Integrand size = 22, antiderivative size = 202 \[ \int x^4 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\frac {29 x}{3780 a^4}-\frac {67 x^3}{11340 a^2}-\frac {23 x^5}{3780}+\frac {a^2 x^7}{252}-\frac {29 \text {arctanh}(a x)}{3780 a^5}+\frac {8 x^2 \text {arctanh}(a x)}{315 a^3}+\frac {4 x^4 \text {arctanh}(a x)}{315 a}-\frac {11}{189} a x^6 \text {arctanh}(a x)+\frac {1}{36} a^3 x^8 \text {arctanh}(a x)+\frac {8 \text {arctanh}(a x)^2}{315 a^5}+\frac {1}{5} x^5 \text {arctanh}(a x)^2-\frac {2}{7} a^2 x^7 \text {arctanh}(a x)^2+\frac {1}{9} a^4 x^9 \text {arctanh}(a x)^2-\frac {16 \text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{315 a^5}-\frac {8 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{315 a^5} \] Output:
29/3780*x/a^4-67/11340*x^3/a^2-23/3780*x^5+1/252*a^2*x^7-29/3780*arctanh(a *x)/a^5+8/315*x^2*arctanh(a*x)/a^3+4/315*x^4*arctanh(a*x)/a-11/189*a*x^6*a rctanh(a*x)+1/36*a^3*x^8*arctanh(a*x)+8/315*arctanh(a*x)^2/a^5+1/5*x^5*arc tanh(a*x)^2-2/7*a^2*x^7*arctanh(a*x)^2+1/9*a^4*x^9*arctanh(a*x)^2-16/315*a rctanh(a*x)*ln(2/(-a*x+1))/a^5-8/315*polylog(2,1-2/(-a*x+1))/a^5
Time = 1.32 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.68 \[ \int x^4 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\frac {a x \left (87-67 a^2 x^2-69 a^4 x^4+45 a^6 x^6\right )+36 \left (-8+63 a^5 x^5-90 a^7 x^7+35 a^9 x^9\right ) \text {arctanh}(a x)^2+3 \text {arctanh}(a x) \left (-29+96 a^2 x^2+48 a^4 x^4-220 a^6 x^6+105 a^8 x^8-192 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )\right )+288 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )}{11340 a^5} \] Input:
Integrate[x^4*(1 - a^2*x^2)^2*ArcTanh[a*x]^2,x]
Output:
(a*x*(87 - 67*a^2*x^2 - 69*a^4*x^4 + 45*a^6*x^6) + 36*(-8 + 63*a^5*x^5 - 9 0*a^7*x^7 + 35*a^9*x^9)*ArcTanh[a*x]^2 + 3*ArcTanh[a*x]*(-29 + 96*a^2*x^2 + 48*a^4*x^4 - 220*a^6*x^6 + 105*a^8*x^8 - 192*Log[1 + E^(-2*ArcTanh[a*x]) ]) + 288*PolyLog[2, -E^(-2*ArcTanh[a*x])])/(11340*a^5)
Time = 1.22 (sec) , antiderivative size = 202, 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.
| \(\displaystyle \int x^4 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx\) |
| \(\Big \downarrow \) 6574 |
| \(\displaystyle \int \left (a^4 x^8 \text {arctanh}(a x)^2-2 a^2 x^6 \text {arctanh}(a x)^2+x^4 \text {arctanh}(a x)^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8 \text {arctanh}(a x)^2}{315 a^5}-\frac {29 \text {arctanh}(a x)}{3780 a^5}-\frac {16 \text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{315 a^5}-\frac {8 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{315 a^5}+\frac {1}{9} a^4 x^9 \text {arctanh}(a x)^2+\frac {29 x}{3780 a^4}+\frac {1}{36} a^3 x^8 \text {arctanh}(a x)+\frac {8 x^2 \text {arctanh}(a x)}{315 a^3}-\frac {2}{7} a^2 x^7 \text {arctanh}(a x)^2+\frac {a^2 x^7}{252}-\frac {67 x^3}{11340 a^2}-\frac {11}{189} a x^6 \text {arctanh}(a x)+\frac {1}{5} x^5 \text {arctanh}(a x)^2+\frac {4 x^4 \text {arctanh}(a x)}{315 a}-\frac {23 x^5}{3780}\) |
Input:
Int[x^4*(1 - a^2*x^2)^2*ArcTanh[a*x]^2,x]
Output:
(29*x)/(3780*a^4) - (67*x^3)/(11340*a^2) - (23*x^5)/3780 + (a^2*x^7)/252 - (29*ArcTanh[a*x])/(3780*a^5) + (8*x^2*ArcTanh[a*x])/(315*a^3) + (4*x^4*Ar cTanh[a*x])/(315*a) - (11*a*x^6*ArcTanh[a*x])/189 + (a^3*x^8*ArcTanh[a*x]) /36 + (8*ArcTanh[a*x]^2)/(315*a^5) + (x^5*ArcTanh[a*x]^2)/5 - (2*a^2*x^7*A rcTanh[a*x]^2)/7 + (a^4*x^9*ArcTanh[a*x]^2)/9 - (16*ArcTanh[a*x]*Log[2/(1 - a*x)])/(315*a^5) - (8*PolyLog[2, 1 - 2/(1 - a*x)])/(315*a^5)
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 = 1.16 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.15
| method | result | size |
| derivativedivides | \(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{9} x^{9}}{9}-\frac {2 \operatorname {arctanh}\left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{5} x^{5}}{5}+\frac {\operatorname {arctanh}\left (a x \right ) a^{8} x^{8}}{36}-\frac {11 \,\operatorname {arctanh}\left (a x \right ) a^{6} x^{6}}{189}+\frac {4 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )}{315}+\frac {8 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{315}+\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{315}+\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{315}+\frac {a^{7} x^{7}}{252}-\frac {23 a^{5} x^{5}}{3780}-\frac {67 a^{3} x^{3}}{11340}+\frac {29 a x}{3780}+\frac {29 \ln \left (a x -1\right )}{7560}-\frac {29 \ln \left (a x +1\right )}{7560}+\frac {2 \ln \left (a x -1\right )^{2}}{315}-\frac {8 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{315}-\frac {4 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{315}+\frac {4 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{315}-\frac {2 \ln \left (a x +1\right )^{2}}{315}}{a^{5}}\) | \(233\) |
| default | \(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{9} x^{9}}{9}-\frac {2 \operatorname {arctanh}\left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{5} x^{5}}{5}+\frac {\operatorname {arctanh}\left (a x \right ) a^{8} x^{8}}{36}-\frac {11 \,\operatorname {arctanh}\left (a x \right ) a^{6} x^{6}}{189}+\frac {4 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )}{315}+\frac {8 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{315}+\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{315}+\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{315}+\frac {a^{7} x^{7}}{252}-\frac {23 a^{5} x^{5}}{3780}-\frac {67 a^{3} x^{3}}{11340}+\frac {29 a x}{3780}+\frac {29 \ln \left (a x -1\right )}{7560}-\frac {29 \ln \left (a x +1\right )}{7560}+\frac {2 \ln \left (a x -1\right )^{2}}{315}-\frac {8 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{315}-\frac {4 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{315}+\frac {4 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{315}-\frac {2 \ln \left (a x +1\right )^{2}}{315}}{a^{5}}\) | \(233\) |
| parts | \(\frac {a^{4} x^{9} \operatorname {arctanh}\left (a x \right )^{2}}{9}-\frac {2 a^{2} x^{7} \operatorname {arctanh}\left (a x \right )^{2}}{7}+\frac {x^{5} \operatorname {arctanh}\left (a x \right )^{2}}{5}+\frac {a^{3} x^{8} \operatorname {arctanh}\left (a x \right )}{36}-\frac {11 a \,x^{6} \operatorname {arctanh}\left (a x \right )}{189}+\frac {4 x^{4} \operatorname {arctanh}\left (a x \right )}{315 a}+\frac {8 x^{2} \operatorname {arctanh}\left (a x \right )}{315 a^{3}}+\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{315 a^{5}}+\frac {8 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{315 a^{5}}-\frac {-24 \ln \left (a x -1\right )^{2}+96 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )+48 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )-48 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )+24 \ln \left (a x +1\right )^{2}-15 a^{7} x^{7}+23 a^{5} x^{5}+\frac {67 a^{3} x^{3}}{3}-29 a x -\frac {29 \ln \left (a x -1\right )}{2}+\frac {29 \ln \left (a x +1\right )}{2}}{3780 a^{5}}\) | \(236\) |
| risch | \(-\frac {\left (\left (-\frac {1}{25}+\frac {\ln \left (a x +1\right )}{5}\right ) \left (a x +1\right )^{5}+\left (\frac {1}{4}-\ln \left (a x +1\right )\right ) \left (a x +1\right )^{4}+\left (-\frac {2}{3}+2 \ln \left (a x +1\right )\right ) \left (a x +1\right )^{3}+\left (1-2 \ln \left (a x +1\right )\right ) \left (a x +1\right )^{2}+\left (-1+\ln \left (a x +1\right )\right ) \left (a x +1\right )\right ) \ln \left (-a x +1\right )}{2 a^{5}}+\frac {\left (a x +1\right ) \ln \left (a x +1\right )}{10 a^{5}}+\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 )}{5 a^{5}}-\frac {\left (a x +1\right )^{2} \ln \left (a x +1\right )}{10 a^{5}}-\frac {3 \left (a x +1\right )^{4} \ln \left (a x +1\right )}{40 a^{5}}+\frac {\left (a x +1\right )^{5} \ln \left (a x +1\right )}{50 a^{5}}+\frac {2 \left (a x +1\right )^{3} \ln \left (a x +1\right )}{15 a^{5}}+\frac {11 \ln \left (-a x +1\right ) \ln \left (a x +1\right )}{126 a^{5}}-\frac {11 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{63 a^{5}}+\frac {11 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{63 a^{5}}-\frac {2 \ln \left (-a x +1\right )^{2}}{315 a^{5}}-\frac {78647 \ln \left (-a x +1\right )}{396900 a^{5}}+\frac {\ln \left (-a x +1\right )^{2} x^{5}}{20}-\frac {1556839}{62511750 a^{5}}+\frac {a^{2} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{7}}{7}-\frac {23 x^{5}}{3780}+\frac {a^{2} x^{7}}{252}-\frac {2614 \ln \left (a x -1\right )}{99225 a^{5}}-\frac {1553 \ln \left (a x +1\right )}{18900 a^{5}}-\frac {x^{5} \ln \left (-a x +1\right )}{50}+\frac {29 x}{3780 a^{4}}-\frac {67 x^{3}}{11340 a^{2}}+\frac {a^{4} \ln \left (-a x +1\right )^{2} x^{9}}{36}-\frac {47 \ln \left (a x +1\right ) x^{2}}{1260 a^{3}}-\frac {a^{3} \ln \left (-a x +1\right ) x^{8}}{72}-\frac {8 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{315 a^{5}}+\frac {2 \ln \left (a x +1\right )^{2}}{315 a^{5}}+\frac {\ln \left (a x +1\right )^{2} x^{5}}{20}-\frac {\ln \left (a x +1\right ) x^{5}}{50}-\frac {\ln \left (a x +1\right ) x}{10 a^{4}}-\frac {a^{2} \ln \left (a x +1\right )^{2} x^{7}}{14}-\frac {11 a \ln \left (a x +1\right ) x^{6}}{378}-\frac {47 \ln \left (a x +1\right ) x^{4}}{2520 a}-\frac {\ln \left (a x +1\right ) x^{3}}{30 a^{2}}+\frac {11 a \ln \left (-a x +1\right ) x^{6}}{378}+\frac {47 \ln \left (-a x +1\right ) x^{4}}{2520 a}-\frac {\ln \left (-a x +1\right ) x^{3}}{30 a^{2}}+\frac {47 \ln \left (-a x +1\right ) x^{2}}{1260 a^{3}}-\frac {\ln \left (-a x +1\right ) x}{10 a^{4}}-\frac {a^{2} \ln \left (-a x +1\right )^{2} x^{7}}{14}+\frac {a^{4} \ln \left (a x +1\right )^{2} x^{9}}{36}+\frac {a^{3} \ln \left (a x +1\right ) x^{8}}{72}-\frac {a^{4} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{9}}{18}\) | \(701\) |
Input:
int(x^4*(-a^2*x^2+1)^2*arctanh(a*x)^2,x,method=_RETURNVERBOSE)
Output:
1/a^5*(1/9*arctanh(a*x)^2*a^9*x^9-2/7*arctanh(a*x)^2*a^7*x^7+1/5*arctanh(a *x)^2*a^5*x^5+1/36*arctanh(a*x)*a^8*x^8-11/189*arctanh(a*x)*a^6*x^6+4/315* a^4*x^4*arctanh(a*x)+8/315*a^2*x^2*arctanh(a*x)+8/315*arctanh(a*x)*ln(a*x- 1)+8/315*arctanh(a*x)*ln(a*x+1)+1/252*a^7*x^7-23/3780*a^5*x^5-67/11340*a^3 *x^3+29/3780*a*x+29/7560*ln(a*x-1)-29/7560*ln(a*x+1)+2/315*ln(a*x-1)^2-8/3 15*dilog(1/2*a*x+1/2)-4/315*ln(a*x-1)*ln(1/2*a*x+1/2)+4/315*(ln(a*x+1)-ln( 1/2*a*x+1/2))*ln(-1/2*a*x+1/2)-2/315*ln(a*x+1)^2)
\[ \int x^4 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\int { {\left (a^{2} x^{2} - 1\right )}^{2} x^{4} \operatorname {artanh}\left (a x\right )^{2} \,d x } \] Input:
integrate(x^4*(-a^2*x^2+1)^2*arctanh(a*x)^2,x, algorithm="fricas")
Output:
integral((a^4*x^8 - 2*a^2*x^6 + x^4)*arctanh(a*x)^2, x)
\[ \int x^4 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\int x^{4} \left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{2}{\left (a x \right )}\, dx \] Input:
integrate(x**4*(-a**2*x**2+1)**2*atanh(a*x)**2,x)
Output:
Integral(x**4*(a*x - 1)**2*(a*x + 1)**2*atanh(a*x)**2, x)
Time = 0.03 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.06 \[ \int x^4 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\frac {1}{22680} \, a^{2} {\left (\frac {90 \, a^{7} x^{7} - 138 \, a^{5} x^{5} - 134 \, a^{3} x^{3} + 174 \, a x - 144 \, \log \left (a x + 1\right )^{2} + 288 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + 144 \, \log \left (a x - 1\right )^{2} + 87 \, \log \left (a x - 1\right )}{a^{7}} - \frac {576 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{7}} - \frac {87 \, \log \left (a x + 1\right )}{a^{7}}\right )} + \frac {1}{3780} \, a {\left (\frac {105 \, a^{6} x^{8} - 220 \, a^{4} x^{6} + 48 \, a^{2} x^{4} + 96 \, x^{2}}{a^{4}} + \frac {96 \, \log \left (a x + 1\right )}{a^{6}} + \frac {96 \, \log \left (a x - 1\right )}{a^{6}}\right )} \operatorname {artanh}\left (a x\right ) + \frac {1}{315} \, {\left (35 \, a^{4} x^{9} - 90 \, a^{2} x^{7} + 63 \, x^{5}\right )} \operatorname {artanh}\left (a x\right )^{2} \] Input:
integrate(x^4*(-a^2*x^2+1)^2*arctanh(a*x)^2,x, algorithm="maxima")
Output:
1/22680*a^2*((90*a^7*x^7 - 138*a^5*x^5 - 134*a^3*x^3 + 174*a*x - 144*log(a *x + 1)^2 + 288*log(a*x + 1)*log(a*x - 1) + 144*log(a*x - 1)^2 + 87*log(a* x - 1))/a^7 - 576*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2) )/a^7 - 87*log(a*x + 1)/a^7) + 1/3780*a*((105*a^6*x^8 - 220*a^4*x^6 + 48*a ^2*x^4 + 96*x^2)/a^4 + 96*log(a*x + 1)/a^6 + 96*log(a*x - 1)/a^6)*arctanh( a*x) + 1/315*(35*a^4*x^9 - 90*a^2*x^7 + 63*x^5)*arctanh(a*x)^2
\[ \int x^4 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\int { {\left (a^{2} x^{2} - 1\right )}^{2} x^{4} \operatorname {artanh}\left (a x\right )^{2} \,d x } \] Input:
integrate(x^4*(-a^2*x^2+1)^2*arctanh(a*x)^2,x, algorithm="giac")
Output:
integrate((a^2*x^2 - 1)^2*x^4*arctanh(a*x)^2, x)
Timed out. \[ \int x^4 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\int x^4\,{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^2 \,d x \] Input:
int(x^4*atanh(a*x)^2*(a^2*x^2 - 1)^2,x)
Output:
int(x^4*atanh(a*x)^2*(a^2*x^2 - 1)^2, x)
\[ \int x^4 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2 \, dx=\frac {1260 \mathit {atanh} \left (a x \right )^{2} a^{9} x^{9}-3240 \mathit {atanh} \left (a x \right )^{2} a^{7} x^{7}+2268 \mathit {atanh} \left (a x \right )^{2} a^{5} x^{5}-288 \mathit {atanh} \left (a x \right )^{2} a x +315 \mathit {atanh} \left (a x \right ) a^{8} x^{8}-660 \mathit {atanh} \left (a x \right ) a^{6} x^{6}+144 \mathit {atanh} \left (a x \right ) a^{4} x^{4}+288 \mathit {atanh} \left (a x \right ) a^{2} x^{2}-87 \mathit {atanh} \left (a x \right )+288 \left (\int \mathit {atanh} \left (a x \right )^{2}d x \right ) a +45 a^{7} x^{7}-69 a^{5} x^{5}-67 a^{3} x^{3}+87 a x}{11340 a^{5}} \] Input:
int(x^4*(-a^2*x^2+1)^2*atanh(a*x)^2,x)
Output:
(1260*atanh(a*x)**2*a**9*x**9 - 3240*atanh(a*x)**2*a**7*x**7 + 2268*atanh( a*x)**2*a**5*x**5 - 288*atanh(a*x)**2*a*x + 315*atanh(a*x)*a**8*x**8 - 660 *atanh(a*x)*a**6*x**6 + 144*atanh(a*x)*a**4*x**4 + 288*atanh(a*x)*a**2*x** 2 - 87*atanh(a*x) + 288*int(atanh(a*x)**2,x)*a + 45*a**7*x**7 - 69*a**5*x* *5 - 67*a**3*x**3 + 87*a*x)/(11340*a**5)