Integrand size = 20, antiderivative size = 138 \[ \int x^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx=\frac {x}{30 a^2}-\frac {x^3}{30}-\frac {\text {arctanh}(a x)}{30 a^3}+\frac {2 x^2 \text {arctanh}(a x)}{15 a}-\frac {1}{10} a x^4 \text {arctanh}(a x)+\frac {2 \text {arctanh}(a x)^2}{15 a^3}+\frac {1}{3} x^3 \text {arctanh}(a x)^2-\frac {1}{5} a^2 x^5 \text {arctanh}(a x)^2-\frac {4 \text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{15 a^3}-\frac {2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{15 a^3} \] Output:
1/30*x/a^2-1/30*x^3-1/30*arctanh(a*x)/a^3+2/15*x^2*arctanh(a*x)/a-1/10*a*x ^4*arctanh(a*x)+2/15*arctanh(a*x)^2/a^3+1/3*x^3*arctanh(a*x)^2-1/5*a^2*x^5 *arctanh(a*x)^2-4/15*arctanh(a*x)*ln(2/(-a*x+1))/a^3-2/15*polylog(2,1-2/(- a*x+1))/a^3
Time = 0.19 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.69 \[ \int x^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx=-\frac {-a x+a^3 x^3+2 \left (2-5 a^3 x^3+3 a^5 x^5\right ) \text {arctanh}(a x)^2+\text {arctanh}(a x) \left (1-4 a^2 x^2+3 a^4 x^4+8 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )}{30 a^3} \] Input:
Integrate[x^2*(1 - a^2*x^2)*ArcTanh[a*x]^2,x]
Output:
-1/30*(-(a*x) + a^3*x^3 + 2*(2 - 5*a^3*x^3 + 3*a^5*x^5)*ArcTanh[a*x]^2 + A rcTanh[a*x]*(1 - 4*a^2*x^2 + 3*a^4*x^4 + 8*Log[1 + E^(-2*ArcTanh[a*x])]) - 4*PolyLog[2, -E^(-2*ArcTanh[a*x])])/a^3
Leaf count is larger than twice the leaf count of optimal. \(297\) vs. \(2(138)=276\).
Time = 1.88 (sec) , antiderivative size = 297, normalized size of antiderivative = 2.15, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6576, 6452, 6542, 6452, 254, 262, 219, 2009, 6542, 6452, 262, 219, 6546, 6470, 2849, 2752}
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.
| \(\displaystyle \int x^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx\) |
| \(\Big \downarrow \) 6576 |
| \(\displaystyle \int x^2 \text {arctanh}(a x)^2dx-a^2 \int x^4 \text {arctanh}(a x)^2dx\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle -a^2 \left (\frac {1}{5} x^5 \text {arctanh}(a x)^2-\frac {2}{5} a \int \frac {x^5 \text {arctanh}(a x)}{1-a^2 x^2}dx\right )-\frac {2}{3} a \int \frac {x^3 \text {arctanh}(a x)}{1-a^2 x^2}dx+\frac {1}{3} x^3 \text {arctanh}(a x)^2\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle -\frac {2}{3} a \left (\frac {\int \frac {x \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int x \text {arctanh}(a x)dx}{a^2}\right )-a^2 \left (\frac {1}{5} x^5 \text {arctanh}(a x)^2-\frac {2}{5} a \left (\frac {\int \frac {x^3 \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int x^3 \text {arctanh}(a x)dx}{a^2}\right )\right )+\frac {1}{3} x^3 \text {arctanh}(a x)^2\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle -\frac {2}{3} a \left (\frac {\int \frac {x \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \int \frac {x^2}{1-a^2 x^2}dx}{a^2}\right )-a^2 \left (\frac {1}{5} x^5 \text {arctanh}(a x)^2-\frac {2}{5} a \left (\frac {\int \frac {x^3 \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{4} x^4 \text {arctanh}(a x)-\frac {1}{4} a \int \frac {x^4}{1-a^2 x^2}dx}{a^2}\right )\right )+\frac {1}{3} x^3 \text {arctanh}(a x)^2\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle -\frac {2}{3} a \left (\frac {\int \frac {x \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \int \frac {x^2}{1-a^2 x^2}dx}{a^2}\right )-a^2 \left (\frac {1}{5} x^5 \text {arctanh}(a x)^2-\frac {2}{5} a \left (\frac {\int \frac {x^3 \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{4} x^4 \text {arctanh}(a x)-\frac {1}{4} a \int \left (-\frac {x^2}{a^2}+\frac {1}{a^4 \left (1-a^2 x^2\right )}-\frac {1}{a^4}\right )dx}{a^2}\right )\right )+\frac {1}{3} x^3 \text {arctanh}(a x)^2\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle -\frac {2}{3} a \left (\frac {\int \frac {x \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \left (\frac {\int \frac {1}{1-a^2 x^2}dx}{a^2}-\frac {x}{a^2}\right )}{a^2}\right )-a^2 \left (\frac {1}{5} x^5 \text {arctanh}(a x)^2-\frac {2}{5} a \left (\frac {\int \frac {x^3 \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{4} x^4 \text {arctanh}(a x)-\frac {1}{4} a \int \left (-\frac {x^2}{a^2}+\frac {1}{a^4 \left (1-a^2 x^2\right )}-\frac {1}{a^4}\right )dx}{a^2}\right )\right )+\frac {1}{3} x^3 \text {arctanh}(a x)^2\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -a^2 \left (\frac {1}{5} x^5 \text {arctanh}(a x)^2-\frac {2}{5} a \left (\frac {\int \frac {x^3 \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{4} x^4 \text {arctanh}(a x)-\frac {1}{4} a \int \left (-\frac {x^2}{a^2}+\frac {1}{a^4 \left (1-a^2 x^2\right )}-\frac {1}{a^4}\right )dx}{a^2}\right )\right )-\frac {2}{3} a \left (\frac {\int \frac {x \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )+\frac {1}{3} x^3 \text {arctanh}(a x)^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2}{3} a \left (\frac {\int \frac {x \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )-a^2 \left (\frac {1}{5} x^5 \text {arctanh}(a x)^2-\frac {2}{5} a \left (\frac {\int \frac {x^3 \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{4} x^4 \text {arctanh}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )\right )+\frac {1}{3} x^3 \text {arctanh}(a x)^2\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle -\frac {2}{3} a \left (\frac {\int \frac {x \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )-a^2 \left (\frac {1}{5} x^5 \text {arctanh}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\int \frac {x \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int x \text {arctanh}(a x)dx}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \text {arctanh}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )\right )+\frac {1}{3} x^3 \text {arctanh}(a x)^2\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle -\frac {2}{3} a \left (\frac {\int \frac {x \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )-a^2 \left (\frac {1}{5} x^5 \text {arctanh}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\int \frac {x \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \int \frac {x^2}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \text {arctanh}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )\right )+\frac {1}{3} x^3 \text {arctanh}(a x)^2\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle -\frac {2}{3} a \left (\frac {\int \frac {x \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )-a^2 \left (\frac {1}{5} x^5 \text {arctanh}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\int \frac {x \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \left (\frac {\int \frac {1}{1-a^2 x^2}dx}{a^2}-\frac {x}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \text {arctanh}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )\right )+\frac {1}{3} x^3 \text {arctanh}(a x)^2\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2}{3} a \left (\frac {\int \frac {x \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )-a^2 \left (\frac {1}{5} x^5 \text {arctanh}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\int \frac {x \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \text {arctanh}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )\right )+\frac {1}{3} x^3 \text {arctanh}(a x)^2\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle -\frac {2}{3} a \left (\frac {\frac {\int \frac {\text {arctanh}(a x)}{1-a x}dx}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )-a^2 \left (\frac {1}{5} x^5 \text {arctanh}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\frac {\int \frac {\text {arctanh}(a x)}{1-a x}dx}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \text {arctanh}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )\right )+\frac {1}{3} x^3 \text {arctanh}(a x)^2\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle -\frac {2}{3} a \left (\frac {\frac {\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}-\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )-a^2 \left (\frac {1}{5} x^5 \text {arctanh}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\frac {\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}-\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \text {arctanh}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )\right )+\frac {1}{3} x^3 \text {arctanh}(a x)^2\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle -\frac {2}{3} a \left (\frac {\frac {\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-\frac {2}{1-a x}}d\frac {1}{1-a x}}{a}+\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )-a^2 \left (\frac {1}{5} x^5 \text {arctanh}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\frac {\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-\frac {2}{1-a x}}d\frac {1}{1-a x}}{a}+\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \text {arctanh}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )\right )+\frac {1}{3} x^3 \text {arctanh}(a x)^2\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle -\frac {2}{3} a \left (\frac {\frac {\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )-a^2 \left (\frac {1}{5} x^5 \text {arctanh}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\frac {\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \text {arctanh}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \text {arctanh}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )\right )+\frac {1}{3} x^3 \text {arctanh}(a x)^2\) |
Input:
Int[x^2*(1 - a^2*x^2)*ArcTanh[a*x]^2,x]
Output:
(x^3*ArcTanh[a*x]^2)/3 - (2*a*(-(((x^2*ArcTanh[a*x])/2 - (a*(-(x/a^2) + Ar cTanh[a*x]/a^3))/2)/a^2) + (-1/2*ArcTanh[a*x]^2/a^2 + ((ArcTanh[a*x]*Log[2 /(1 - a*x)])/a + PolyLog[2, 1 - 2/(1 - a*x)]/(2*a))/a)/a^2))/3 - a^2*((x^5 *ArcTanh[a*x]^2)/5 - (2*a*(-(((x^4*ArcTanh[a*x])/4 - (a*(-(x/a^4) - x^3/(3 *a^2) + ArcTanh[a*x]/a^5))/4)/a^2) + (-(((x^2*ArcTanh[a*x])/2 - (a*(-(x/a^ 2) + ArcTanh[a*x]/a^3))/2)/a^2) + (-1/2*ArcTanh[a*x]^2/a^2 + ((ArcTanh[a*x ]*Log[2/(1 - a*x)])/a + PolyLog[2, 1 - 2/(1 - a*x)]/(2*a))/a)/a^2)/a^2))/5 )
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[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol ] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c *(p/e) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[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_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTanh[c* x])^p, x], x] - Simp[d*(f^2/e) Int[(f*x)^(m - 2)*((a + b*ArcTanh[c*x])^p/ (d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ (c*d) Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] - Simp[c^2*(d/f^2) Int[(f*x)^(m + 2)*(d + e*x ^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ [p, 1] && IntegerQ[q]))
Time = 0.43 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.30
| method | result | size |
| derivativedivides | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{5} x^{5}}{5}+\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{3} x^{3}}{3}-\frac {a^{4} x^{4} \operatorname {arctanh}\left (a x \right )}{10}+\frac {2 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{15}+\frac {2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{15}+\frac {2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{15}+\frac {\ln \left (a x -1\right )^{2}}{30}-\frac {2 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{15}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {\ln \left (a x +1\right )^{2}}{30}-\frac {a^{3} x^{3}}{30}+\frac {a x}{30}+\frac {\ln \left (a x -1\right )}{60}-\frac {\ln \left (a x +1\right )}{60}}{a^{3}}\) | \(179\) |
| default | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{5} x^{5}}{5}+\frac {\operatorname {arctanh}\left (a x \right )^{2} a^{3} x^{3}}{3}-\frac {a^{4} x^{4} \operatorname {arctanh}\left (a x \right )}{10}+\frac {2 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )}{15}+\frac {2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{15}+\frac {2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{15}+\frac {\ln \left (a x -1\right )^{2}}{30}-\frac {2 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{15}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {\ln \left (a x +1\right )^{2}}{30}-\frac {a^{3} x^{3}}{30}+\frac {a x}{30}+\frac {\ln \left (a x -1\right )}{60}-\frac {\ln \left (a x +1\right )}{60}}{a^{3}}\) | \(179\) |
| parts | \(-\frac {a^{2} x^{5} \operatorname {arctanh}\left (a x \right )^{2}}{5}+\frac {x^{3} \operatorname {arctanh}\left (a x \right )^{2}}{3}-\frac {a \,x^{4} \operatorname {arctanh}\left (a x \right )}{10}+\frac {2 x^{2} \operatorname {arctanh}\left (a x \right )}{15 a}+\frac {2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{15 a^{3}}+\frac {2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{15 a^{3}}+\frac {\ln \left (a x -1\right )^{2}-4 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )-2 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )+2 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )-\ln \left (a x +1\right )^{2}-a^{3} x^{3}+a x +\frac {\ln \left (a x -1\right )}{2}-\frac {\ln \left (a x +1\right )}{2}}{30 a^{3}}\) | \(179\) |
| risch | \(-\frac {x^{3}}{30}-\frac {443}{3375 a^{3}}-\frac {a \ln \left (a x +1\right ) x^{4}}{20}-\frac {\ln \left (a x +1\right ) x^{2}}{60 a}-\frac {\ln \left (a x +1\right ) x}{6 a^{2}}-\frac {a^{2} \ln \left (a x +1\right )^{2} x^{5}}{20}-\frac {x^{3} \ln \left (-a x +1\right )}{18}-\frac {7 \ln \left (a x +1\right )}{45 a^{3}}-\frac {\ln \left (a x +1\right ) x^{3}}{18}+\frac {\ln \left (a x +1\right )^{2}}{30 a^{3}}+\frac {\ln \left (a x +1\right )^{2} x^{3}}{12}+\frac {a \ln \left (-a x +1\right ) x^{4}}{20}+\frac {\ln \left (-a x +1\right ) x^{2}}{60 a}-\frac {\ln \left (-a x +1\right ) x}{6 a^{2}}-\frac {a^{2} \ln \left (-a x +1\right )^{2} x^{5}}{20}-\frac {34 \ln \left (-a x +1\right )}{225 a^{3}}+\frac {\ln \left (-a x +1\right )^{2} x^{3}}{12}-\frac {\ln \left (-a x +1\right )^{2}}{30 a^{3}}+\frac {x}{30 a^{2}}+\frac {a^{2} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{5}}{10}-\frac {2 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{15 a^{3}}-\frac {31 \ln \left (a x -1\right )}{225 a^{3}}+\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{5 a^{3}}-\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{5 a^{3}}+\frac {\ln \left (-a x +1\right ) \ln \left (a x +1\right )}{10 a^{3}}-\frac {\left (\left (-\frac {1}{9}+\frac {\ln \left (a x +1\right )}{3}\right ) \left (a x +1\right )^{3}+\left (\frac {1}{2}-\ln \left (a x +1\right )\right ) \left (a x +1\right )^{2}+\left (-1+\ln \left (a x +1\right )\right ) \left (a x +1\right )\right ) \ln \left (-a x +1\right )}{2 a^{3}}+\frac {\left (a x +1\right ) \ln \left (a x +1\right )}{6 a^{3}}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{3 a^{3}}-\frac {\left (a x +1\right )^{2} \ln \left (a x +1\right )}{12 a^{3}}+\frac {\left (a x +1\right )^{3} \ln \left (a x +1\right )}{18 a^{3}}\) | \(472\) |
Input:
int(x^2*(-a^2*x^2+1)*arctanh(a*x)^2,x,method=_RETURNVERBOSE)
Output:
1/a^3*(-1/5*arctanh(a*x)^2*a^5*x^5+1/3*arctanh(a*x)^2*a^3*x^3-1/10*a^4*x^4 *arctanh(a*x)+2/15*a^2*x^2*arctanh(a*x)+2/15*arctanh(a*x)*ln(a*x-1)+2/15*a rctanh(a*x)*ln(a*x+1)+1/30*ln(a*x-1)^2-2/15*dilog(1/2*a*x+1/2)-1/15*ln(a*x -1)*ln(1/2*a*x+1/2)+1/15*(ln(a*x+1)-ln(1/2*a*x+1/2))*ln(-1/2*a*x+1/2)-1/30 *ln(a*x+1)^2-1/30*a^3*x^3+1/30*a*x+1/60*ln(a*x-1)-1/60*ln(a*x+1))
\[ \int x^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx=\int { -{\left (a^{2} x^{2} - 1\right )} x^{2} \operatorname {artanh}\left (a x\right )^{2} \,d x } \] Input:
integrate(x^2*(-a^2*x^2+1)*arctanh(a*x)^2,x, algorithm="fricas")
Output:
integral(-(a^2*x^4 - x^2)*arctanh(a*x)^2, x)
\[ \int x^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx=- \int \left (- x^{2} \operatorname {atanh}^{2}{\left (a x \right )}\right )\, dx - \int a^{2} x^{4} \operatorname {atanh}^{2}{\left (a x \right )}\, dx \] Input:
integrate(x**2*(-a**2*x**2+1)*atanh(a*x)**2,x)
Output:
-Integral(-x**2*atanh(a*x)**2, x) - Integral(a**2*x**4*atanh(a*x)**2, x)
Time = 0.04 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.25 \[ \int x^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx=-\frac {1}{60} \, a^{2} {\left (\frac {2 \, a^{3} x^{3} - 2 \, a x + 2 \, \log \left (a x + 1\right )^{2} - 4 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 2 \, \log \left (a x - 1\right )^{2} - \log \left (a x - 1\right )}{a^{5}} + \frac {8 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{5}} + \frac {\log \left (a x + 1\right )}{a^{5}}\right )} - \frac {1}{30} \, a {\left (\frac {3 \, a^{2} x^{4} - 4 \, x^{2}}{a^{2}} - \frac {4 \, \log \left (a x + 1\right )}{a^{4}} - \frac {4 \, \log \left (a x - 1\right )}{a^{4}}\right )} \operatorname {artanh}\left (a x\right ) - \frac {1}{15} \, {\left (3 \, a^{2} x^{5} - 5 \, x^{3}\right )} \operatorname {artanh}\left (a x\right )^{2} \] Input:
integrate(x^2*(-a^2*x^2+1)*arctanh(a*x)^2,x, algorithm="maxima")
Output:
-1/60*a^2*((2*a^3*x^3 - 2*a*x + 2*log(a*x + 1)^2 - 4*log(a*x + 1)*log(a*x - 1) - 2*log(a*x - 1)^2 - log(a*x - 1))/a^5 + 8*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a^5 + log(a*x + 1)/a^5) - 1/30*a*((3*a^2*x ^4 - 4*x^2)/a^2 - 4*log(a*x + 1)/a^4 - 4*log(a*x - 1)/a^4)*arctanh(a*x) - 1/15*(3*a^2*x^5 - 5*x^3)*arctanh(a*x)^2
\[ \int x^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx=\int { -{\left (a^{2} x^{2} - 1\right )} x^{2} \operatorname {artanh}\left (a x\right )^{2} \,d x } \] Input:
integrate(x^2*(-a^2*x^2+1)*arctanh(a*x)^2,x, algorithm="giac")
Output:
integrate(-(a^2*x^2 - 1)*x^2*arctanh(a*x)^2, x)
Timed out. \[ \int x^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx=-\int x^2\,{\mathrm {atanh}\left (a\,x\right )}^2\,\left (a^2\,x^2-1\right ) \,d x \] Input:
int(-x^2*atanh(a*x)^2*(a^2*x^2 - 1),x)
Output:
-int(x^2*atanh(a*x)^2*(a^2*x^2 - 1), x)
\[ \int x^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2 \, dx=\frac {-6 \mathit {atanh} \left (a x \right )^{2} a^{5} x^{5}+10 \mathit {atanh} \left (a x \right )^{2} a^{3} x^{3}-4 \mathit {atanh} \left (a x \right )^{2} a x -3 \mathit {atanh} \left (a x \right ) a^{4} x^{4}+4 \mathit {atanh} \left (a x \right ) a^{2} x^{2}-\mathit {atanh} \left (a x \right )+4 \left (\int \mathit {atanh} \left (a x \right )^{2}d x \right ) a -a^{3} x^{3}+a x}{30 a^{3}} \] Input:
int(x^2*(-a^2*x^2+1)*atanh(a*x)^2,x)
Output:
( - 6*atanh(a*x)**2*a**5*x**5 + 10*atanh(a*x)**2*a**3*x**3 - 4*atanh(a*x)* *2*a*x - 3*atanh(a*x)*a**4*x**4 + 4*atanh(a*x)*a**2*x**2 - atanh(a*x) + 4* int(atanh(a*x)**2,x)*a - a**3*x**3 + a*x)/(30*a**3)