Integrand size = 19, antiderivative size = 227 \[ \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2 \, dx=-\frac {38 x}{105}+\frac {19 a^2 x^3}{315}-\frac {a^4 x^5}{105}+\frac {8 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}{35 a}+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{35 a}+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {16 \text {arctanh}(a x)^2}{35 a}+\frac {16}{35} x \text {arctanh}(a x)^2+\frac {8}{35} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2-\frac {32 \text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{35 a}-\frac {16 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{35 a} \]
-38/105*x+19/315*a^2*x^3-1/105*a^4*x^5+8/35*(-a^2*x^2+1)*arctanh(a*x)/a+3/ 35*(-a^2*x^2+1)^2*arctanh(a*x)/a+1/21*(-a^2*x^2+1)^3*arctanh(a*x)/a+16/35* arctanh(a*x)^2/a+16/35*x*arctanh(a*x)^2+8/35*x*(-a^2*x^2+1)*arctanh(a*x)^2 +6/35*x*(-a^2*x^2+1)^2*arctanh(a*x)^2+1/7*x*(-a^2*x^2+1)^3*arctanh(a*x)^2- 32/35*arctanh(a*x)*ln(2/(-a*x+1))/a-16/35*polylog(2,1-2/(-a*x+1))/a
Time = 0.84 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.55 \[ \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2 \, dx=-\frac {114 a x-19 a^3 x^3+3 a^5 x^5+9 (-1+a x)^4 \left (16+29 a x+20 a^2 x^2+5 a^3 x^3\right ) \text {arctanh}(a x)^2+3 \text {arctanh}(a x) \left (-38+57 a^2 x^2-24 a^4 x^4+5 a^6 x^6+96 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )\right )-144 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )}{315 a} \]
-1/315*(114*a*x - 19*a^3*x^3 + 3*a^5*x^5 + 9*(-1 + a*x)^4*(16 + 29*a*x + 2 0*a^2*x^2 + 5*a^3*x^3)*ArcTanh[a*x]^2 + 3*ArcTanh[a*x]*(-38 + 57*a^2*x^2 - 24*a^4*x^4 + 5*a^6*x^6 + 96*Log[1 + E^(-2*ArcTanh[a*x])]) - 144*PolyLog[2 , -E^(-2*ArcTanh[a*x])])/a
Time = 1.13 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.19, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {6506, 210, 2009, 6506, 2009, 6506, 24, 6436, 6546, 6470, 2849, 2752}
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.
\(\displaystyle \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2 \, dx\) |
\(\Big \downarrow \) 6506 |
\(\displaystyle \frac {6}{7} \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2dx-\frac {1}{21} \int \left (1-a^2 x^2\right )^2dx+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}\) |
\(\Big \downarrow \) 210 |
\(\displaystyle \frac {6}{7} \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2dx-\frac {1}{21} \int \left (a^4 x^4-2 a^2 x^2+1\right )dx+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {6}{7} \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2dx+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\) |
\(\Big \downarrow \) 6506 |
\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2dx-\frac {1}{10} \int \left (1-a^2 x^2\right )dx+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2dx+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\) |
\(\Big \downarrow \) 6506 |
\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \text {arctanh}(a x)^2dx-\frac {\int 1dx}{3}+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \text {arctanh}(a x)^2dx+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}-\frac {x}{3}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\) |
\(\Big \downarrow \) 6436 |
\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^2-2 a \int \frac {x \text {arctanh}(a x)}{1-a^2 x^2}dx\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}-\frac {x}{3}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\) |
\(\Big \downarrow \) 6546 |
\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^2-2 a \left (\frac {\int \frac {\text {arctanh}(a x)}{1-a x}dx}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}-\frac {x}{3}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\) |
\(\Big \downarrow \) 6470 |
\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^2-2 a \left (\frac {\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}-\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}-\frac {x}{3}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\) |
\(\Big \downarrow \) 2849 |
\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^2-2 a \left (\frac {\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-\frac {2}{1-a x}}d\frac {1}{1-a x}}{a}+\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}-\frac {x}{3}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\) |
\(\Big \downarrow \) 2752 |
\(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^2-2 a \left (\frac {\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{3 a}-\frac {x}{3}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}-x\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}{21 a}+\frac {1}{21} \left (-\frac {1}{5} a^4 x^5+\frac {2 a^2 x^3}{3}-x\right )\) |
(-x + (2*a^2*x^3)/3 - (a^4*x^5)/5)/21 + ((1 - a^2*x^2)^3*ArcTanh[a*x])/(21 *a) + (x*(1 - a^2*x^2)^3*ArcTanh[a*x]^2)/7 + (6*((-x + (a^2*x^3)/3)/10 + ( (1 - a^2*x^2)^2*ArcTanh[a*x])/(10*a) + (x*(1 - a^2*x^2)^2*ArcTanh[a*x]^2)/ 5 + (4*(-1/3*x + ((1 - a^2*x^2)*ArcTanh[a*x])/(3*a) + (x*(1 - a^2*x^2)*Arc Tanh[a*x]^2)/3 + (2*(x*ArcTanh[a*x]^2 - 2*a*(-1/2*ArcTanh[a*x]^2/a^2 + ((A rcTanh[a*x]*Log[2/(1 - a*x)])/a + PolyLog[2, 1 - 2/(1 - a*x)]/(2*a))/a)))/ 3))/5))/7
3.3.25.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 )^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 0]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTanh[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTanh[c*x^n]) ^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol ] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c *(p/e) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^2*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2 , 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_.), x _Symbol] :> Simp[b*p*(d + e*x^2)^q*((a + b*ArcTanh[c*x])^(p - 1)/(2*c*q*(2* q + 1))), x] + (Simp[x*(d + e*x^2)^q*((a + b*ArcTanh[c*x])^p/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] - Simp[b^2*d*p*((p - 1)/(2*q*(2*q + 1))) Int[(d + e*x^2)^(q - 1)* (a + b*ArcTanh[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c ^2*d + e, 0] && GtQ[q, 0] && GtQ[p, 1]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ (c*d) Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Time = 0.48 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} a^{5} x^{5}}{5}-\operatorname {arctanh}\left (a x \right )^{2} a^{3} x^{3}+\operatorname {arctanh}\left (a x \right )^{2} a x -\frac {\operatorname {arctanh}\left (a x \right ) a^{6} x^{6}}{21}+\frac {8 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )}{35}-\frac {19 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{35}+\frac {16 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{35}+\frac {16 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{35}-\frac {16 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {8 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{35}+\frac {4 \ln \left (a x -1\right )^{2}}{35}-\frac {4 \ln \left (a x +1\right )^{2}}{35}+\frac {8 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {a^{5} x^{5}}{105}+\frac {19 a^{3} x^{3}}{315}-\frac {38 a x}{105}-\frac {19 \ln \left (a x -1\right )}{105}+\frac {19 \ln \left (a x +1\right )}{105}}{a}\) | \(222\) |
default | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} a^{5} x^{5}}{5}-\operatorname {arctanh}\left (a x \right )^{2} a^{3} x^{3}+\operatorname {arctanh}\left (a x \right )^{2} a x -\frac {\operatorname {arctanh}\left (a x \right ) a^{6} x^{6}}{21}+\frac {8 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )}{35}-\frac {19 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{35}+\frac {16 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{35}+\frac {16 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{35}-\frac {16 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {8 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{35}+\frac {4 \ln \left (a x -1\right )^{2}}{35}-\frac {4 \ln \left (a x +1\right )^{2}}{35}+\frac {8 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{35}-\frac {a^{5} x^{5}}{105}+\frac {19 a^{3} x^{3}}{315}-\frac {38 a x}{105}-\frac {19 \ln \left (a x -1\right )}{105}+\frac {19 \ln \left (a x +1\right )}{105}}{a}\) | \(222\) |
parts | \(-\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{6} x^{7}}{7}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} a^{4} x^{5}}{5}-\operatorname {arctanh}\left (a x \right )^{2} a^{2} x^{3}+x \operatorname {arctanh}\left (a x \right )^{2}-\frac {a^{5} \operatorname {arctanh}\left (a x \right ) x^{6}}{21}+\frac {8 a^{3} \operatorname {arctanh}\left (a x \right ) x^{4}}{35}-\frac {19 a \,\operatorname {arctanh}\left (a x \right ) x^{2}}{35}+\frac {16 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{35 a}+\frac {16 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{35 a}+\frac {-48 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )-24 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )+12 \ln \left (a x -1\right )^{2}-12 \ln \left (a x +1\right )^{2}+24 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )-a^{5} x^{5}+\frac {19 a^{3} x^{3}}{3}-38 a x -19 \ln \left (a x -1\right )+19 \ln \left (a x +1\right )}{105 a}\) | \(227\) |
risch | \(-\frac {38 x}{105}-\frac {20469}{42875 a}+\frac {a^{2} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{3}}{2}-\frac {3 a^{4} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{5}}{10}-\frac {\left (-1+\ln \left (a x +1\right )\right ) \left (a x +1\right ) \ln \left (-a x +1\right )}{2 a}-\frac {4 \ln \left (-a x +1\right )^{2}}{35 a}+\frac {\ln \left (-a x +1\right )^{2} x}{4}+\frac {4 \ln \left (a x +1\right )^{2}}{35 a}+\frac {\ln \left (a x +1\right )^{2} x}{4}-\frac {16 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{35 a}-\frac {\ln \left (a x +1\right ) x}{2}-\frac {2453 \ln \left (-a x +1\right )}{7350 a}+\frac {19 a^{2} x^{3}}{315}-\frac {a^{4} x^{5}}{105}-\frac {x \ln \left (-a x +1\right )}{2}-\frac {1276 \ln \left (a x -1\right )}{3675 a}-\frac {67 \ln \left (a x +1\right )}{210 a}+\frac {3 a^{4} \ln \left (-a x +1\right )^{2} x^{5}}{20}-\frac {a^{2} \ln \left (-a x +1\right )^{2} x^{3}}{4}+\frac {3 a^{4} \ln \left (a x +1\right )^{2} x^{5}}{20}-\frac {a^{2} \ln \left (a x +1\right )^{2} x^{3}}{4}+\frac {19 \ln \left (-a x +1\right ) \ln \left (a x +1\right )}{70 a}-\frac {19 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{35 a}+\frac {19 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{35 a}+\frac {\left (a x +1\right ) \ln \left (a x +1\right )}{2 a}+\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 )}{a}+\frac {4 a^{3} \ln \left (a x +1\right ) x^{4}}{35}-\frac {19 a \ln \left (a x +1\right ) x^{2}}{70}-\frac {4 a^{3} \ln \left (-a x +1\right ) x^{4}}{35}+\frac {19 a \ln \left (-a x +1\right ) x^{2}}{70}+\frac {a^{6} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{7}}{14}-\frac {a^{5} \ln \left (a x +1\right ) x^{6}}{42}+\frac {a^{5} \ln \left (-a x +1\right ) x^{6}}{42}-\frac {a^{6} \ln \left (a x +1\right )^{2} x^{7}}{28}-\frac {a^{6} \ln \left (-a x +1\right )^{2} x^{7}}{28}\) | \(509\) |
1/a*(-1/7*arctanh(a*x)^2*a^7*x^7+3/5*arctanh(a*x)^2*a^5*x^5-arctanh(a*x)^2 *a^3*x^3+arctanh(a*x)^2*a*x-1/21*arctanh(a*x)*a^6*x^6+8/35*a^4*x^4*arctanh (a*x)-19/35*a^2*x^2*arctanh(a*x)+16/35*arctanh(a*x)*ln(a*x-1)+16/35*arctan h(a*x)*ln(a*x+1)-16/35*dilog(1/2*a*x+1/2)-8/35*ln(a*x-1)*ln(1/2*a*x+1/2)+4 /35*ln(a*x-1)^2-4/35*ln(a*x+1)^2+8/35*(ln(a*x+1)-ln(1/2*a*x+1/2))*ln(-1/2* a*x+1/2)-1/105*a^5*x^5+19/315*a^3*x^3-38/105*a*x-19/105*ln(a*x-1)+19/105*l n(a*x+1))
\[ \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2 \, dx=\int { -{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{2} \,d x } \]
\[ \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2 \, dx=- \int 3 a^{2} x^{2} \operatorname {atanh}^{2}{\left (a x \right )}\, dx - \int \left (- 3 a^{4} x^{4} \operatorname {atanh}^{2}{\left (a x \right )}\right )\, dx - \int a^{6} x^{6} \operatorname {atanh}^{2}{\left (a x \right )}\, dx - \int \left (- \operatorname {atanh}^{2}{\left (a x \right )}\right )\, dx \]
-Integral(3*a**2*x**2*atanh(a*x)**2, x) - Integral(-3*a**4*x**4*atanh(a*x) **2, x) - Integral(a**6*x**6*atanh(a*x)**2, x) - Integral(-atanh(a*x)**2, x)
Time = 0.20 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.88 \[ \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2 \, dx=-\frac {1}{315} \, a^{2} {\left (\frac {3 \, a^{5} x^{5} - 19 \, a^{3} x^{3} + 114 \, a x + 36 \, \log \left (a x + 1\right )^{2} - 72 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 36 \, \log \left (a x - 1\right )^{2} + 57 \, \log \left (a x - 1\right )}{a^{3}} + \frac {144 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{3}} - \frac {57 \, \log \left (a x + 1\right )}{a^{3}}\right )} - \frac {1}{105} \, {\left (5 \, a^{4} x^{6} - 24 \, a^{2} x^{4} + 57 \, x^{2} - \frac {48 \, \log \left (a x + 1\right )}{a^{2}} - \frac {48 \, \log \left (a x - 1\right )}{a^{2}}\right )} a \operatorname {artanh}\left (a x\right ) - \frac {1}{35} \, {\left (5 \, a^{6} x^{7} - 21 \, a^{4} x^{5} + 35 \, a^{2} x^{3} - 35 \, x\right )} \operatorname {artanh}\left (a x\right )^{2} \]
-1/315*a^2*((3*a^5*x^5 - 19*a^3*x^3 + 114*a*x + 36*log(a*x + 1)^2 - 72*log (a*x + 1)*log(a*x - 1) - 36*log(a*x - 1)^2 + 57*log(a*x - 1))/a^3 + 144*(l og(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a^3 - 57*log(a*x + 1)/a^3) - 1/105*(5*a^4*x^6 - 24*a^2*x^4 + 57*x^2 - 48*log(a*x + 1)/a^2 - 48*log(a*x - 1)/a^2)*a*arctanh(a*x) - 1/35*(5*a^6*x^7 - 21*a^4*x^5 + 35*a^ 2*x^3 - 35*x)*arctanh(a*x)^2
\[ \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2 \, dx=\int { -{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{2} \,d x } \]
Timed out. \[ \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2 \, dx=-\int {\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^3 \,d x \]