Integrand size = 22, antiderivative size = 100 \[ \int \frac {x^4}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=-\frac {x^4}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}-\frac {2 x}{a^4 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}+\frac {2 x}{a^4 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}-\frac {\text {Chi}(2 \text {arctanh}(a x))}{a^5}+\frac {\text {Chi}(4 \text {arctanh}(a x))}{a^5} \] Output:
-1/2*x^4/a/(-a^2*x^2+1)^2/arctanh(a*x)^2-2*x/a^4/(-a^2*x^2+1)^2/arctanh(a* x)+2*x/a^4/(-a^2*x^2+1)/arctanh(a*x)-Chi(2*arctanh(a*x))/a^5+Chi(4*arctanh (a*x))/a^5
Time = 0.21 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.60 \[ \int \frac {x^4}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=-\frac {\frac {a^3 x^3 (a x+4 \text {arctanh}(a x))}{\left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^2}+2 \text {Chi}(2 \text {arctanh}(a x))-2 \text {Chi}(4 \text {arctanh}(a x))}{2 a^5} \] Input:
Integrate[x^4/((1 - a^2*x^2)^3*ArcTanh[a*x]^3),x]
Output:
-1/2*((a^3*x^3*(a*x + 4*ArcTanh[a*x]))/((-1 + a^2*x^2)^2*ArcTanh[a*x]^2) + 2*CoshIntegral[2*ArcTanh[a*x]] - 2*CoshIntegral[4*ArcTanh[a*x]])/a^5
Leaf count is larger than twice the leaf count of optimal. \(206\) vs. \(2(100)=200\).
Time = 2.11 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.06, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {6568, 6590, 6594, 6530, 3042, 3793, 2009, 6596, 3042, 25, 3793, 2009, 5971, 2009}
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.
| \(\displaystyle \int \frac {x^4}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx\) |
| \(\Big \downarrow \) 6568 |
| \(\displaystyle \frac {2 \int \frac {x^3}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx}{a}-\frac {x^4}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 6590 |
| \(\displaystyle \frac {2 \left (\frac {\int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx}{a^2}-\frac {\int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx}{a^2}\right )}{a}-\frac {x^4}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 6594 |
| \(\displaystyle \frac {2 \left (\frac {\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx}{a}+3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx}{a}+a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )}{a}-\frac {x^4}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 6530 |
| \(\displaystyle \frac {2 \left (\frac {3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \frac {1}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )}{a}-\frac {x^4}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {x^4}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}+\frac {2 \left (\frac {3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\int \frac {\sin \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )^4}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \frac {\sin \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )}{a}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {2 \left (\frac {3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\int \left (\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}+\frac {\cosh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}+\frac {3}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \left (\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}+\frac {1}{2 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )}{a}-\frac {x^4}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (\frac {3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )}{a}-\frac {x^4}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 6596 |
| \(\displaystyle \frac {2 \left (\frac {\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {a^2 x^2}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )}{a}-\frac {x^4}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {x^4}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}+\frac {2 \left (\frac {\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int -\frac {\sin (i \text {arctanh}(a x))^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^4}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}+\frac {2 \left (\frac {\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {-\frac {\int \frac {\sin (i \text {arctanh}(a x))^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )}{a}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {2 \left (\frac {\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {-\frac {\int \left (\frac {1}{2 \text {arctanh}(a x)}-\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )}{a}-\frac {x^4}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (\frac {\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )}{a}-\frac {x^4}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {2 \left (\frac {\frac {3 \int \left (\frac {\cosh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}-\frac {1}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )}{a}-\frac {x^4}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (\frac {\frac {3 \left (\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))-\frac {1}{8} \log (\text {arctanh}(a x))\right )}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )}{a}-\frac {x^4}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\) |
Input:
Int[x^4/((1 - a^2*x^2)^3*ArcTanh[a*x]^3),x]
Output:
-1/2*x^4/(a*(1 - a^2*x^2)^2*ArcTanh[a*x]^2) + (2*((-(x/(a*(1 - a^2*x^2)^2* ArcTanh[a*x])) + (3*(CoshIntegral[4*ArcTanh[a*x]]/8 - Log[ArcTanh[a*x]]/8) )/a^2 + (CoshIntegral[2*ArcTanh[a*x]]/2 + CoshIntegral[4*ArcTanh[a*x]]/8 + (3*Log[ArcTanh[a*x]])/8)/a^2)/a^2 - (-(x/(a*(1 - a^2*x^2)*ArcTanh[a*x])) + (CoshIntegral[2*ArcTanh[a*x]]/2 - Log[ArcTanh[a*x]]/2)/a^2 + (CoshIntegr al[2*ArcTanh[a*x]]/2 + Log[ArcTanh[a*x]]/2)/a^2)/a^2))/a
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
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^q/c Subst[Int[(a + b*x)^p/Cosh[x]^(2*(q + 1)), x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && I LtQ[2*(q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x^2)^(q + 1)*((a + b*Arc Tanh[c*x])^(p + 1)/(b*c*d*(p + 1))), x] - Simp[f*(m/(b*c*(p + 1))) Int[(f *x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && Lt Q[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 = 1.37 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.90
| method | result | size |
| derivativedivides | \(\frac {-\frac {3}{16 \operatorname {arctanh}\left (a x \right )^{2}}+\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \operatorname {arctanh}\left (a x \right )^{2}}+\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2 \,\operatorname {arctanh}\left (a x \right )}-\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{16 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\sinh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Chi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{a^{5}}\) | \(90\) |
| default | \(\frac {-\frac {3}{16 \operatorname {arctanh}\left (a x \right )^{2}}+\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \operatorname {arctanh}\left (a x \right )^{2}}+\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2 \,\operatorname {arctanh}\left (a x \right )}-\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{16 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\sinh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Chi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{a^{5}}\) | \(90\) |
Input:
int(x^4/(-a^2*x^2+1)^3/arctanh(a*x)^3,x,method=_RETURNVERBOSE)
Output:
1/a^5*(-3/16/arctanh(a*x)^2+1/4/arctanh(a*x)^2*cosh(2*arctanh(a*x))+1/2/ar ctanh(a*x)*sinh(2*arctanh(a*x))-Chi(2*arctanh(a*x))-1/16/arctanh(a*x)^2*co sh(4*arctanh(a*x))-1/4*sinh(4*arctanh(a*x))/arctanh(a*x)+Chi(4*arctanh(a*x )))
Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (95) = 190\).
Time = 0.09 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.56 \[ \int \frac {x^4}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=-\frac {4 \, a^{4} x^{4} + 8 \, a^{3} x^{3} \log \left (-\frac {a x + 1}{a x - 1}\right ) - {\left ({\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}}{2 \, {\left (a^{9} x^{4} - 2 \, a^{7} x^{2} + a^{5}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}} \] Input:
integrate(x^4/(-a^2*x^2+1)^3/arctanh(a*x)^3,x, algorithm="fricas")
Output:
-1/2*(4*a^4*x^4 + 8*a^3*x^3*log(-(a*x + 1)/(a*x - 1)) - ((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^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)/((a^9*x^4 - 2*a^7*x^2 + a^5)*log(-(a*x + 1)/(a*x - 1))^2)
\[ \int \frac {x^4}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=- \int \frac {x^{4}}{a^{6} x^{6} \operatorname {atanh}^{3}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname {atanh}^{3}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atanh}^{3}{\left (a x \right )} - \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \] Input:
integrate(x**4/(-a**2*x**2+1)**3/atanh(a*x)**3,x)
Output:
-Integral(x**4/(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)
\[ \int \frac {x^4}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=\int { -\frac {x^{4}}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{3}} \,d x } \] Input:
integrate(x^4/(-a^2*x^2+1)^3/arctanh(a*x)^3,x, algorithm="maxima")
Output:
-2*(a*x^4 + 2*x^3*log(a*x + 1) - 2*x^3*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(-4*(a^ 2*x^4 + 3*x^2)/((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)
\[ \int \frac {x^4}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=\int { -\frac {x^{4}}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{3}} \,d x } \] Input:
integrate(x^4/(-a^2*x^2+1)^3/arctanh(a*x)^3,x, algorithm="giac")
Output:
integrate(-x^4/((a^2*x^2 - 1)^3*arctanh(a*x)^3), x)
Timed out. \[ \int \frac {x^4}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=-\int \frac {x^4}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (a^2\,x^2-1\right )}^3} \,d x \] Input:
int(-x^4/(atanh(a*x)^3*(a^2*x^2 - 1)^3),x)
Output:
-int(x^4/(atanh(a*x)^3*(a^2*x^2 - 1)^3), x)
\[ \int \frac {x^4}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=\frac {-4 \mathit {atanh} \left (a x \right )^{2} \left (\int \frac {x^{3}}{\mathit {atanh} \left (a x \right )^{2} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{2}}d x \right ) a^{6} x^{2}+4 \mathit {atanh} \left (a x \right )^{2} \left (\int \frac {x^{3}}{\mathit {atanh} \left (a x \right )^{2} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{2}}d x \right ) a^{4}+4 \mathit {atanh} \left (a x \right )^{2} \left (\int \frac {x}{\mathit {atanh} \left (a x \right )^{2} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{2}}d x \right ) a^{4} x^{2}-4 \mathit {atanh} \left (a x \right )^{2} \left (\int \frac {x}{\mathit {atanh} \left (a x \right )^{2} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{2}}d x \right ) a^{2}-2 \mathit {atanh} \left (a x \right )^{2} \left (\int \frac {1}{\mathit {atanh} \left (a x \right )^{3} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{3} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{3} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{3}}d x \right ) a^{3} x^{2}+2 \mathit {atanh} \left (a x \right )^{2} \left (\int \frac {1}{\mathit {atanh} \left (a x \right )^{3} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{3} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{3} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{3}}d x \right ) a -a^{2} x^{2}-1}{2 \mathit {atanh} \left (a x \right )^{2} a^{5} \left (a^{2} x^{2}-1\right )} \] Input:
int(x^4/(-a^2*x^2+1)^3/atanh(a*x)^3,x)
Output:
( - 4*atanh(a*x)**2*int(x**3/(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*x**2 + 4*atan h(a*x)**2*int(x**3/(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**4 + 4*atanh(a*x)**2*int(x /(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**4*x**2 - 4*atanh(a*x)**2*int(x/(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 - ata nh(a*x)**2),x)*a**2 - 2*atanh(a*x)**2*int(1/(atanh(a*x)**3*a**6*x**6 - 3*a tanh(a*x)**3*a**4*x**4 + 3*atanh(a*x)**3*a**2*x**2 - atanh(a*x)**3),x)*a** 3*x**2 + 2*atanh(a*x)**2*int(1/(atanh(a*x)**3*a**6*x**6 - 3*atanh(a*x)**3* a**4*x**4 + 3*atanh(a*x)**3*a**2*x**2 - atanh(a*x)**3),x)*a - a**2*x**2 - 1)/(2*atanh(a*x)**2*a**5*(a**2*x**2 - 1))