Integrand size = 22, antiderivative size = 136 \[ \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )^2} \, dx=\frac {1}{4 \left (1-a^2 x^2\right )}-\frac {a x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4} \text {arctanh}(a x)^2+\frac {\text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right ) \]
1/4/(-a^2*x^2+1)-1/2*a*x*arctanh(a*x)/(-a^2*x^2+1)-1/4*arctanh(a*x)^2+1/2* arctanh(a*x)^2/(-a^2*x^2+1)+1/3*arctanh(a*x)^3+arctanh(a*x)^2*ln(2-2/(a*x+ 1))-arctanh(a*x)*polylog(2,-1+2/(a*x+1))-1/2*polylog(3,-1+2/(a*x+1))
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.78 \[ \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )^2} \, dx=\frac {1}{24} \left (i \pi ^3-8 \text {arctanh}(a x)^3+3 \cosh (2 \text {arctanh}(a x))+6 \text {arctanh}(a x)^2 \cosh (2 \text {arctanh}(a x))+24 \text {arctanh}(a x)^2 \log \left (1-e^{2 \text {arctanh}(a x)}\right )+24 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )-12 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )-6 \text {arctanh}(a x) \sinh (2 \text {arctanh}(a x))\right ) \]
(I*Pi^3 - 8*ArcTanh[a*x]^3 + 3*Cosh[2*ArcTanh[a*x]] + 6*ArcTanh[a*x]^2*Cos h[2*ArcTanh[a*x]] + 24*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] + 24*Arc Tanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] - 12*PolyLog[3, E^(2*ArcTanh[a*x] )] - 6*ArcTanh[a*x]*Sinh[2*ArcTanh[a*x]])/24
Time = 1.19 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.23, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6592, 6550, 6494, 6556, 6518, 241, 6618, 7164}
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.
\(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 6592 |
\(\displaystyle a^2 \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )}dx\) |
\(\Big \downarrow \) 6550 |
\(\displaystyle a^2 \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\int \frac {\text {arctanh}(a x)^2}{x (a x+1)}dx+\frac {1}{3} \text {arctanh}(a x)^3\) |
\(\Big \downarrow \) 6494 |
\(\displaystyle a^2 \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx-2 a \int \frac {\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\) |
\(\Big \downarrow \) 6556 |
\(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx}{a}\right )-2 a \int \frac {\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\) |
\(\Big \downarrow \) 6518 |
\(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {-\frac {1}{2} a \int \frac {x}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )-2 a \int \frac {\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\) |
\(\Big \downarrow \) 241 |
\(\displaystyle -2 a \int \frac {\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+a^2 \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\) |
\(\Big \downarrow \) 6618 |
\(\displaystyle -2 a \left (\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx\right )+a^2 \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\) |
\(\Big \downarrow \) 7164 |
\(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )-2 a \left (\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{4 a}\right )+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\) |
ArcTanh[a*x]^3/3 + a^2*(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) + ArcTanh[a*x]^2*Log[2 - 2/(1 + a*x)] - 2*a*((ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)])/(2*a) + PolyLog[3, -1 + 2/(1 + a*x)]/(4*a))
3.3.70.3.1 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_.)/((x_)*((d_) + (e_.)*(x_))), x _Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Simp[b*c*(p/d) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))] /(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c ^2*d^2 - e^2, 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_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*d*(p + 1)), x] + Simp[1/ d Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), 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_.)*(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/d Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh [c*x])^p, x], x] - Simp[e/d Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTanh[c* x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Integers Q[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x ] - Simp[b*(p/2) Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[2, 1 - u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 + c*x))^2, 0]
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /; !FalseQ[w]] /; FreeQ[n, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.66 (sec) , antiderivative size = 1203, normalized size of antiderivative = 8.85
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1203\) |
default | \(\text {Expression too large to display}\) | \(1203\) |
parts | \(\text {Expression too large to display}\) | \(1602\) |
arctanh(a*x)^2*ln(a*x)-1/4*arctanh(a*x)^2/(a*x-1)-1/2*arctanh(a*x)^2*ln(a* x-1)+1/4*arctanh(a*x)^2/(a*x+1)-1/2*arctanh(a*x)^2*ln(a*x+1)+arctanh(a*x)^ 2*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-1/3*arctanh(a*x)^3-1/8*arctanh(a*x)*(a*x- 1)/(a*x+1)-1/16/(a*x+1)*(a*x-1)+1/8*(a*x+1)*arctanh(a*x)/(a*x-1)-1/16*(a*x +1)/(a*x-1)-arctanh(a*x)^2*ln((a*x+1)^2/(-a^2*x^2+1)-1)+arctanh(a*x)^2*ln( 1-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1 )^(1/2))-2*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+arctanh(a*x)^2*ln(1+(a*x+ 1)/(-a^2*x^2+1)^(1/2))+2*arctanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2 ))-2*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*(-I*Pi*csgn(I*(a*x+1)^2/(a ^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))-2*I*P i*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1))*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1)/(1- (a*x+1)^2/(a^2*x^2-1)))^2+I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3+2*I*Pi*csgn (I/(1-(a*x+1)^2/(a^2*x^2-1)))^3+I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+ 1)^2/(a^2*x^2-1)))^2*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))+I*Pi*csgn(I*(a*x+1) ^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^3-I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2 -1)/(1-(a*x+1)^2/(a^2*x^2-1)))*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))*csgn(I*(a *x+1)^2/(a^2*x^2-1))+2*I*Pi*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1))*csgn(I*(-(a *x+1)^2/(a^2*x^2-1)-1)/(1-(a*x+1)^2/(a^2*x^2-1)))*csgn(I/(1-(a*x+1)^2/(a^2 *x^2-1)))+I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2* x^2-1))-2*I*Pi*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))^2-2*I*Pi*csgn(I*(-(a*x...
\[ \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )^2} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (a^{2} x^{2} - 1\right )}^{2} x} \,d x } \]
\[ \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )^2} \, dx=\int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{x \left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \]
\[ \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )^2} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (a^{2} x^{2} - 1\right )}^{2} x} \,d x } \]
1/4*a^4*integrate(x^4*log(a*x + 1)*log(-a*x + 1)/(a^4*x^5 - 2*a^2*x^3 + x) , x) + 1/4*a^3*integrate(x^3*log(a*x + 1)*log(-a*x + 1)/(a^4*x^5 - 2*a^2*x ^3 + x), x) - 1/32*(a*(2/(a^4*x - a^3) - log(a*x + 1)/a^3 + log(a*x - 1)/a ^3) + 4*log(-a*x + 1)/(a^4*x^2 - a^2))*a^2 - 1/4*a^2*integrate(x^2*log(a*x + 1)*log(-a*x + 1)/(a^4*x^5 - 2*a^2*x^3 + x), x) - 1/4*a*integrate(x*log( a*x + 1)*log(-a*x + 1)/(a^4*x^5 - 2*a^2*x^3 + x), x) + 1/4*a*integrate(x*l og(-a*x + 1)/(a^4*x^5 - 2*a^2*x^3 + x), x) - 1/24*((a^2*x^2 - 1)*log(-a*x + 1)^3 + 3*((a^2*x^2 - 1)*log(a*x + 1) + 1)*log(-a*x + 1)^2)/(a^2*x^2 - 1) + 1/4*integrate(log(a*x + 1)^2/(a^4*x^5 - 2*a^2*x^3 + x), x) - 1/2*integr ate(log(a*x + 1)*log(-a*x + 1)/(a^4*x^5 - 2*a^2*x^3 + x), x)
\[ \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )^2} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (a^{2} x^{2} - 1\right )}^{2} x} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )^2} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{x\,{\left (a^2\,x^2-1\right )}^2} \,d x \]