Integrand size = 20, antiderivative size = 100 \[ \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}-\frac {2}{a^2 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}+\frac {3}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}+\frac {\text {Shi}(2 \text {arctanh}(a x))}{2 a^2}+\frac {\text {Shi}(4 \text {arctanh}(a x))}{a^2} \] Output:
-1/2*x/a/(-a^2*x^2+1)^2/arctanh(a*x)^2-2/a^2/(-a^2*x^2+1)^2/arctanh(a*x)+3 /2/a^2/(-a^2*x^2+1)/arctanh(a*x)+1/2*Shi(2*arctanh(a*x))/a^2+Shi(4*arctanh (a*x))/a^2
Time = 0.18 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.96 \[ \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=-\frac {a x+\text {arctanh}(a x)+3 a^2 x^2 \text {arctanh}(a x)-\left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^2 \text {Shi}(2 \text {arctanh}(a x))-2 \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^2 \text {Shi}(4 \text {arctanh}(a x))}{2 a^2 \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^2} \] Input:
Integrate[x/((1 - a^2*x^2)^3*ArcTanh[a*x]^3),x]
Output:
-1/2*(a*x + ArcTanh[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*(-1 + a^2*x^2)^2*ArcTanh[a*x]^2 *SinhIntegral[4*ArcTanh[a*x]])/(a^2*(-1 + a^2*x^2)^2*ArcTanh[a*x]^2)
Time = 1.58 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.88, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {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.
\(\displaystyle \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx\) |
\(\Big \downarrow \) 6594 |
\(\displaystyle \frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx}{2 a}+\frac {3}{2} a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 6528 |
\(\displaystyle \frac {3}{2} a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx+\frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 6590 |
\(\displaystyle \frac {3}{2} a \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx}{a^2}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx}{a^2}\right )+\frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 6528 |
\(\displaystyle \frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}+\frac {3}{2} a \left (\frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {2 a \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 6596 |
\(\displaystyle \frac {\frac {4 \int \frac {a x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}+\frac {3}{2} a \left (\frac {\frac {4 \int \frac {a x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {2 \int \frac {a x}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle \frac {\frac {4 \int \left (\frac {\sinh (2 \text {arctanh}(a x))}{4 \text {arctanh}(a x)}+\frac {\sinh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}+\frac {3}{2} a \left (\frac {\frac {4 \int \left (\frac {\sinh (2 \text {arctanh}(a x))}{4 \text {arctanh}(a x)}+\frac {\sinh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {2 \int \frac {\sinh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {4 \int \left (\frac {\sinh (2 \text {arctanh}(a x))}{4 \text {arctanh}(a x)}+\frac {\sinh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}+\frac {3}{2} a \left (\frac {\frac {4 \int \left (\frac {\sinh (2 \text {arctanh}(a x))}{4 \text {arctanh}(a x)}+\frac {\sinh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {\sinh (2 \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3}{2} a \left (\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {\sinh (2 \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )+\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{2} a \left (\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}+\frac {\int -\frac {i \sin (2 i \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a}}{a^2}\right )+\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {3}{2} a \left (\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}-\frac {i \int \frac {\sin (2 i \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a}}{a^2}\right )+\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 3779 |
\(\displaystyle \frac {3}{2} a \left (\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\text {Shi}(2 \text {arctanh}(a x))}{a}-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )+\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
Input:
Int[x/((1 - a^2*x^2)^3*ArcTanh[a*x]^3),x]
Output:
-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*ArcTanh[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*(SinhIntegral[2*ArcTanh[a*x]]/4 + SinhIntegral[4*ArcTanh[a*x]]/8))/a)/(2 *a)
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_)^(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 = 1.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {-\frac {\sinh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{16 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Shi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{8 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \,\operatorname {arctanh}\left (a x \right )}+\frac {\operatorname {Shi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2}}{a^{2}}\) | \(82\) |
default | \(\frac {-\frac {\sinh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{16 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Shi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{8 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \,\operatorname {arctanh}\left (a x \right )}+\frac {\operatorname {Shi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2}}{a^{2}}\) | \(82\) |
Input:
int(x/(-a^2*x^2+1)^3/arctanh(a*x)^3,x,method=_RETURNVERBOSE)
Output:
1/a^2*(-1/16*sinh(4*arctanh(a*x))/arctanh(a*x)^2-1/4/arctanh(a*x)*cosh(4*a rctanh(a*x))+Shi(4*arctanh(a*x))-1/8/arctanh(a*x)^2*sinh(2*arctanh(a*x))-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. 256 vs. \(2 (91) = 182\).
Time = 0.08 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.56 \[ \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=\frac {{\left (2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 8 \, a x - 4 \, {\left (3 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{4 \, {\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}} \] Input:
integrate(x/(-a^2*x^2+1)^3/arctanh(a*x)^3,x, algorithm="fricas")
Output:
1/4*((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)*log_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 - 8*a*x - 4*(3*a^2*x^2 + 1)*log(-(a *x + 1)/(a*x - 1)))/((a^6*x^4 - 2*a^4*x^2 + a^2)*log(-(a*x + 1)/(a*x - 1)) ^2)
\[ \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=- \int \frac {x}{a^{6} x^{6} \operatorname {atanh}^{3}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname {atanh}^{3}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atanh}^{3}{\left (a x \right )} - \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \] Input:
integrate(x/(-a**2*x**2+1)**3/atanh(a*x)**3,x)
Output:
-Integral(x/(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)
\[ \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=\int { -\frac {x}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{3}} \,d x } \] Input:
integrate(x/(-a^2*x^2+1)^3/arctanh(a*x)^3,x, algorithm="maxima")
Output:
-(2*a*x + (3*a^2*x^2 + 1)*log(a*x + 1) - (3*a^2*x^2 + 1)*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*(3*a^2*x^3 + 5*x)/((a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*lo g(a*x + 1) - (a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(-a*x + 1)), x)
\[ \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=\int { -\frac {x}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{3}} \,d x } \] Input:
integrate(x/(-a^2*x^2+1)^3/arctanh(a*x)^3,x, algorithm="giac")
Output:
integrate(-x/((a^2*x^2 - 1)^3*arctanh(a*x)^3), x)
Timed out. \[ \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=-\int \frac {x}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (a^2\,x^2-1\right )}^3} \,d x \] Input:
int(-x/(atanh(a*x)^3*(a^2*x^2 - 1)^3),x)
Output:
-int(x/(atanh(a*x)^3*(a^2*x^2 - 1)^3), x)
\[ \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx =\text {Too large to display} \] Input:
int(x/(-a^2*x^2+1)^3/atanh(a*x)^3,x)
Output:
( - 12*atanh(a*x)**2*int(x**3/(atanh(a*x)*a**6*x**6 - 3*atanh(a*x)*a**4*x* *4 + 3*atanh(a*x)*a**2*x**2 - atanh(a*x)),x)*a**8*x**4 + 24*atanh(a*x)**2* int(x**3/(atanh(a*x)*a**6*x**6 - 3*atanh(a*x)*a**4*x**4 + 3*atanh(a*x)*a** 2*x**2 - atanh(a*x)),x)*a**6*x**2 - 12*atanh(a*x)**2*int(x**3/(atanh(a*x)* a**6*x**6 - 3*atanh(a*x)*a**4*x**4 + 3*atanh(a*x)*a**2*x**2 - atanh(a*x)), x)*a**4 - 8*atanh(a*x)**2*int(x/(atanh(a*x)*a**6*x**6 - 3*atanh(a*x)*a**4* x**4 + 3*atanh(a*x)*a**2*x**2 - atanh(a*x)),x)*a**6*x**4 + 16*atanh(a*x)** 2*int(x/(atanh(a*x)*a**6*x**6 - 3*atanh(a*x)*a**4*x**4 + 3*atanh(a*x)*a**2 *x**2 - atanh(a*x)),x)*a**4*x**2 - 8*atanh(a*x)**2*int(x/(atanh(a*x)*a**6* x**6 - 3*atanh(a*x)*a**4*x**4 + 3*atanh(a*x)*a**2*x**2 - atanh(a*x)),x)*a* *2 - 3*atanh(a*x)**2*int(1/(atanh(a*x)**2*a**6*x**6 - 3*atanh(a*x)**2*a**4 *x**4 + 3*atanh(a*x)**2*a**2*x**2 - atanh(a*x)**2),x)*a**5*x**4 + 6*atanh( a*x)**2*int(1/(atanh(a*x)**2*a**6*x**6 - 3*atanh(a*x)**2*a**4*x**4 + 3*ata nh(a*x)**2*a**2*x**2 - atanh(a*x)**2),x)*a**3*x**2 - 3*atanh(a*x)**2*int(1 /(atanh(a*x)**2*a**6*x**6 - 3*atanh(a*x)**2*a**4*x**4 + 3*atanh(a*x)**2*a* *2*x**2 - atanh(a*x)**2),x)*a - 6*atanh(a*x)*a**2*x**2 + atanh(a*x) - 2*a* x)/(4*atanh(a*x)**2*a**2*(a**4*x**4 - 2*a**2*x**2 + 1))