Integrand size = 22, antiderivative size = 215 \[ \int \frac {x^2 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {3}{128 a^3 \left (1-a^2 x^2\right )^2}+\frac {3}{128 a^3 \left (1-a^2 x^2\right )}+\frac {3 x \text {arctanh}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 x \text {arctanh}(a x)}{64 a^2 \left (1-a^2 x^2\right )}-\frac {3 \text {arctanh}(a x)^2}{128 a^3}-\frac {3 \text {arctanh}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )^2}+\frac {3 \text {arctanh}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )}+\frac {x \text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \text {arctanh}(a x)^3}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\text {arctanh}(a x)^4}{32 a^3} \]
-3/128/a^3/(-a^2*x^2+1)^2+3/128/a^3/(-a^2*x^2+1)+3/32*x*arctanh(a*x)/a^2/( -a^2*x^2+1)^2-3/64*x*arctanh(a*x)/a^2/(-a^2*x^2+1)-3/128*arctanh(a*x)^2/a^ 3-3/16*arctanh(a*x)^2/a^3/(-a^2*x^2+1)^2+3/16*arctanh(a*x)^2/a^3/(-a^2*x^2 +1)+1/4*x*arctanh(a*x)^3/a^2/(-a^2*x^2+1)^2-1/8*x*arctanh(a*x)^3/a^2/(-a^2 *x^2+1)-1/32*arctanh(a*x)^4/a^3
Time = 0.14 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.50 \[ \int \frac {x^2 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=\frac {-3 a^2 x^2+6 \left (a x+a^3 x^3\right ) \text {arctanh}(a x)-3 \left (1+6 a^2 x^2+a^4 x^4\right ) \text {arctanh}(a x)^2+16 \left (a x+a^3 x^3\right ) \text {arctanh}(a x)^3-4 \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^4}{128 a^3 \left (-1+a^2 x^2\right )^2} \]
(-3*a^2*x^2 + 6*(a*x + a^3*x^3)*ArcTanh[a*x] - 3*(1 + 6*a^2*x^2 + a^4*x^4) *ArcTanh[a*x]^2 + 16*(a*x + a^3*x^3)*ArcTanh[a*x]^3 - 4*(-1 + a^2*x^2)^2*A rcTanh[a*x]^4)/(128*a^3*(-1 + a^2*x^2)^2)
Time = 1.77 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.95, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {6590, 6518, 6526, 6518, 6522, 6518, 241, 6556, 6518, 241}
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.
| \(\displaystyle \int \frac {x^2 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6590 |
| \(\displaystyle \frac {\int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3}dx}{a^2}-\frac {\int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2}dx}{a^2}\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle \frac {\int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3}dx}{a^2}-\frac {-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}}{a^2}\) |
| \(\Big \downarrow \) 6526 |
| \(\displaystyle \frac {\frac {3}{8} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3}dx+\frac {3}{4} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}}{a^2}-\frac {-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}}{a^2}\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle \frac {\frac {3}{8} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3}dx+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}}{a^2}-\frac {-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}}{a^2}\) |
| \(\Big \downarrow \) 6522 |
| \(\displaystyle \frac {\frac {3}{8} \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}}{a^2}-\frac {-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}}{a^2}\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle \frac {\frac {3}{8} \left (\frac {3}{4} \left (-\frac {1}{2} a \int \frac {x}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )+\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}}{a^2}-\frac {-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}}{a^2}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )}{a^2}-\frac {-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}}{a^2}\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle \frac {\frac {3}{4} \left (-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx}{a}\right )+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )}{a^2}-\frac {-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx}{a}\right )+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}}{a^2}\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle \frac {\frac {3}{4} \left (-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {-\frac {1}{2} a \int \frac {x}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )}{a^2}-\frac {-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {-\frac {1}{2} a \int \frac {x}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}}{a^2}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )}{a^2}-\frac {\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}}{a^2}\) |
-(((x*ArcTanh[a*x]^3)/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^4/(8*a) - (3*a*(Arc Tanh[a*x]^2/(2*a^2*(1 - a^2*x^2)) - (-1/4*1/(a*(1 - a^2*x^2)) + (x*ArcTanh [a*x])/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^2/(4*a))/a))/2)/a^2) + ((-3*ArcTan h[a*x]^2)/(16*a*(1 - a^2*x^2)^2) + (x*ArcTanh[a*x]^3)/(4*(1 - a^2*x^2)^2) + (3*(-1/16*1/(a*(1 - a^2*x^2)^2) + (x*ArcTanh[a*x])/(4*(1 - a^2*x^2)^2) + (3*(-1/4*1/(a*(1 - a^2*x^2)) + (x*ArcTanh[a*x])/(2*(1 - a^2*x^2)) + ArcTa nh[a*x]^2/(4*a)))/4))/8 + (3*((x*ArcTanh[a*x]^3)/(2*(1 - a^2*x^2)) + ArcTa nh[a*x]^4/(8*a) - (3*a*(ArcTanh[a*x]^2/(2*a^2*(1 - a^2*x^2)) - (-1/4*1/(a* (1 - a^2*x^2)) + (x*ArcTanh[a*x])/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^2/(4*a) )/a))/2))/4)/a^2
3.4.15.3.1 Defintions of rubi rules used
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2)^2, x_Sy mbol] :> Simp[x*((a + b*ArcTanh[c*x])^p/(2*d*(d + e*x^2))), x] + (Simp[(a + b*ArcTanh[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x] - Simp[b*c*(p/2) Int[x*( (a + b*ArcTanh[c*x])^(p - 1)/(d + e*x^2)^2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbo l] :> Simp[(-b)*((d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])/(2*d*(q + 1))), x] + Simp[(2*q + 3)/( 2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x]) /; Fre eQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_ Symbol] :> Simp[(-b)*p*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^(p - 1)/(4 *c*d*(q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p /(2*d*(q + 1))), x] + Simp[(2*q + 3)/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1 )*(a + b*ArcTanh[c*x])^p, x], x] + Simp[b^2*p*((p - 1)/(4*(q + 1)^2)) Int [(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q _.), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p/(2*e*(q + 1))), x] + Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTanh[c* x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && NeQ[q, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[1/e Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*A rcTanh[c*x])^p, x], x] - Simp[d/e Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcT anh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && In tegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]
Time = 0.53 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.66
| method | result | size |
| parallelrisch | \(-\frac {4 \operatorname {arctanh}\left (a x \right )^{4} a^{4} x^{4}+3 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}-16 \operatorname {arctanh}\left (a x \right )^{3} a^{3} x^{3}-8 \operatorname {arctanh}\left (a x \right )^{4} a^{2} x^{2}-6 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )+18 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}-16 \operatorname {arctanh}\left (a x \right )^{3} a x +3 a^{2} x^{2}+4 \operatorname {arctanh}\left (a x \right )^{4}-6 a x \,\operatorname {arctanh}\left (a x \right )+3 \operatorname {arctanh}\left (a x \right )^{2}}{128 \left (a^{2} x^{2}-1\right )^{2} a^{3}}\) | \(142\) |
| risch | \(-\frac {\ln \left (a x +1\right )^{4}}{512 a^{3}}+\frac {\left (a^{4} x^{4} \ln \left (-a x +1\right )+2 a^{3} x^{3}-2 x^{2} \ln \left (-a x +1\right ) a^{2}+2 a x +\ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )^{3}}{128 a^{3} \left (a^{2} x^{2}-1\right )^{2}}-\frac {3 \left (2 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+a^{4} x^{4}+8 a^{3} x^{3} \ln \left (-a x +1\right )-4 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+6 a^{2} x^{2}+8 a x \ln \left (-a x +1\right )+2 \ln \left (-a x +1\right )^{2}+1\right ) \ln \left (a x +1\right )^{2}}{512 a^{3} \left (a x -1\right ) \left (a x +1\right ) \left (a^{2} x^{2}-1\right )}+\frac {\left (2 a^{4} x^{4} \ln \left (-a x +1\right )^{3}+3 a^{4} x^{4} \ln \left (-a x +1\right )+12 a^{3} x^{3} \ln \left (-a x +1\right )^{2}-4 a^{2} x^{2} \ln \left (-a x +1\right )^{3}+6 a^{3} x^{3}+18 x^{2} \ln \left (-a x +1\right ) a^{2}+12 a \ln \left (-a x +1\right )^{2} x +2 \ln \left (-a x +1\right )^{3}+6 a x +3 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{256 a^{3} \left (a x -1\right ) \left (a x +1\right ) \left (a^{2} x^{2}-1\right )}-\frac {a^{4} x^{4} \ln \left (-a x +1\right )^{4}+3 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+8 a^{3} x^{3} \ln \left (-a x +1\right )^{3}-2 a^{2} x^{2} \ln \left (-a x +1\right )^{4}+12 a^{3} x^{3} \ln \left (-a x +1\right )+18 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+8 a x \ln \left (-a x +1\right )^{3}+12 a^{2} x^{2}+\ln \left (-a x +1\right )^{4}+12 a x \ln \left (-a x +1\right )+3 \ln \left (-a x +1\right )^{2}}{512 a^{3} \left (a x -1\right ) \left (a x +1\right ) \left (a^{2} x^{2}-1\right )}\) | \(559\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(809\) |
| default | \(\text {Expression too large to display}\) | \(809\) |
| parts | \(\text {Expression too large to display}\) | \(881\) |
-1/128*(4*arctanh(a*x)^4*a^4*x^4+3*a^4*x^4*arctanh(a*x)^2-16*arctanh(a*x)^ 3*a^3*x^3-8*arctanh(a*x)^4*a^2*x^2-6*a^3*x^3*arctanh(a*x)+18*a^2*x^2*arcta nh(a*x)^2-16*arctanh(a*x)^3*a*x+3*a^2*x^2+4*arctanh(a*x)^4-6*a*x*arctanh(a *x)+3*arctanh(a*x)^2)/(a^2*x^2-1)^2/a^3
Time = 0.25 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.75 \[ \int \frac {x^2 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{4} + 12 \, a^{2} x^{2} - 8 \, {\left (a^{3} x^{3} + a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 3 \, {\left (a^{4} x^{4} + 6 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 12 \, {\left (a^{3} x^{3} + a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{512 \, {\left (a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}\right )}} \]
-1/512*((a^4*x^4 - 2*a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1))^4 + 12*a^2*x^2 - 8*(a^3*x^3 + a*x)*log(-(a*x + 1)/(a*x - 1))^3 + 3*(a^4*x^4 + 6*a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1))^2 - 12*(a^3*x^3 + a*x)*log(-(a*x + 1)/(a*x - 1)))/(a^7*x^4 - 2*a^5*x^2 + a^3)
\[ \int \frac {x^2 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=- \int \frac {x^{2} \operatorname {atanh}^{3}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 657 vs. \(2 (187) = 374\).
Time = 0.20 (sec) , antiderivative size = 657, normalized size of antiderivative = 3.06 \[ \int \frac {x^2 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=\frac {1}{16} \, {\left (\frac {2 \, {\left (a^{2} x^{3} + x\right )}}{a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}} - \frac {\log \left (a x + 1\right )}{a^{3}} + \frac {\log \left (a x - 1\right )}{a^{3}}\right )} \operatorname {artanh}\left (a x\right )^{3} - \frac {3 \, {\left (4 \, a^{2} x^{2} - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2}\right )} a \operatorname {artanh}\left (a x\right )^{2}}{64 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} + \frac {1}{512} \, {\left (\frac {{\left ({\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{4} - 4 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{3} \log \left (a x - 1\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{4} - 12 \, a^{2} x^{2} + 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 2 \, {\left (2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{3} + 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right )\right )} a^{2}}{a^{10} x^{4} - 2 \, a^{8} x^{2} + a^{6}} + \frac {4 \, {\left (6 \, a^{3} x^{3} - 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{3} + 6 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{3} + 6 \, a x - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} + 1\right )} \log \left (a x + 1\right ) + 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} a \operatorname {artanh}\left (a x\right )}{a^{9} x^{4} - 2 \, a^{7} x^{2} + a^{5}}\right )} a \]
1/16*(2*(a^2*x^3 + x)/(a^6*x^4 - 2*a^4*x^2 + a^2) - log(a*x + 1)/a^3 + log (a*x - 1)/a^3)*arctanh(a*x)^3 - 3/64*(4*a^2*x^2 - (a^4*x^4 - 2*a^2*x^2 + 1 )*log(a*x + 1)^2 + 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)*log(a*x - 1) - (a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2)*a*arctanh(a*x)^2/(a^8*x^4 - 2*a ^6*x^2 + a^4) + 1/512*(((a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^4 - 4*(a^4* x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^3*log(a*x - 1) + (a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^4 - 12*a^2*x^2 + 3*(a^4*x^4 - 2*a^2*x^2 + 2*(a^4*x^4 - 2*a ^2*x^2 + 1)*log(a*x - 1)^2 + 1)*log(a*x + 1)^2 + 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 2*(2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^3 + 3*(a^4 *x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))*log(a*x + 1))*a^2/(a^10*x^4 - 2*a^8*x^ 2 + a^6) + 4*(6*a^3*x^3 - 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^3 + 6*( a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2*log(a*x - 1) + 2*(a^4*x^4 - 2*a^2* x^2 + 1)*log(a*x - 1)^3 + 6*a*x - 3*(a^4*x^4 - 2*a^2*x^2 + 2*(a^4*x^4 - 2* a^2*x^2 + 1)*log(a*x - 1)^2 + 1)*log(a*x + 1) + 3*(a^4*x^4 - 2*a^2*x^2 + 1 )*log(a*x - 1))*a*arctanh(a*x)/(a^9*x^4 - 2*a^7*x^2 + a^5))*a
\[ \int \frac {x^2 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=\int { -\frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{3}} \,d x } \]
Time = 6.61 (sec) , antiderivative size = 831, normalized size of antiderivative = 3.87 \[ \int \frac {x^2 \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx=\frac {3\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{4\,\left (64\,a^7\,x^4-128\,a^5\,x^2+64\,a^3\right )}-\frac {3\,{\ln \left (1-a\,x\right )}^2}{512\,a^3}-\frac {{\ln \left (a\,x+1\right )}^4}{512\,a^3}-\frac {{\ln \left (1-a\,x\right )}^4}{512\,a^3}-\frac {3\,x^2}{2\,\left (64\,a^5\,x^4-128\,a^3\,x^2+64\,a\right )}-\frac {x\,{\ln \left (1-a\,x\right )}^3}{8\,\left (8\,a^6\,x^4-16\,a^4\,x^2+8\,a^2\right )}-\frac {6\,x^2\,{\ln \left (1-a\,x\right )}^2}{128\,a^5\,x^4-256\,a^3\,x^2+128\,a}-\frac {3\,{\ln \left (a\,x+1\right )}^2}{512\,a^3}+\frac {x^3\,{\ln \left (a\,x+1\right )}^3}{64\,\left (a^4\,x^4-2\,a^2\,x^2+1\right )}-\frac {x^3\,{\ln \left (1-a\,x\right )}^3}{8\,\left (8\,a^4\,x^4-16\,a^2\,x^2+8\right )}+\frac {3\,x\,\ln \left (a\,x+1\right )}{128\,\left (a^6\,x^4-2\,a^4\,x^2+a^2\right )}+\frac {\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^3}{128\,a^3}+\frac {{\ln \left (a\,x+1\right )}^3\,\ln \left (1-a\,x\right )}{128\,a^3}-\frac {3\,x\,\ln \left (1-a\,x\right )}{128\,a^6\,x^4-256\,a^4\,x^2+128\,a^2}-\frac {3\,x^2\,{\ln \left (a\,x+1\right )}^2}{64\,\left (a^5\,x^4-2\,a^3\,x^2+a\right )}+\frac {x\,{\ln \left (a\,x+1\right )}^3}{64\,\left (a^6\,x^4-2\,a^4\,x^2+a^2\right )}-\frac {3\,{\ln \left (a\,x+1\right )}^2\,{\ln \left (1-a\,x\right )}^2}{256\,a^3}+\frac {3\,x^3\,\ln \left (a\,x+1\right )}{128\,\left (a^4\,x^4-2\,a^2\,x^2+1\right )}-\frac {3\,a\,x^3\,\ln \left (1-a\,x\right )}{128\,a^5\,x^4-256\,a^3\,x^2+128\,a}+\frac {6\,x\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2}{128\,a^6\,x^4-256\,a^4\,x^2+128\,a^2}-\frac {6\,x\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )}{128\,a^6\,x^4-256\,a^4\,x^2+128\,a^2}+\frac {6\,x^2\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{64\,a^5\,x^4-128\,a^3\,x^2+64\,a}+\frac {6\,a^2\,x^3\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2}{128\,a^6\,x^4-256\,a^4\,x^2+128\,a^2}-\frac {6\,a^2\,x^3\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )}{128\,a^6\,x^4-256\,a^4\,x^2+128\,a^2}-\frac {3\,a^2\,x^2\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{2\,\left (64\,a^7\,x^4-128\,a^5\,x^2+64\,a^3\right )}+\frac {3\,a^4\,x^4\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{4\,\left (64\,a^7\,x^4-128\,a^5\,x^2+64\,a^3\right )} \]
(3*log(a*x + 1)*log(1 - a*x))/(4*(64*a^3 - 128*a^5*x^2 + 64*a^7*x^4)) - (3 *log(1 - a*x)^2)/(512*a^3) - log(a*x + 1)^4/(512*a^3) - log(1 - a*x)^4/(51 2*a^3) - (3*x^2)/(2*(64*a - 128*a^3*x^2 + 64*a^5*x^4)) - (x*log(1 - a*x)^3 )/(8*(8*a^2 - 16*a^4*x^2 + 8*a^6*x^4)) - (6*x^2*log(1 - a*x)^2)/(128*a - 2 56*a^3*x^2 + 128*a^5*x^4) - (3*log(a*x + 1)^2)/(512*a^3) + (x^3*log(a*x + 1)^3)/(64*(a^4*x^4 - 2*a^2*x^2 + 1)) - (x^3*log(1 - a*x)^3)/(8*(8*a^4*x^4 - 16*a^2*x^2 + 8)) + (3*x*log(a*x + 1))/(128*(a^2 - 2*a^4*x^2 + a^6*x^4)) + (log(a*x + 1)*log(1 - a*x)^3)/(128*a^3) + (log(a*x + 1)^3*log(1 - a*x))/ (128*a^3) - (3*x*log(1 - a*x))/(128*a^2 - 256*a^4*x^2 + 128*a^6*x^4) - (3* x^2*log(a*x + 1)^2)/(64*(a - 2*a^3*x^2 + a^5*x^4)) + (x*log(a*x + 1)^3)/(6 4*(a^2 - 2*a^4*x^2 + a^6*x^4)) - (3*log(a*x + 1)^2*log(1 - a*x)^2)/(256*a^ 3) + (3*x^3*log(a*x + 1))/(128*(a^4*x^4 - 2*a^2*x^2 + 1)) - (3*a*x^3*log(1 - a*x))/(128*a - 256*a^3*x^2 + 128*a^5*x^4) + (6*x*log(a*x + 1)*log(1 - a *x)^2)/(128*a^2 - 256*a^4*x^2 + 128*a^6*x^4) - (6*x*log(a*x + 1)^2*log(1 - a*x))/(128*a^2 - 256*a^4*x^2 + 128*a^6*x^4) + (6*x^2*log(a*x + 1)*log(1 - a*x))/(64*a - 128*a^3*x^2 + 64*a^5*x^4) + (6*a^2*x^3*log(a*x + 1)*log(1 - a*x)^2)/(128*a^2 - 256*a^4*x^2 + 128*a^6*x^4) - (6*a^2*x^3*log(a*x + 1)^2 *log(1 - a*x))/(128*a^2 - 256*a^4*x^2 + 128*a^6*x^4) - (3*a^2*x^2*log(a*x + 1)*log(1 - a*x))/(2*(64*a^3 - 128*a^5*x^2 + 64*a^7*x^4)) + (3*a^4*x^4*lo g(a*x + 1)*log(1 - a*x))/(4*(64*a^3 - 128*a^5*x^2 + 64*a^7*x^4))