Integrand size = 22, antiderivative size = 186 \[ \int x^3 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx=\frac {3 x \sqrt {1-a^2 x^2}}{112 a^3}+\frac {23 x^3 \sqrt {1-a^2 x^2}}{840 a}-\frac {1}{42} a x^5 \sqrt {1-a^2 x^2}+\frac {17 \arcsin (a x)}{560 a^4}-\frac {2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{35 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{35 a^2}+\frac {8}{35} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{7} a^2 x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x) \] Output:
3/112*x*(-a^2*x^2+1)^(1/2)/a^3+23/840*x^3*(-a^2*x^2+1)^(1/2)/a-1/42*a*x^5* (-a^2*x^2+1)^(1/2)+17/560*arcsin(a*x)/a^4-2/35*(-a^2*x^2+1)^(1/2)*arctanh( a*x)/a^4-1/35*x^2*(-a^2*x^2+1)^(1/2)*arctanh(a*x)/a^2+8/35*x^4*(-a^2*x^2+1 )^(1/2)*arctanh(a*x)-1/7*a^2*x^6*(-a^2*x^2+1)^(1/2)*arctanh(a*x)
Time = 0.06 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.42 \[ \int x^3 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx=\frac {a x \sqrt {1-a^2 x^2} \left (45+46 a^2 x^2-40 a^4 x^4\right )+51 \arcsin (a x)-48 \left (1-a^2 x^2\right )^{5/2} \left (2+5 a^2 x^2\right ) \text {arctanh}(a x)}{1680 a^4} \] Input:
Integrate[x^3*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x],x]
Output:
(a*x*Sqrt[1 - a^2*x^2]*(45 + 46*a^2*x^2 - 40*a^4*x^4) + 51*ArcSin[a*x] - 4 8*(1 - a^2*x^2)^(5/2)*(2 + 5*a^2*x^2)*ArcTanh[a*x])/(1680*a^4)
Leaf count is larger than twice the leaf count of optimal. \(566\) vs. \(2(186)=372\).
Time = 2.14 (sec) , antiderivative size = 566, normalized size of antiderivative = 3.04, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.864, Rules used = {6576, 6572, 262, 262, 223, 262, 223, 6578, 262, 223, 262, 223, 6556, 223, 6578, 262, 223, 6556, 223}
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.
| \(\displaystyle \int x^3 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx\) |
| \(\Big \downarrow \) 6576 |
| \(\displaystyle \int x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)dx-a^2 \int x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)dx\) |
| \(\Big \downarrow \) 6572 |
| \(\displaystyle \frac {1}{5} \int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-a^2 \left (\frac {1}{7} \int \frac {x^5 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-\frac {1}{7} a \int \frac {x^6}{\sqrt {1-a^2 x^2}}dx+\frac {1}{7} x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x)\right )-\frac {1}{5} a \int \frac {x^4}{\sqrt {1-a^2 x^2}}dx+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{5} \int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-a^2 \left (\frac {1}{7} \int \frac {x^5 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-\frac {1}{7} a \left (\frac {5 \int \frac {x^4}{\sqrt {1-a^2 x^2}}dx}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2}}{6 a^2}\right )+\frac {1}{7} x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x)\right )-\frac {1}{5} a \left (\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{5} \int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-a^2 \left (\frac {1}{7} \int \frac {x^5 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-\frac {1}{7} a \left (\frac {5 \left (\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2}}{6 a^2}\right )+\frac {1}{7} x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x)\right )-\frac {1}{5} a \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{5} \int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-a^2 \left (\frac {1}{7} \int \frac {x^5 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-\frac {1}{7} a \left (\frac {5 \left (\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2}}{6 a^2}\right )+\frac {1}{7} x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x)\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{5} \int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-a^2 \left (\frac {1}{7} \int \frac {x^5 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-\frac {1}{7} a \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2}}{6 a^2}\right )+\frac {1}{7} x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x)\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{5} \int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-a^2 \left (\frac {1}{7} \int \frac {x^5 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx+\frac {1}{7} x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{7} a \left (\frac {5 \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2}}{6 a^2}\right )\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\) |
| \(\Big \downarrow \) 6578 |
| \(\displaystyle \frac {1}{5} \left (\frac {2 \int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\int \frac {x^2}{\sqrt {1-a^2 x^2}}dx}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}\right )-a^2 \left (\frac {1}{7} \left (\frac {4 \int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}+\frac {\int \frac {x^4}{\sqrt {1-a^2 x^2}}dx}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 a^2}\right )+\frac {1}{7} x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{7} a \left (\frac {5 \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2}}{6 a^2}\right )\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{5} \left (\frac {2 \int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}\right )-a^2 \left (\frac {1}{7} \left (\frac {4 \int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}+\frac {\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 a^2}\right )+\frac {1}{7} x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{7} a \left (\frac {5 \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2}}{6 a^2}\right )\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{5} \left (\frac {2 \int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}\right )-a^2 \left (\frac {1}{7} \left (\frac {4 \int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}+\frac {\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 a^2}\right )+\frac {1}{7} x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{7} a \left (\frac {5 \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2}}{6 a^2}\right )\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{5} \left (\frac {2 \int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}\right )-a^2 \left (\frac {1}{7} \left (\frac {4 \int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}+\frac {\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 a^2}\right )+\frac {1}{7} x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{7} a \left (\frac {5 \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2}}{6 a^2}\right )\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{5} \left (\frac {2 \int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}\right )-a^2 \left (\frac {1}{7} \left (\frac {4 \int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 a^2}+\frac {\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}}{5 a}\right )+\frac {1}{7} x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{7} a \left (\frac {5 \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2}}{6 a^2}\right )\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle \frac {1}{5} \left (\frac {2 \left (\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}\right )-a^2 \left (\frac {1}{7} \left (\frac {4 \int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 a^2}+\frac {\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}}{5 a}\right )+\frac {1}{7} x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{7} a \left (\frac {5 \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2}}{6 a^2}\right )\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -a^2 \left (\frac {1}{7} \left (\frac {4 \int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 a^2}+\frac {\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}}{5 a}\right )+\frac {1}{7} x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{7} a \left (\frac {5 \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2}}{6 a^2}\right )\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)+\frac {1}{5} \left (\frac {2 \left (\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}\right )-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\) |
| \(\Big \downarrow \) 6578 |
| \(\displaystyle -a^2 \left (\frac {1}{7} \left (\frac {4 \left (\frac {2 \int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\int \frac {x^2}{\sqrt {1-a^2 x^2}}dx}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 a^2}+\frac {\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}}{5 a}\right )+\frac {1}{7} x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{7} a \left (\frac {5 \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2}}{6 a^2}\right )\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)+\frac {1}{5} \left (\frac {2 \left (\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}\right )-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle -a^2 \left (\frac {1}{7} \left (\frac {4 \left (\frac {2 \int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 a^2}+\frac {\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}}{5 a}\right )+\frac {1}{7} x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{7} a \left (\frac {5 \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2}}{6 a^2}\right )\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)+\frac {1}{5} \left (\frac {2 \left (\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}\right )-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -a^2 \left (\frac {1}{7} \left (\frac {4 \left (\frac {2 \int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 a^2}+\frac {\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}}{5 a}\right )+\frac {1}{7} x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{7} a \left (\frac {5 \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2}}{6 a^2}\right )\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)+\frac {1}{5} \left (\frac {2 \left (\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}\right )-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle -a^2 \left (\frac {1}{7} \left (\frac {4 \left (\frac {2 \left (\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 a^2}+\frac {\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}}{5 a}\right )+\frac {1}{7} x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{7} a \left (\frac {5 \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2}}{6 a^2}\right )\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)+\frac {1}{5} \left (\frac {2 \left (\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}\right )-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)+\frac {1}{5} \left (\frac {2 \left (\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}\right )-a^2 \left (\frac {1}{7} x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x)+\frac {1}{7} \left (-\frac {x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 a^2}+\frac {4 \left (\frac {2 \left (\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}\right )}{5 a^2}+\frac {\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}}{5 a}\right )-\frac {1}{7} a \left (\frac {5 \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2}}{6 a^2}\right )\right )-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\) |
Input:
Int[x^3*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x],x]
Output:
-1/5*(a*(-1/4*(x^3*Sqrt[1 - a^2*x^2])/a^2 + (3*(-1/2*(x*Sqrt[1 - a^2*x^2]) /a^2 + ArcSin[a*x]/(2*a^3)))/(4*a^2))) + (x^4*Sqrt[1 - a^2*x^2]*ArcTanh[a* x])/5 + ((-1/2*(x*Sqrt[1 - a^2*x^2])/a^2 + ArcSin[a*x]/(2*a^3))/(3*a) - (x ^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(3*a^2) + (2*(ArcSin[a*x]/a^2 - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/a^2))/(3*a^2))/5 - a^2*(-1/7*(a*(-1/6*(x^5*Sqrt[ 1 - a^2*x^2])/a^2 + (5*(-1/4*(x^3*Sqrt[1 - a^2*x^2])/a^2 + (3*(-1/2*(x*Sqr t[1 - a^2*x^2])/a^2 + ArcSin[a*x]/(2*a^3)))/(4*a^2)))/(6*a^2))) + (x^6*Sqr t[1 - a^2*x^2]*ArcTanh[a*x])/7 + ((-1/4*(x^3*Sqrt[1 - a^2*x^2])/a^2 + (3*( -1/2*(x*Sqrt[1 - a^2*x^2])/a^2 + ArcSin[a*x]/(2*a^3)))/(4*a^2))/(5*a) - (x ^4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(5*a^2) + (4*((-1/2*(x*Sqrt[1 - a^2*x^2 ])/a^2 + ArcSin[a*x]/(2*a^3))/(3*a) - (x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]) /(3*a^2) + (2*(ArcSin[a*x]/a^2 - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/a^2))/(3 *a^2)))/(5*a^2))/7)
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
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[((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_.))*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.) *(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcTanh[c *x])/(f*(m + 2))), x] + (Simp[d/(m + 2) Int[(f*x)^m*((a + b*ArcTanh[c*x]) /Sqrt[d + e*x^2]), x], x] - Simp[b*c*(d/(f*(m + 2))) Int[(f*x)^(m + 1)/Sq rt[d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && NeQ[m, -2]
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]))
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(-f)*(f*x)^(m - 1)*Sqrt[d + e*x^2]*((a + b*ArcTanh[c*x])^p/(c^2*d*m)), x] + (Simp[b*f*(p/(c*m)) Int[(f*x)^(m - 1 )*((a + b*ArcTanh[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] + Simp[f^2*((m - 1 )/(c^2*m)) Int[(f*x)^(m - 2)*((a + b*ArcTanh[c*x])^p/Sqrt[d + e*x^2]), x] , x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && GtQ[m, 1]
Result contains complex when optimal does not.
Time = 0.96 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.75
| method | result | size |
| default | \(-\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (240 \,\operatorname {arctanh}\left (a x \right ) a^{6} x^{6}+40 a^{5} x^{5}-384 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )-46 a^{3} x^{3}+48 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )-45 a x +96 \,\operatorname {arctanh}\left (a x \right )\right )}{1680 a^{4}}+\frac {17 i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}+i\right )}{560 a^{4}}-\frac {17 i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-i\right )}{560 a^{4}}\) | \(140\) |
Input:
int(x^3*(-a^2*x^2+1)^(3/2)*arctanh(a*x),x,method=_RETURNVERBOSE)
Output:
-1/1680/a^4*(-(a*x-1)*(a*x+1))^(1/2)*(240*arctanh(a*x)*a^6*x^6+40*a^5*x^5- 384*a^4*x^4*arctanh(a*x)-46*a^3*x^3+48*a^2*x^2*arctanh(a*x)-45*a*x+96*arct anh(a*x))+17/560*I*ln((a*x+1)/(-a^2*x^2+1)^(1/2)+I)/a^4-17/560*I*ln((a*x+1 )/(-a^2*x^2+1)^(1/2)-I)/a^4
Time = 0.11 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.57 \[ \int x^3 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx=-\frac {{\left (40 \, a^{5} x^{5} - 46 \, a^{3} x^{3} - 45 \, a x + 24 \, {\left (5 \, a^{6} x^{6} - 8 \, a^{4} x^{4} + a^{2} x^{2} + 2\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {-a^{2} x^{2} + 1} + 102 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{1680 \, a^{4}} \] Input:
integrate(x^3*(-a^2*x^2+1)^(3/2)*arctanh(a*x),x, algorithm="fricas")
Output:
-1/1680*((40*a^5*x^5 - 46*a^3*x^3 - 45*a*x + 24*(5*a^6*x^6 - 8*a^4*x^4 + a ^2*x^2 + 2)*log(-(a*x + 1)/(a*x - 1)))*sqrt(-a^2*x^2 + 1) + 102*arctan((sq rt(-a^2*x^2 + 1) - 1)/(a*x)))/a^4
\[ \int x^3 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx=\int x^{3} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}{\left (a x \right )}\, dx \] Input:
integrate(x**3*(-a**2*x**2+1)**(3/2)*atanh(a*x),x)
Output:
Integral(x**3*(-(a*x - 1)*(a*x + 1))**(3/2)*atanh(a*x), x)
Time = 0.12 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.88 \[ \int x^3 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx=-\frac {1}{1680} \, a {\left (\frac {5 \, {\left (\frac {8 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} x}{a^{2}} - \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{a^{2}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{a^{2}} - \frac {3 \, \arcsin \left (a x\right )}{a^{3}}\right )}}{a^{2}} - \frac {12 \, {\left (2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x + 3 \, \sqrt {-a^{2} x^{2} + 1} x + \frac {3 \, \arcsin \left (a x\right )}{a}\right )}}{a^{4}}\right )} - \frac {1}{35} \, {\left (\frac {5 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{a^{2}} + \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{a^{4}}\right )} \operatorname {artanh}\left (a x\right ) \] Input:
integrate(x^3*(-a^2*x^2+1)^(3/2)*arctanh(a*x),x, algorithm="maxima")
Output:
-1/1680*a*(5*(8*(-a^2*x^2 + 1)^(5/2)*x/a^2 - 2*(-a^2*x^2 + 1)^(3/2)*x/a^2 - 3*sqrt(-a^2*x^2 + 1)*x/a^2 - 3*arcsin(a*x)/a^3)/a^2 - 12*(2*(-a^2*x^2 + 1)^(3/2)*x + 3*sqrt(-a^2*x^2 + 1)*x + 3*arcsin(a*x)/a)/a^4) - 1/35*(5*(-a^ 2*x^2 + 1)^(5/2)*x^2/a^2 + 2*(-a^2*x^2 + 1)^(5/2)/a^4)*arctanh(a*x)
Exception generated. \[ \int x^3 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(-a^2*x^2+1)^(3/2)*arctanh(a*x),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int x^3 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx=\int x^3\,\mathrm {atanh}\left (a\,x\right )\,{\left (1-a^2\,x^2\right )}^{3/2} \,d x \] Input:
int(x^3*atanh(a*x)*(1 - a^2*x^2)^(3/2),x)
Output:
int(x^3*atanh(a*x)*(1 - a^2*x^2)^(3/2), x)
\[ \int x^3 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx=-\left (\int \sqrt {-a^{2} x^{2}+1}\, \mathit {atanh} \left (a x \right ) x^{5}d x \right ) a^{2}+\int \sqrt {-a^{2} x^{2}+1}\, \mathit {atanh} \left (a x \right ) x^{3}d x \] Input:
int(x^3*(-a^2*x^2+1)^(3/2)*atanh(a*x),x)
Output:
- int(sqrt( - a**2*x**2 + 1)*atanh(a*x)*x**5,x)*a**2 + int(sqrt( - a**2*x **2 + 1)*atanh(a*x)*x**3,x)