Integrand size = 22, antiderivative size = 156 \[ \int x^3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=-\frac {5 x^2}{504 a^2}-\frac {x^4}{84}+\frac {a^2 x^6}{168}+\frac {x \text {arctanh}(a x)}{12 a^3}+\frac {x^3 \text {arctanh}(a x)}{36 a}-\frac {1}{12} a x^5 \text {arctanh}(a x)+\frac {1}{28} a^3 x^7 \text {arctanh}(a x)-\frac {\text {arctanh}(a x)^2}{24 a^4}+\frac {1}{4} x^4 \text {arctanh}(a x)^2-\frac {1}{3} a^2 x^6 \text {arctanh}(a x)^2+\frac {1}{8} a^4 x^8 \text {arctanh}(a x)^2+\frac {2 \log \left (1-a^2 x^2\right )}{63 a^4} \] Output:
-5/504*x^2/a^2-1/84*x^4+1/168*a^2*x^6+1/12*x*arctanh(a*x)/a^3+1/36*x^3*arc tanh(a*x)/a-1/12*a*x^5*arctanh(a*x)+1/28*a^3*x^7*arctanh(a*x)-1/24*arctanh (a*x)^2/a^4+1/4*x^4*arctanh(a*x)^2-1/3*a^2*x^6*arctanh(a*x)^2+1/8*a^4*x^8* arctanh(a*x)^2+2/63*ln(-a^2*x^2+1)/a^4
Time = 0.04 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.69 \[ \int x^3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\frac {-5 a^2 x^2-6 a^4 x^4+3 a^6 x^6+2 a x \left (21+7 a^2 x^2-21 a^4 x^4+9 a^6 x^6\right ) \text {arctanh}(a x)+21 \left (-1+a^2 x^2\right )^3 \left (1+3 a^2 x^2\right ) \text {arctanh}(a x)^2+16 \log \left (1-a^2 x^2\right )}{504 a^4} \] Input:
Integrate[x^3*(1 - a^2*x^2)^2*ArcTanh[a*x]^2,x]
Output:
(-5*a^2*x^2 - 6*a^4*x^4 + 3*a^6*x^6 + 2*a*x*(21 + 7*a^2*x^2 - 21*a^4*x^4 + 9*a^6*x^6)*ArcTanh[a*x] + 21*(-1 + a^2*x^2)^3*(1 + 3*a^2*x^2)*ArcTanh[a*x ]^2 + 16*Log[1 - a^2*x^2])/(504*a^4)
Time = 1.07 (sec) , antiderivative size = 156, 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.
| \(\displaystyle \int x^3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx\) |
| \(\Big \downarrow \) 6574 |
| \(\displaystyle \int \left (a^4 x^7 \text {arctanh}(a x)^2-2 a^2 x^5 \text {arctanh}(a x)^2+x^3 \text {arctanh}(a x)^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{8} a^4 x^8 \text {arctanh}(a x)^2-\frac {\text {arctanh}(a x)^2}{24 a^4}+\frac {1}{28} a^3 x^7 \text {arctanh}(a x)+\frac {x \text {arctanh}(a x)}{12 a^3}-\frac {1}{3} a^2 x^6 \text {arctanh}(a x)^2+\frac {a^2 x^6}{168}-\frac {5 x^2}{504 a^2}+\frac {2 \log \left (1-a^2 x^2\right )}{63 a^4}-\frac {1}{12} a x^5 \text {arctanh}(a x)+\frac {1}{4} x^4 \text {arctanh}(a x)^2+\frac {x^3 \text {arctanh}(a x)}{36 a}-\frac {x^4}{84}\) |
Input:
Int[x^3*(1 - a^2*x^2)^2*ArcTanh[a*x]^2,x]
Output:
(-5*x^2)/(504*a^2) - x^4/84 + (a^2*x^6)/168 + (x*ArcTanh[a*x])/(12*a^3) + (x^3*ArcTanh[a*x])/(36*a) - (a*x^5*ArcTanh[a*x])/12 + (a^3*x^7*ArcTanh[a*x ])/28 - ArcTanh[a*x]^2/(24*a^4) + (x^4*ArcTanh[a*x]^2)/4 - (a^2*x^6*ArcTan h[a*x]^2)/3 + (a^4*x^8*ArcTanh[a*x]^2)/8 + (2*Log[1 - a^2*x^2])/(63*a^4)
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 = 0.58 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.90
| method | result | size |
| parallelrisch | \(-\frac {-63 \operatorname {arctanh}\left (a x \right )^{2} a^{8} x^{8}-18 \,\operatorname {arctanh}\left (a x \right ) a^{7} x^{7}+168 \operatorname {arctanh}\left (a x \right )^{2} a^{6} x^{6}-3 a^{6} x^{6}+42 \,\operatorname {arctanh}\left (a x \right ) a^{5} x^{5}-126 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}+6 a^{4} x^{4}-14 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )+5+5 a^{2} x^{2}-42 a x \,\operatorname {arctanh}\left (a x \right )+21 \operatorname {arctanh}\left (a x \right )^{2}-32 \ln \left (a x -1\right )-32 \,\operatorname {arctanh}\left (a x \right )}{504 a^{4}}\) | \(140\) |
| derivativedivides | \(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{8} x^{8}}{8}-\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{6} x^{6}}{3}+\frac {a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}}{4}+\frac {\operatorname {arctanh}\left (a x \right ) a^{7} x^{7}}{28}-\frac {\operatorname {arctanh}\left (a x \right ) a^{5} x^{5}}{12}+\frac {a^{3} x^{3} \operatorname {arctanh}\left (a x \right )}{36}+\frac {a x \,\operatorname {arctanh}\left (a x \right )}{12}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{24}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{24}+\frac {\ln \left (a x -1\right )^{2}}{96}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{48}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{48}+\frac {\ln \left (a x +1\right )^{2}}{96}+\frac {a^{6} x^{6}}{168}-\frac {a^{4} x^{4}}{84}-\frac {5 a^{2} x^{2}}{504}+\frac {2 \ln \left (a x -1\right )}{63}+\frac {2 \ln \left (a x +1\right )}{63}}{a^{4}}\) | \(216\) |
| default | \(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{8} x^{8}}{8}-\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{6} x^{6}}{3}+\frac {a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}}{4}+\frac {\operatorname {arctanh}\left (a x \right ) a^{7} x^{7}}{28}-\frac {\operatorname {arctanh}\left (a x \right ) a^{5} x^{5}}{12}+\frac {a^{3} x^{3} \operatorname {arctanh}\left (a x \right )}{36}+\frac {a x \,\operatorname {arctanh}\left (a x \right )}{12}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{24}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{24}+\frac {\ln \left (a x -1\right )^{2}}{96}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{48}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{48}+\frac {\ln \left (a x +1\right )^{2}}{96}+\frac {a^{6} x^{6}}{168}-\frac {a^{4} x^{4}}{84}-\frac {5 a^{2} x^{2}}{504}+\frac {2 \ln \left (a x -1\right )}{63}+\frac {2 \ln \left (a x +1\right )}{63}}{a^{4}}\) | \(216\) |
| parts | \(\frac {a^{4} x^{8} \operatorname {arctanh}\left (a x \right )^{2}}{8}-\frac {a^{2} x^{6} \operatorname {arctanh}\left (a x \right )^{2}}{3}+\frac {x^{4} \operatorname {arctanh}\left (a x \right )^{2}}{4}+\frac {a^{3} x^{7} \operatorname {arctanh}\left (a x \right )}{28}-\frac {a \,x^{5} \operatorname {arctanh}\left (a x \right )}{12}+\frac {x^{3} \operatorname {arctanh}\left (a x \right )}{36 a}+\frac {x \,\operatorname {arctanh}\left (a x \right )}{12 a^{3}}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{24 a^{4}}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{24 a^{4}}-\frac {-\frac {21 \ln \left (a x -1\right )^{2}}{4}+\frac {21 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {21 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {21 \ln \left (a x +1\right )^{2}}{4}-3 a^{6} x^{6}+6 a^{4} x^{4}+5 a^{2} x^{2}-16 \ln \left (a x -1\right )-16 \ln \left (a x +1\right )}{504 a^{4}}\) | \(221\) |
| risch | \(\frac {\left (3 a^{8} x^{8}-8 a^{6} x^{6}+6 a^{4} x^{4}-1\right ) \ln \left (a x +1\right )^{2}}{96 a^{4}}-\frac {\left (63 a^{8} x^{8} \ln \left (-a x +1\right )-18 a^{7} x^{7}-168 a^{6} x^{6} \ln \left (-a x +1\right )+42 a^{5} x^{5}+126 x^{4} \ln \left (-a x +1\right ) a^{4}-14 a^{3} x^{3}-42 a x -21 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{1008 a^{4}}+\frac {\ln \left (-a x +1\right )^{2} a^{4} x^{8}}{32}-\frac {a^{3} x^{7} \ln \left (-a x +1\right )}{56}-\frac {\ln \left (-a x +1\right )^{2} a^{2} x^{6}}{12}+\frac {a^{2} x^{6}}{168}+\frac {a \,x^{5} \ln \left (-a x +1\right )}{24}+\frac {x^{4} \ln \left (-a x +1\right )^{2}}{16}-\frac {x^{4}}{84}-\frac {x^{3} \ln \left (-a x +1\right )}{72 a}-\frac {5 x^{2}}{504 a^{2}}-\frac {x \ln \left (-a x +1\right )}{24 a^{3}}-\frac {\ln \left (-a x +1\right )^{2}}{96 a^{4}}+\frac {2 \ln \left (a^{2} x^{2}-1\right )}{63 a^{4}}+\frac {25}{1134 a^{4}}\) | \(294\) |
Input:
int(x^3*(-a^2*x^2+1)^2*arctanh(a*x)^2,x,method=_RETURNVERBOSE)
Output:
-1/504*(-63*arctanh(a*x)^2*a^8*x^8-18*arctanh(a*x)*a^7*x^7+168*arctanh(a*x )^2*a^6*x^6-3*a^6*x^6+42*arctanh(a*x)*a^5*x^5-126*a^4*x^4*arctanh(a*x)^2+6 *a^4*x^4-14*a^3*x^3*arctanh(a*x)+5+5*a^2*x^2-42*a*x*arctanh(a*x)+21*arctan h(a*x)^2-32*ln(a*x-1)-32*arctanh(a*x))/a^4
Time = 0.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.85 \[ \int x^3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\frac {12 \, a^{6} x^{6} - 24 \, a^{4} x^{4} - 20 \, a^{2} x^{2} + 21 \, {\left (3 \, a^{8} x^{8} - 8 \, a^{6} x^{6} + 6 \, a^{4} x^{4} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, {\left (9 \, a^{7} x^{7} - 21 \, a^{5} x^{5} + 7 \, a^{3} x^{3} + 21 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + 64 \, \log \left (a^{2} x^{2} - 1\right )}{2016 \, a^{4}} \] Input:
integrate(x^3*(-a^2*x^2+1)^2*arctanh(a*x)^2,x, algorithm="fricas")
Output:
1/2016*(12*a^6*x^6 - 24*a^4*x^4 - 20*a^2*x^2 + 21*(3*a^8*x^8 - 8*a^6*x^6 + 6*a^4*x^4 - 1)*log(-(a*x + 1)/(a*x - 1))^2 + 4*(9*a^7*x^7 - 21*a^5*x^5 + 7*a^3*x^3 + 21*a*x)*log(-(a*x + 1)/(a*x - 1)) + 64*log(a^2*x^2 - 1))/a^4
Time = 0.68 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.98 \[ \int x^3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\begin {cases} \frac {a^{4} x^{8} \operatorname {atanh}^{2}{\left (a x \right )}}{8} + \frac {a^{3} x^{7} \operatorname {atanh}{\left (a x \right )}}{28} - \frac {a^{2} x^{6} \operatorname {atanh}^{2}{\left (a x \right )}}{3} + \frac {a^{2} x^{6}}{168} - \frac {a x^{5} \operatorname {atanh}{\left (a x \right )}}{12} + \frac {x^{4} \operatorname {atanh}^{2}{\left (a x \right )}}{4} - \frac {x^{4}}{84} + \frac {x^{3} \operatorname {atanh}{\left (a x \right )}}{36 a} - \frac {5 x^{2}}{504 a^{2}} + \frac {x \operatorname {atanh}{\left (a x \right )}}{12 a^{3}} + \frac {4 \log {\left (x - \frac {1}{a} \right )}}{63 a^{4}} - \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{24 a^{4}} + \frac {4 \operatorname {atanh}{\left (a x \right )}}{63 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x**3*(-a**2*x**2+1)**2*atanh(a*x)**2,x)
Output:
Piecewise((a**4*x**8*atanh(a*x)**2/8 + a**3*x**7*atanh(a*x)/28 - a**2*x**6 *atanh(a*x)**2/3 + a**2*x**6/168 - a*x**5*atanh(a*x)/12 + x**4*atanh(a*x)* *2/4 - x**4/84 + x**3*atanh(a*x)/(36*a) - 5*x**2/(504*a**2) + x*atanh(a*x) /(12*a**3) + 4*log(x - 1/a)/(63*a**4) - atanh(a*x)**2/(24*a**4) + 4*atanh( a*x)/(63*a**4), Ne(a, 0)), (0, True))
Time = 0.03 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.09 \[ \int x^3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\frac {1}{504} \, a {\left (\frac {2 \, {\left (9 \, a^{6} x^{7} - 21 \, a^{4} x^{5} + 7 \, a^{2} x^{3} + 21 \, x\right )}}{a^{4}} - \frac {21 \, \log \left (a x + 1\right )}{a^{5}} + \frac {21 \, \log \left (a x - 1\right )}{a^{5}}\right )} \operatorname {artanh}\left (a x\right ) + \frac {1}{24} \, {\left (3 \, a^{4} x^{8} - 8 \, a^{2} x^{6} + 6 \, x^{4}\right )} \operatorname {artanh}\left (a x\right )^{2} + \frac {12 \, a^{6} x^{6} - 24 \, a^{4} x^{4} - 20 \, a^{2} x^{2} - 2 \, {\left (21 \, \log \left (a x - 1\right ) - 32\right )} \log \left (a x + 1\right ) + 21 \, \log \left (a x + 1\right )^{2} + 21 \, \log \left (a x - 1\right )^{2} + 64 \, \log \left (a x - 1\right )}{2016 \, a^{4}} \] Input:
integrate(x^3*(-a^2*x^2+1)^2*arctanh(a*x)^2,x, algorithm="maxima")
Output:
1/504*a*(2*(9*a^6*x^7 - 21*a^4*x^5 + 7*a^2*x^3 + 21*x)/a^4 - 21*log(a*x + 1)/a^5 + 21*log(a*x - 1)/a^5)*arctanh(a*x) + 1/24*(3*a^4*x^8 - 8*a^2*x^6 + 6*x^4)*arctanh(a*x)^2 + 1/2016*(12*a^6*x^6 - 24*a^4*x^4 - 20*a^2*x^2 - 2* (21*log(a*x - 1) - 32)*log(a*x + 1) + 21*log(a*x + 1)^2 + 21*log(a*x - 1)^ 2 + 64*log(a*x - 1))/a^4
Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (132) = 264\).
Time = 0.14 (sec) , antiderivative size = 683, normalized size of antiderivative = 4.38 \[ \int x^3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx =\text {Too large to display} \] Input:
integrate(x^3*(-a^2*x^2+1)^2*arctanh(a*x)^2,x, algorithm="giac")
Output:
2/63*(84*((a*x + 1)^5/(a*x - 1)^5 + (a*x + 1)^4/(a*x - 1)^4 + (a*x + 1)^3/ (a*x - 1)^3)*log(-(a*x + 1)/(a*x - 1))^2/((a*x + 1)^8*a^5/(a*x - 1)^8 - 8* (a*x + 1)^7*a^5/(a*x - 1)^7 + 28*(a*x + 1)^6*a^5/(a*x - 1)^6 - 56*(a*x + 1 )^5*a^5/(a*x - 1)^5 + 70*(a*x + 1)^4*a^5/(a*x - 1)^4 - 56*(a*x + 1)^3*a^5/ (a*x - 1)^3 + 28*(a*x + 1)^2*a^5/(a*x - 1)^2 - 8*(a*x + 1)*a^5/(a*x - 1) + a^5) + 2*(28*(a*x + 1)^4/(a*x - 1)^4 - 7*(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)*log(-(a*x + 1)/(a*x - 1) )/((a*x + 1)^7*a^5/(a*x - 1)^7 - 7*(a*x + 1)^6*a^5/(a*x - 1)^6 + 21*(a*x + 1)^5*a^5/(a*x - 1)^5 - 35*(a*x + 1)^4*a^5/(a*x - 1)^4 + 35*(a*x + 1)^3*a^ 5/(a*x - 1)^3 - 21*(a*x + 1)^2*a^5/(a*x - 1)^2 + 7*(a*x + 1)*a^5/(a*x - 1) - a^5) - (2*(a*x + 1)^5/(a*x - 1)^5 - 11*(a*x + 1)^4/(a*x - 1)^4 + 6*(a*x + 1)^3/(a*x - 1)^3 - 11*(a*x + 1)^2/(a*x - 1)^2 + 2*(a*x + 1)/(a*x - 1))/ ((a*x + 1)^6*a^5/(a*x - 1)^6 - 6*(a*x + 1)^5*a^5/(a*x - 1)^5 + 15*(a*x + 1 )^4*a^5/(a*x - 1)^4 - 20*(a*x + 1)^3*a^5/(a*x - 1)^3 + 15*(a*x + 1)^2*a^5/ (a*x - 1)^2 - 6*(a*x + 1)*a^5/(a*x - 1) + a^5) - 2*log(-(a*x + 1)/(a*x - 1 ) + 1)/a^5 + 2*log(-(a*x + 1)/(a*x - 1))/a^5)*a
Time = 4.05 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.42 \[ \int x^3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\frac {2\,\ln \left (a^2\,x^2-1\right )}{63\,a^4}-{\ln \left (1-a\,x\right )}^2\,\left (\frac {1}{96\,a^4}-\frac {x^4}{16}+\frac {a^2\,x^6}{12}-\frac {a^4\,x^8}{32}\right )-\frac {x^4}{84}-{\ln \left (a\,x+1\right )}^2\,\left (\frac {1}{96\,a^4}-\frac {x^4}{16}+\frac {a^2\,x^6}{12}-\frac {a^4\,x^8}{32}\right )-\ln \left (1-a\,x\right )\,\left (\frac {x}{24\,a^3}-\ln \left (a\,x+1\right )\,\left (\frac {1}{48\,a^4}-\frac {x^4}{8}+\frac {a^2\,x^6}{6}-\frac {a^4\,x^8}{16}\right )-\frac {a\,x^5}{24}+\frac {x^3}{72\,a}+\frac {a^3\,x^7}{56}\right )-\frac {5\,x^2}{504\,a^2}+\frac {a^2\,x^6}{168}+a\,\ln \left (a\,x+1\right )\,\left (\frac {x}{24\,a^4}-\frac {x^5}{24}+\frac {x^3}{72\,a^2}+\frac {a^2\,x^7}{56}\right ) \] Input:
int(x^3*atanh(a*x)^2*(a^2*x^2 - 1)^2,x)
Output:
(2*log(a^2*x^2 - 1))/(63*a^4) - log(1 - a*x)^2*(1/(96*a^4) - x^4/16 + (a^2 *x^6)/12 - (a^4*x^8)/32) - x^4/84 - log(a*x + 1)^2*(1/(96*a^4) - x^4/16 + (a^2*x^6)/12 - (a^4*x^8)/32) - log(1 - a*x)*(x/(24*a^3) - log(a*x + 1)*(1/ (48*a^4) - x^4/8 + (a^2*x^6)/6 - (a^4*x^8)/16) - (a*x^5)/24 + x^3/(72*a) + (a^3*x^7)/56) - (5*x^2)/(504*a^2) + (a^2*x^6)/168 + a*log(a*x + 1)*(x/(24 *a^4) - x^5/24 + x^3/(72*a^2) + (a^2*x^7)/56)
Time = 0.17 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.91 \[ \int x^3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\frac {63 \mathit {atanh} \left (a x \right )^{2} a^{8} x^{8}-168 \mathit {atanh} \left (a x \right )^{2} a^{6} x^{6}+126 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}-21 \mathit {atanh} \left (a x \right )^{2}+18 \mathit {atanh} \left (a x \right ) a^{7} x^{7}-42 \mathit {atanh} \left (a x \right ) a^{5} x^{5}+14 \mathit {atanh} \left (a x \right ) a^{3} x^{3}+42 \mathit {atanh} \left (a x \right ) a x +32 \mathit {atanh} \left (a x \right )+32 \,\mathrm {log}\left (a^{2} x -a \right )+3 a^{6} x^{6}-6 a^{4} x^{4}-5 a^{2} x^{2}}{504 a^{4}} \] Input:
int(x^3*(-a^2*x^2+1)^2*atanh(a*x)^2,x)
Output:
(63*atanh(a*x)**2*a**8*x**8 - 168*atanh(a*x)**2*a**6*x**6 + 126*atanh(a*x) **2*a**4*x**4 - 21*atanh(a*x)**2 + 18*atanh(a*x)*a**7*x**7 - 42*atanh(a*x) *a**5*x**5 + 14*atanh(a*x)*a**3*x**3 + 42*atanh(a*x)*a*x + 32*atanh(a*x) + 32*log(a**2*x - a) + 3*a**6*x**6 - 6*a**4*x**4 - 5*a**2*x**2)/(504*a**4)