \left (a x \right )^{2} a^{6} x^{6}-90 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}+90 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-30 \mathit {atanh} \left (a x \right )^{2}-32 \mathit {atanh} \left (a x \right ) a^{6} x^{6}-60 \mathit {atanh} \left (a x \right ) a^{5} x^{5}+40 \mathit {atanh} \left (a x \right ) a^{3} x^{3}-12 \mathit {atanh} \left (a x \right ) a x -32 \,\mathrm {log}\left (a^{2} x -a \right ) a^{6} x^{6}+32 \,\mathrm {log}\left (x \right ) a^{6} x^{6}+14 a^{4} x^{4}-3 a^{2} x^{2}}{180 x^{6}} \] Input:
int((-a^2*x^2+1)^2*atanh(a*x)^2/x^7,x)
Output:
(30*atanh(a*x)**2*a**6*x**6 - 90*atanh(a*x)**2*a**4*x**4 + 90*atanh(a*x)** 2*a**2*x**2 - 30*atanh(a*x)**2 - 32*atanh(a*x)*a**6*x**6 - 60*atanh(a*x)*a **5*x**5 + 40*atanh(a*x)*a**3*x**3 - 12*atanh(a*x)*a*x - 32*log(a**2*x - a )*a**6*x**6 + 32*log(x)*a**6*x**6 + 14*a**4*x**4 - 3*a**2*x**2)/(180*x**6)