Integrand size = 22, antiderivative size = 107 \[ \int \frac {x^3}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=-\frac {x^3}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}-\frac {3 x^2}{2 a^2 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}-\frac {x^4}{2 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}-\frac {\text {Shi}(2 \text {arctanh}(a x))}{2 a^4}+\frac {\text {Shi}(4 \text {arctanh}(a x))}{a^4} \]
-1/2*x^3/a/(-a^2*x^2+1)^2/arctanh(a*x)^2-3/2*x^2/a^2/(-a^2*x^2+1)^2/arctan h(a*x)-1/2*x^4/(-a^2*x^2+1)^2/arctanh(a*x)-1/2*Shi(2*arctanh(a*x))/a^4+Shi (4*arctanh(a*x))/a^4
Time = 0.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.62 \[ \int \frac {x^3}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=-\frac {\frac {a^2 x^2 \left (a x+\left (3+a^2 x^2\right ) \text {arctanh}(a x)\right )}{\left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^2}+\text {Shi}(2 \text {arctanh}(a x))-2 \text {Shi}(4 \text {arctanh}(a x))}{2 a^4} \]
-1/2*((a^2*x^2*(a*x + (3 + a^2*x^2)*ArcTanh[a*x]))/((-1 + a^2*x^2)^2*ArcTa nh[a*x]^2) + SinhIntegral[2*ArcTanh[a*x]] - 2*SinhIntegral[4*ArcTanh[a*x]] )/a^4
Leaf count is larger than twice the leaf count of optimal. \(270\) vs. \(2(107)=214\).
Time = 2.26 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.52, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {6590, 6558, 6594, 6528, 6590, 6528, 6596, 5971, 27, 2009, 3042, 26, 3779}
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.
| \(\displaystyle \int \frac {x^3}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx\) |
| \(\Big \downarrow \) 6590 |
| \(\displaystyle \frac {\int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3}dx}{a^2}-\frac {\int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}dx}{a^2}\) |
| \(\Big \downarrow \) 6558 |
| \(\displaystyle \frac {\int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3}dx}{a^2}-\frac {2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\) |
| \(\Big \downarrow \) 6594 |
| \(\displaystyle \frac {\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx}{2 a}+\frac {3}{2} a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\) |
| \(\Big \downarrow \) 6528 |
| \(\displaystyle \frac {\frac {3}{2} a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx+\frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\) |
| \(\Big \downarrow \) 6590 |
| \(\displaystyle \frac {\frac {3}{2} a \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx}{a^2}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx}{a^2}\right )+\frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\) |
| \(\Big \downarrow \) 6528 |
| \(\displaystyle \frac {\frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}+\frac {3}{2} a \left (\frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {2 a \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\) |
| \(\Big \downarrow \) 6596 |
| \(\displaystyle \frac {\frac {\frac {4 \int \frac {a x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}+\frac {3}{2} a \left (\frac {\frac {4 \int \frac {a x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {2 \int \frac {a x}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {\frac {2 \int \frac {a x}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {\frac {\frac {4 \int \left (\frac {\sinh (2 \text {arctanh}(a x))}{4 \text {arctanh}(a x)}+\frac {\sinh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}+\frac {3}{2} a \left (\frac {\frac {4 \int \left (\frac {\sinh (2 \text {arctanh}(a x))}{4 \text {arctanh}(a x)}+\frac {\sinh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {2 \int \frac {\sinh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {\frac {2 \int \frac {\sinh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {4 \int \left (\frac {\sinh (2 \text {arctanh}(a x))}{4 \text {arctanh}(a x)}+\frac {\sinh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}+\frac {3}{2} a \left (\frac {\frac {4 \int \left (\frac {\sinh (2 \text {arctanh}(a x))}{4 \text {arctanh}(a x)}+\frac {\sinh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {\sinh (2 \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {\frac {\int \frac {\sinh (2 \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3}{2} a \left (\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {\sinh (2 \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )+\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {\frac {\int \frac {\sinh (2 \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{2} a \left (\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}+\frac {\int -\frac {i \sin (2 i \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a}}{a^2}\right )+\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {\frac {\int -\frac {i \sin (2 i \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {3}{2} a \left (\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}-\frac {i \int \frac {\sin (2 i \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a}}{a^2}\right )+\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {-\frac {i \int \frac {\sin (2 i \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle \frac {\frac {3}{2} a \left (\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\text {Shi}(2 \text {arctanh}(a x))}{a}-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )+\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {\frac {\text {Shi}(2 \text {arctanh}(a x))}{a^2}-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\) |
-((-1/2*x/(a*(1 - a^2*x^2)*ArcTanh[a*x]^2) - (1 + a^2*x^2)/(2*a^2*(1 - a^2 *x^2)*ArcTanh[a*x]) + SinhIntegral[2*ArcTanh[a*x]]/a^2)/a^2) + (-1/2*x/(a* (1 - a^2*x^2)^2*ArcTanh[a*x]^2) + (3*a*(-((-(1/(a*(1 - a^2*x^2)*ArcTanh[a* x])) + SinhIntegral[2*ArcTanh[a*x]]/a)/a^2) + (-(1/(a*(1 - a^2*x^2)^2*ArcT anh[a*x])) + (4*(SinhIntegral[2*ArcTanh[a*x]]/4 + SinhIntegral[4*ArcTanh[a *x]]/8))/a)/a^2))/2 + (-(1/(a*(1 - a^2*x^2)^2*ArcTanh[a*x])) + (4*(SinhInt egral[2*ArcTanh[a*x]]/4 + SinhIntegral[4*ArcTanh[a*x]]/8))/a)/(2*a))/a^2
3.4.37.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f , fz}, x] && EqQ[d*e - c*f*fz*I, 0]
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_ Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1))), x] + Simp[2*c*((q + 1)/(b*(p + 1))) Int[x*(d + e*x^2)^q*(a + b*A rcTanh[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && LtQ[p, -1]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*(x_))/((d_) + (e_.)*(x_)^2)^2 , x_Symbol] :> Simp[x*((a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)*(d + e*x ^2))), x] + (Simp[(1 + c^2*x^2)*((a + b*ArcTanh[c*x])^(p + 2)/(b^2*e*(p + 1 )*(p + 2)*(d + e*x^2))), x] + Simp[4/(b^2*(p + 1)*(p + 2)) Int[x*((a + b* ArcTanh[c*x])^(p + 2)/(d + e*x^2)^2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1] && NeQ[p, -2]
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]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(q_), x_Symbol] :> Simp[x^m*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^( p + 1)/(b*c*d*(p + 1))), x] + (Simp[c*((m + 2*q + 2)/(b*(p + 1))) Int[x^( m + 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x], x] - Simp[m/(b*c*(p + 1)) Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x], x]) / ; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && LtQ[q, - 1] && LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(q_), x_Symbol] :> Simp[d^q/c^(m + 1) Subst[Int[(a + b*x)^p*(Sinh[x]^ m/Cosh[x]^(m + 2*(q + 1))), x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (In tegerQ[q] || GtQ[d, 0])
Time = 0.37 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.77
| method | result | size |
| derivativedivides | \(\frac {-\frac {\sinh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{16 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Shi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )+\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{8 \operatorname {arctanh}\left (a x \right )^{2}}+\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \,\operatorname {arctanh}\left (a x \right )}-\frac {\operatorname {Shi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2}}{a^{4}}\) | \(82\) |
| default | \(\frac {-\frac {\sinh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{16 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Shi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )+\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{8 \operatorname {arctanh}\left (a x \right )^{2}}+\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \,\operatorname {arctanh}\left (a x \right )}-\frac {\operatorname {Shi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2}}{a^{4}}\) | \(82\) |
1/a^4*(-1/16/arctanh(a*x)^2*sinh(4*arctanh(a*x))-1/4/arctanh(a*x)*cosh(4*a rctanh(a*x))+Shi(4*arctanh(a*x))+1/8*sinh(2*arctanh(a*x))/arctanh(a*x)^2+1 /4/arctanh(a*x)*cosh(2*arctanh(a*x))-1/2*Shi(2*arctanh(a*x)))
Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (96) = 192\).
Time = 0.25 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.50 \[ \int \frac {x^3}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=-\frac {8 \, a^{3} x^{3} - {\left (2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, {\left (a^{4} x^{4} + 3 \, a^{2} x^{2}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{4 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}} \]
-1/4*(8*a^3*x^3 - (2*(a^4*x^4 - 2*a^2*x^2 + 1)*log_integral((a^2*x^2 + 2*a *x + 1)/(a^2*x^2 - 2*a*x + 1)) - 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log_integral( (a^2*x^2 - 2*a*x + 1)/(a^2*x^2 + 2*a*x + 1)) - (a^4*x^4 - 2*a^2*x^2 + 1)*l og_integral(-(a*x + 1)/(a*x - 1)) + (a^4*x^4 - 2*a^2*x^2 + 1)*log_integral (-(a*x - 1)/(a*x + 1)))*log(-(a*x + 1)/(a*x - 1))^2 + 4*(a^4*x^4 + 3*a^2*x ^2)*log(-(a*x + 1)/(a*x - 1)))/((a^8*x^4 - 2*a^6*x^2 + a^4)*log(-(a*x + 1) /(a*x - 1))^2)
\[ \int \frac {x^3}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=- \int \frac {x^{3}}{a^{6} x^{6} \operatorname {atanh}^{3}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname {atanh}^{3}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atanh}^{3}{\left (a x \right )} - \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \]
-Integral(x**3/(a**6*x**6*atanh(a*x)**3 - 3*a**4*x**4*atanh(a*x)**3 + 3*a* *2*x**2*atanh(a*x)**3 - atanh(a*x)**3), x)
\[ \int \frac {x^3}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=\int { -\frac {x^{3}}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{3}} \,d x } \]
-(2*a*x^3 + (a^2*x^4 + 3*x^2)*log(a*x + 1) - (a^2*x^4 + 3*x^2)*log(-a*x + 1))/((a^6*x^4 - 2*a^4*x^2 + a^2)*log(a*x + 1)^2 - 2*(a^6*x^4 - 2*a^4*x^2 + a^2)*log(a*x + 1)*log(-a*x + 1) + (a^6*x^4 - 2*a^4*x^2 + a^2)*log(-a*x + 1)^2) + integrate(-2*(5*a^2*x^3 + 3*x)/((a^8*x^6 - 3*a^6*x^4 + 3*a^4*x^2 - a^2)*log(a*x + 1) - (a^8*x^6 - 3*a^6*x^4 + 3*a^4*x^2 - a^2)*log(-a*x + 1) ), x)
\[ \int \frac {x^3}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=\int { -\frac {x^{3}}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{3}} \,d x } \]
Timed out. \[ \int \frac {x^3}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=-\int \frac {x^3}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (a^2\,x^2-1\right )}^3} \,d x \]