Integrand size = 19, antiderivative size = 214 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^4} \, dx=\frac {x}{108 \left (1-a^2 x^2\right )^3}+\frac {65 x}{1728 \left (1-a^2 x^2\right )^2}+\frac {245 x}{1152 \left (1-a^2 x^2\right )}+\frac {245 \text {arctanh}(a x)}{1152 a}-\frac {\text {arctanh}(a x)}{18 a \left (1-a^2 x^2\right )^3}-\frac {5 \text {arctanh}(a x)}{48 a \left (1-a^2 x^2\right )^2}-\frac {5 \text {arctanh}(a x)}{16 a \left (1-a^2 x^2\right )}+\frac {x \text {arctanh}(a x)^2}{6 \left (1-a^2 x^2\right )^3}+\frac {5 x \text {arctanh}(a x)^2}{24 \left (1-a^2 x^2\right )^2}+\frac {5 x \text {arctanh}(a x)^2}{16 \left (1-a^2 x^2\right )}+\frac {5 \text {arctanh}(a x)^3}{48 a} \] Output:
1/108*x/(-a^2*x^2+1)^3+65/1728*x/(-a^2*x^2+1)^2+245*x/(-1152*a^2*x^2+1152) +245/1152*arctanh(a*x)/a-1/18*arctanh(a*x)/a/(-a^2*x^2+1)^3-5/48*arctanh(a *x)/a/(-a^2*x^2+1)^2-5/16*arctanh(a*x)/a/(-a^2*x^2+1)+1/6*x*arctanh(a*x)^2 /(-a^2*x^2+1)^3+5/24*x*arctanh(a*x)^2/(-a^2*x^2+1)^2+5*x*arctanh(a*x)^2/(- 16*a^2*x^2+16)+5/48*arctanh(a*x)^3/a
Time = 0.22 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.73 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^4} \, dx=\frac {-\frac {64 x}{\left (-1+a^2 x^2\right )^3}+\frac {260 x}{\left (-1+a^2 x^2\right )^2}-\frac {1470 x}{-1+a^2 x^2}+\frac {48 \left (68-105 a^2 x^2+45 a^4 x^4\right ) \text {arctanh}(a x)}{a \left (-1+a^2 x^2\right )^3}-\frac {144 x \left (33-40 a^2 x^2+15 a^4 x^4\right ) \text {arctanh}(a x)^2}{\left (-1+a^2 x^2\right )^3}+\frac {720 \text {arctanh}(a x)^3}{a}-\frac {735 \log (1-a x)}{a}+\frac {735 \log (1+a x)}{a}}{6912} \] Input:
Integrate[ArcTanh[a*x]^2/(1 - a^2*x^2)^4,x]
Output:
((-64*x)/(-1 + a^2*x^2)^3 + (260*x)/(-1 + a^2*x^2)^2 - (1470*x)/(-1 + a^2* x^2) + (48*(68 - 105*a^2*x^2 + 45*a^4*x^4)*ArcTanh[a*x])/(a*(-1 + a^2*x^2) ^3) - (144*x*(33 - 40*a^2*x^2 + 15*a^4*x^4)*ArcTanh[a*x]^2)/(-1 + a^2*x^2) ^3 + (720*ArcTanh[a*x]^3)/a - (735*Log[1 - a*x])/a + (735*Log[1 + a*x])/a) /6912
Time = 0.94 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.56, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {6526, 215, 215, 215, 219, 6526, 215, 215, 219, 6518, 6556, 215, 219}
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.
| \(\displaystyle \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 6526 |
| \(\displaystyle \frac {5}{6} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3}dx+\frac {1}{18} \int \frac {1}{\left (1-a^2 x^2\right )^4}dx+\frac {x \text {arctanh}(a x)^2}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)}{18 a \left (1-a^2 x^2\right )^3}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {5}{6} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3}dx+\frac {1}{18} \left (\frac {5}{6} \int \frac {1}{\left (1-a^2 x^2\right )^3}dx+\frac {x}{6 \left (1-a^2 x^2\right )^3}\right )+\frac {x \text {arctanh}(a x)^2}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)}{18 a \left (1-a^2 x^2\right )^3}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {5}{6} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3}dx+\frac {1}{18} \left (\frac {5}{6} \left (\frac {3}{4} \int \frac {1}{\left (1-a^2 x^2\right )^2}dx+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x}{6 \left (1-a^2 x^2\right )^3}\right )+\frac {x \text {arctanh}(a x)^2}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)}{18 a \left (1-a^2 x^2\right )^3}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {5}{6} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3}dx+\frac {1}{18} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x}{6 \left (1-a^2 x^2\right )^3}\right )+\frac {x \text {arctanh}(a x)^2}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)}{18 a \left (1-a^2 x^2\right )^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{6} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3}dx+\frac {x \text {arctanh}(a x)^2}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)}{18 a \left (1-a^2 x^2\right )^3}+\frac {1}{18} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x}{6 \left (1-a^2 x^2\right )^3}\right )\) |
| \(\Big \downarrow \) 6526 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {1}{8} \int \frac {1}{\left (1-a^2 x^2\right )^3}dx+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)^2}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)}{18 a \left (1-a^2 x^2\right )^3}+\frac {1}{18} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x}{6 \left (1-a^2 x^2\right )^3}\right )\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {1}{8} \left (\frac {3}{4} \int \frac {1}{\left (1-a^2 x^2\right )^2}dx+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)^2}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)}{18 a \left (1-a^2 x^2\right )^3}+\frac {1}{18} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x}{6 \left (1-a^2 x^2\right )^3}\right )\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {1}{8} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)^2}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)}{18 a \left (1-a^2 x^2\right )^3}+\frac {1}{18} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x}{6 \left (1-a^2 x^2\right )^3}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )\right )+\frac {x \text {arctanh}(a x)^2}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)}{18 a \left (1-a^2 x^2\right )^3}+\frac {1}{18} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x}{6 \left (1-a^2 x^2\right )^3}\right )\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )\right )+\frac {x \text {arctanh}(a x)^2}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)}{18 a \left (1-a^2 x^2\right )^3}+\frac {1}{18} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x}{6 \left (1-a^2 x^2\right )^3}\right )\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2}dx}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )\right )+\frac {x \text {arctanh}(a x)^2}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)}{18 a \left (1-a^2 x^2\right )^3}+\frac {1}{18} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x}{6 \left (1-a^2 x^2\right )^3}\right )\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )\right )+\frac {x \text {arctanh}(a x)^2}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)}{18 a \left (1-a^2 x^2\right )^3}+\frac {1}{18} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x}{6 \left (1-a^2 x^2\right )^3}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {x \text {arctanh}(a x)^2}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)}{18 a \left (1-a^2 x^2\right )^3}+\frac {1}{18} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x}{6 \left (1-a^2 x^2\right )^3}\right )+\frac {5}{6} \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )\) |
Input:
Int[ArcTanh[a*x]^2/(1 - a^2*x^2)^4,x]
Output:
-1/18*ArcTanh[a*x]/(a*(1 - a^2*x^2)^3) + (x*ArcTanh[a*x]^2)/(6*(1 - a^2*x^ 2)^3) + (x/(6*(1 - a^2*x^2)^3) + (5*(x/(4*(1 - a^2*x^2)^2) + (3*(x/(2*(1 - a^2*x^2)) + ArcTanh[a*x]/(2*a)))/4))/6)/18 + (5*(-1/8*ArcTanh[a*x]/(a*(1 - a^2*x^2)^2) + (x*ArcTanh[a*x]^2)/(4*(1 - a^2*x^2)^2) + (x/(4*(1 - a^2*x^ 2)^2) + (3*(x/(2*(1 - a^2*x^2)) + ArcTanh[a*x]/(2*a)))/4)/8 + (3*((x*ArcTa nh[a*x]^2)/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^3/(6*a) - a*(ArcTanh[a*x]/(2*a ^2*(1 - a^2*x^2)) - (x/(2*(1 - a^2*x^2)) + ArcTanh[a*x]/(2*a))/(2*a))))/4) )/6
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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_.))^(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]
Time = 14.47 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.79
| method | result | size |
| parallelrisch | \(-\frac {2376 \operatorname {arctanh}\left (a x \right )^{2} a x -2880 \operatorname {arctanh}\left (a x \right )^{2} a^{3} x^{3}+1080 \operatorname {arctanh}\left (a x \right )^{2} a^{5} x^{5}-1080 \operatorname {arctanh}\left (a x \right )^{3} a^{2} x^{2}-735 \,\operatorname {arctanh}\left (a x \right ) a^{6} x^{6}+897 a x +735 a^{5} x^{5}-1600 a^{3} x^{3}+360 \operatorname {arctanh}\left (a x \right )^{3}+1125 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )+315 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )-897 \,\operatorname {arctanh}\left (a x \right )+1080 \operatorname {arctanh}\left (a x \right )^{3} a^{4} x^{4}-360 x^{6} \operatorname {arctanh}\left (a x \right )^{3} a^{6}}{3456 \left (a^{2} x^{2}-1\right )^{3} a}\) | \(168\) |
| risch | \(\frac {5 \ln \left (a x +1\right )^{3}}{384 a}-\frac {\left (15 a^{6} x^{6} \ln \left (-a x +1\right )+30 a^{5} x^{5}-45 x^{4} \ln \left (-a x +1\right ) a^{4}-80 a^{3} x^{3}+45 x^{2} \ln \left (-a x +1\right ) a^{2}+66 a x -15 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )^{2}}{384 \left (a^{2} x^{2}-1\right )^{3} a}+\frac {\left (45 a^{6} x^{6} \ln \left (-a x +1\right )^{2}+180 x^{5} \ln \left (-a x +1\right ) a^{5}-135 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+180 a^{4} x^{4}-480 a^{3} x^{3} \ln \left (-a x +1\right )+135 a^{2} x^{2} \ln \left (-a x +1\right )^{2}-420 a^{2} x^{2}+396 a x \ln \left (-a x +1\right )-45 \ln \left (-a x +1\right )^{2}+272\right ) \ln \left (a x +1\right )}{1152 \left (a x +1\right ) a \left (a x -1\right ) \left (a^{2} x^{2}-1\right )^{2}}-\frac {90 a^{6} x^{6} \ln \left (-a x +1\right )^{3}+735 \ln \left (a x -1\right ) a^{6} x^{6}-735 \ln \left (-a x -1\right ) a^{6} x^{6}+540 a^{5} x^{5} \ln \left (-a x +1\right )^{2}-270 a^{4} x^{4} \ln \left (-a x +1\right )^{3}+1470 a^{5} x^{5}-2205 \ln \left (a x -1\right ) x^{4} a^{4}+2205 \ln \left (-a x -1\right ) a^{4} x^{4}+1080 x^{4} \ln \left (-a x +1\right ) a^{4}-1440 a^{3} x^{3} \ln \left (-a x +1\right )^{2}+270 a^{2} x^{2} \ln \left (-a x +1\right )^{3}-3200 a^{3} x^{3}+2205 \ln \left (a x -1\right ) a^{2} x^{2}-2205 \ln \left (-a x -1\right ) a^{2} x^{2}-2520 x^{2} \ln \left (-a x +1\right ) a^{2}+1188 a \ln \left (-a x +1\right )^{2} x -90 \ln \left (-a x +1\right )^{3}+1794 a x -735 \ln \left (a x -1\right )+735 \ln \left (-a x -1\right )+1632 \ln \left (-a x +1\right )}{6912 a \left (a x -1\right )^{2} \left (a^{2} x^{2}-1\right ) \left (a x +1\right )^{2}}\) | \(574\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(931\) |
| default | \(\text {Expression too large to display}\) | \(931\) |
| parts | \(\text {Expression too large to display}\) | \(1027\) |
Input:
int(arctanh(a*x)^2/(-a^2*x^2+1)^4,x,method=_RETURNVERBOSE)
Output:
-1/3456*(2376*arctanh(a*x)^2*a*x-2880*arctanh(a*x)^2*a^3*x^3+1080*arctanh( a*x)^2*a^5*x^5-1080*arctanh(a*x)^3*a^2*x^2-735*arctanh(a*x)*a^6*x^6+897*a* x+735*a^5*x^5-1600*a^3*x^3+360*arctanh(a*x)^3+1125*a^4*x^4*arctanh(a*x)+31 5*a^2*x^2*arctanh(a*x)-897*arctanh(a*x)+1080*arctanh(a*x)^3*a^4*x^4-360*x^ 6*arctanh(a*x)^3*a^6)/(a^2*x^2-1)^3/a
Time = 0.08 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.84 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^4} \, dx=-\frac {1470 \, a^{5} x^{5} - 3200 \, a^{3} x^{3} - 90 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 36 \, {\left (15 \, a^{5} x^{5} - 40 \, a^{3} x^{3} + 33 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 1794 \, a x - 3 \, {\left (245 \, a^{6} x^{6} - 375 \, a^{4} x^{4} - 105 \, a^{2} x^{2} + 299\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{6912 \, {\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )}} \] Input:
integrate(arctanh(a*x)^2/(-a^2*x^2+1)^4,x, algorithm="fricas")
Output:
-1/6912*(1470*a^5*x^5 - 3200*a^3*x^3 - 90*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(-(a*x + 1)/(a*x - 1))^3 + 36*(15*a^5*x^5 - 40*a^3*x^3 + 33*a*x)* log(-(a*x + 1)/(a*x - 1))^2 + 1794*a*x - 3*(245*a^6*x^6 - 375*a^4*x^4 - 10 5*a^2*x^2 + 299)*log(-(a*x + 1)/(a*x - 1)))/(a^7*x^6 - 3*a^5*x^4 + 3*a^3*x ^2 - a)
\[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^4} \, dx=\int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4}}\, dx \] Input:
integrate(atanh(a*x)**2/(-a**2*x**2+1)**4,x)
Output:
Integral(atanh(a*x)**2/((a*x - 1)**4*(a*x + 1)**4), x)
Leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (183) = 366\).
Time = 0.04 (sec) , antiderivative size = 516, normalized size of antiderivative = 2.41 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^4} \, dx=-\frac {1}{96} \, {\left (\frac {2 \, {\left (15 \, a^{4} x^{5} - 40 \, a^{2} x^{3} + 33 \, x\right )}}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1} - \frac {15 \, \log \left (a x + 1\right )}{a} + \frac {15 \, \log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right )^{2} - \frac {{\left (1470 \, a^{5} x^{5} - 3200 \, a^{3} x^{3} - 90 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} + 270 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) + 90 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} + 1794 \, a x - 15 \, {\left (49 \, a^{6} x^{6} - 147 \, a^{4} x^{4} + 147 \, a^{2} x^{2} + 18 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 49\right )} \log \left (a x + 1\right ) + 735 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} a^{2}}{6912 \, {\left (a^{9} x^{6} - 3 \, a^{7} x^{4} + 3 \, a^{5} x^{2} - a^{3}\right )}} + \frac {{\left (180 \, a^{4} x^{4} - 420 \, a^{2} x^{2} - 45 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} + 90 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 45 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} + 272\right )} a \operatorname {artanh}\left (a x\right )}{576 \, {\left (a^{8} x^{6} - 3 \, a^{6} x^{4} + 3 \, a^{4} x^{2} - a^{2}\right )}} \] Input:
integrate(arctanh(a*x)^2/(-a^2*x^2+1)^4,x, algorithm="maxima")
Output:
-1/96*(2*(15*a^4*x^5 - 40*a^2*x^3 + 33*x)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1) - 15*log(a*x + 1)/a + 15*log(a*x - 1)/a)*arctanh(a*x)^2 - 1/6912*(14 70*a^5*x^5 - 3200*a^3*x^3 - 90*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a *x + 1)^3 + 270*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1)^2*log(a *x - 1) + 90*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1)^3 + 1794*a *x - 15*(49*a^6*x^6 - 147*a^4*x^4 + 147*a^2*x^2 + 18*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1)^2 - 49)*log(a*x + 1) + 735*(a^6*x^6 - 3*a^4* x^4 + 3*a^2*x^2 - 1)*log(a*x - 1))*a^2/(a^9*x^6 - 3*a^7*x^4 + 3*a^5*x^2 - a^3) + 1/576*(180*a^4*x^4 - 420*a^2*x^2 - 45*(a^6*x^6 - 3*a^4*x^4 + 3*a^2* x^2 - 1)*log(a*x + 1)^2 + 90*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1)*log(a*x - 1) - 45*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1) ^2 + 272)*a*arctanh(a*x)/(a^8*x^6 - 3*a^6*x^4 + 3*a^4*x^2 - a^2)
\[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^4} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (a^{2} x^{2} - 1\right )}^{4}} \,d x } \] Input:
integrate(arctanh(a*x)^2/(-a^2*x^2+1)^4,x, algorithm="giac")
Output:
integrate(arctanh(a*x)^2/(a^2*x^2 - 1)^4, x)
Time = 5.51 (sec) , antiderivative size = 493, normalized size of antiderivative = 2.30 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^4} \, dx={\ln \left (1-a\,x\right )}^2\,\left (\frac {5\,\ln \left (a\,x+1\right )}{128\,a}-\frac {\frac {5\,a^4\,x^5}{16}-\frac {5\,a^2\,x^3}{6}+\frac {11\,x}{16}}{4\,a^6\,x^6-12\,a^4\,x^4+12\,a^2\,x^2-4}\right )-\frac {\frac {245\,a^4\,x^5}{8}-\frac {200\,a^2\,x^3}{3}+\frac {299\,x}{8}}{144\,a^6\,x^6-432\,a^4\,x^4+432\,a^2\,x^2-144}-\ln \left (1-a\,x\right )\,\left (\frac {5\,{\ln \left (a\,x+1\right )}^2}{128\,a}+\frac {\frac {37\,x}{2}-35\,a\,x^2+\frac {68}{3\,a}-\frac {82\,a^2\,x^3}{3}+15\,a^3\,x^4+\frac {23\,a^4\,x^5}{2}}{192\,a^6\,x^6-576\,a^4\,x^4+576\,a^2\,x^2-192}-\frac {\frac {37\,x}{2}+35\,a\,x^2-\frac {68}{3\,a}-\frac {82\,a^2\,x^3}{3}-15\,a^3\,x^4+\frac {23\,a^4\,x^5}{2}}{192\,a^6\,x^6-576\,a^4\,x^4+576\,a^2\,x^2-192}-\frac {\ln \left (a\,x+1\right )\,\left (10\,a^4\,x^5-\frac {80\,a^2\,x^3}{3}+22\,x\right )}{64\,a^6\,x^6-192\,a^4\,x^4+192\,a^2\,x^2-64}\right )+\frac {5\,{\ln \left (a\,x+1\right )}^3}{384\,a}-\frac {5\,{\ln \left (1-a\,x\right )}^3}{384\,a}+\frac {\ln \left (a\,x+1\right )\,\left (\frac {17}{72\,a^2}-\frac {35\,x^2}{96}+\frac {5\,a^2\,x^4}{32}\right )}{3\,a\,x^2-\frac {1}{a}-3\,a^3\,x^4+a^5\,x^6}-\frac {{\ln \left (a\,x+1\right )}^2\,\left (\frac {11\,x}{64\,a}-\frac {5\,a\,x^3}{24}+\frac {5\,a^3\,x^5}{64}\right )}{3\,a\,x^2-\frac {1}{a}-3\,a^3\,x^4+a^5\,x^6}-\frac {\mathrm {atan}\left (a\,x\,1{}\mathrm {i}\right )\,245{}\mathrm {i}}{1152\,a} \] Input:
int(atanh(a*x)^2/(a^2*x^2 - 1)^4,x)
Output:
log(1 - a*x)^2*((5*log(a*x + 1))/(128*a) - ((11*x)/16 - (5*a^2*x^3)/6 + (5 *a^4*x^5)/16)/(12*a^2*x^2 - 12*a^4*x^4 + 4*a^6*x^6 - 4)) - ((299*x)/8 - (2 00*a^2*x^3)/3 + (245*a^4*x^5)/8)/(432*a^2*x^2 - 432*a^4*x^4 + 144*a^6*x^6 - 144) - log(1 - a*x)*((5*log(a*x + 1)^2)/(128*a) + ((37*x)/2 - 35*a*x^2 + 68/(3*a) - (82*a^2*x^3)/3 + 15*a^3*x^4 + (23*a^4*x^5)/2)/(576*a^2*x^2 - 5 76*a^4*x^4 + 192*a^6*x^6 - 192) - ((37*x)/2 + 35*a*x^2 - 68/(3*a) - (82*a^ 2*x^3)/3 - 15*a^3*x^4 + (23*a^4*x^5)/2)/(576*a^2*x^2 - 576*a^4*x^4 + 192*a ^6*x^6 - 192) - (log(a*x + 1)*(22*x - (80*a^2*x^3)/3 + 10*a^4*x^5))/(192*a ^2*x^2 - 192*a^4*x^4 + 64*a^6*x^6 - 64)) + (5*log(a*x + 1)^3)/(384*a) - (5 *log(1 - a*x)^3)/(384*a) - (atan(a*x*1i)*245i)/(1152*a) + (log(a*x + 1)*(1 7/(72*a^2) - (35*x^2)/96 + (5*a^2*x^4)/32))/(3*a*x^2 - 1/a - 3*a^3*x^4 + a ^5*x^6) - (log(a*x + 1)^2*((11*x)/(64*a) - (5*a*x^3)/24 + (5*a^3*x^5)/64)) /(3*a*x^2 - 1/a - 3*a^3*x^4 + a^5*x^6)
Time = 0.16 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.38 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^4} \, dx=\frac {720 \mathit {atanh} \left (a x \right )^{3} a^{6} x^{6}-2160 \mathit {atanh} \left (a x \right )^{3} a^{4} x^{4}+2160 \mathit {atanh} \left (a x \right )^{3} a^{2} x^{2}-720 \mathit {atanh} \left (a x \right )^{3}-2160 \mathit {atanh} \left (a x \right )^{2} a^{5} x^{5}+5760 \mathit {atanh} \left (a x \right )^{2} a^{3} x^{3}-4752 \mathit {atanh} \left (a x \right )^{2} a x +720 \mathit {atanh} \left (a x \right ) a^{6} x^{6}-2880 \mathit {atanh} \left (a x \right ) a^{2} x^{2}+2544 \mathit {atanh} \left (a x \right )-375 \,\mathrm {log}\left (a^{2} x -a \right ) a^{6} x^{6}+1125 \,\mathrm {log}\left (a^{2} x -a \right ) a^{4} x^{4}-1125 \,\mathrm {log}\left (a^{2} x -a \right ) a^{2} x^{2}+375 \,\mathrm {log}\left (a^{2} x -a \right )+375 \,\mathrm {log}\left (a^{2} x +a \right ) a^{6} x^{6}-1125 \,\mathrm {log}\left (a^{2} x +a \right ) a^{4} x^{4}+1125 \,\mathrm {log}\left (a^{2} x +a \right ) a^{2} x^{2}-375 \,\mathrm {log}\left (a^{2} x +a \right )-1470 a^{5} x^{5}+3200 a^{3} x^{3}-1794 a x}{6912 a \left (a^{6} x^{6}-3 a^{4} x^{4}+3 a^{2} x^{2}-1\right )} \] Input:
int(atanh(a*x)^2/(-a^2*x^2+1)^4,x)
Output:
(720*atanh(a*x)**3*a**6*x**6 - 2160*atanh(a*x)**3*a**4*x**4 + 2160*atanh(a *x)**3*a**2*x**2 - 720*atanh(a*x)**3 - 2160*atanh(a*x)**2*a**5*x**5 + 5760 *atanh(a*x)**2*a**3*x**3 - 4752*atanh(a*x)**2*a*x + 720*atanh(a*x)*a**6*x* *6 - 2880*atanh(a*x)*a**2*x**2 + 2544*atanh(a*x) - 375*log(a**2*x - a)*a** 6*x**6 + 1125*log(a**2*x - a)*a**4*x**4 - 1125*log(a**2*x - a)*a**2*x**2 + 375*log(a**2*x - a) + 375*log(a**2*x + a)*a**6*x**6 - 1125*log(a**2*x + a )*a**4*x**4 + 1125*log(a**2*x + a)*a**2*x**2 - 375*log(a**2*x + a) - 1470* a**5*x**5 + 3200*a**3*x**3 - 1794*a*x)/(6912*a*(a**6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 1))