Integrand size = 18, antiderivative size = 304 \[ \int x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\frac {19 b^3 \sqrt {x}}{30 c^5}+\frac {b^3 x^{3/2}}{30 c^3}-\frac {19 b^3 \text {arctanh}\left (c \sqrt {x}\right )}{30 c^6}+\frac {8 b^2 x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{15 c^4}+\frac {b^2 x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{10 c^2}+\frac {23 b \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{15 c^6}+\frac {b \sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{c^5}+\frac {b x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{3 c^3}+\frac {b x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{5 c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{3 c^6}+\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {46 b^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{15 c^6}-\frac {23 b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{15 c^6} \]
1/30*b^3*x^(3/2)/c^3-19/30*b^3*arctanh(c*x^(1/2))/c^6+8/15*b^2*x*(a+b*arct anh(c*x^(1/2)))/c^4+1/10*b^2*x^2*(a+b*arctanh(c*x^(1/2)))/c^2+23/15*b*(a+b *arctanh(c*x^(1/2)))^2/c^6+1/3*b*x^(3/2)*(a+b*arctanh(c*x^(1/2)))^2/c^3+1/ 5*b*x^(5/2)*(a+b*arctanh(c*x^(1/2)))^2/c-1/3*(a+b*arctanh(c*x^(1/2)))^3/c^ 6+1/3*x^3*(a+b*arctanh(c*x^(1/2)))^3-46/15*b^2*(a+b*arctanh(c*x^(1/2)))*ln (2/(1-c*x^(1/2)))/c^6-23/15*b^3*polylog(2,1-2/(1-c*x^(1/2)))/c^6+19/30*b^3 *x^(1/2)/c^5+b*(a+b*arctanh(c*x^(1/2)))^2*x^(1/2)/c^5
Time = 0.53 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.15 \[ \int x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\frac {-19 a b^2+30 a^2 b c \sqrt {x}+19 b^3 c \sqrt {x}+16 a b^2 c^2 x+10 a^2 b c^3 x^{3/2}+b^3 c^3 x^{3/2}+3 a b^2 c^4 x^2+6 a^2 b c^5 x^{5/2}+10 a^3 c^6 x^3+2 b^2 \left (b \left (-23+15 c \sqrt {x}+5 c^3 x^{3/2}+3 c^5 x^{5/2}\right )+15 a \left (-1+c^6 x^3\right )\right ) \text {arctanh}\left (c \sqrt {x}\right )^2+10 b^3 \left (-1+c^6 x^3\right ) \text {arctanh}\left (c \sqrt {x}\right )^3+b \text {arctanh}\left (c \sqrt {x}\right ) \left (30 a^2 c^6 x^3+4 a b c \sqrt {x} \left (15+5 c^2 x+3 c^4 x^2\right )+b^2 \left (-19+16 c^2 x+3 c^4 x^2\right )-92 b^2 \log \left (1+e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )+15 a^2 b \log \left (1-c \sqrt {x}\right )-15 a^2 b \log \left (1+c \sqrt {x}\right )+46 a b^2 \log \left (1-c^2 x\right )+46 b^3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )}{30 c^6} \]
(-19*a*b^2 + 30*a^2*b*c*Sqrt[x] + 19*b^3*c*Sqrt[x] + 16*a*b^2*c^2*x + 10*a ^2*b*c^3*x^(3/2) + b^3*c^3*x^(3/2) + 3*a*b^2*c^4*x^2 + 6*a^2*b*c^5*x^(5/2) + 10*a^3*c^6*x^3 + 2*b^2*(b*(-23 + 15*c*Sqrt[x] + 5*c^3*x^(3/2) + 3*c^5*x ^(5/2)) + 15*a*(-1 + c^6*x^3))*ArcTanh[c*Sqrt[x]]^2 + 10*b^3*(-1 + c^6*x^3 )*ArcTanh[c*Sqrt[x]]^3 + b*ArcTanh[c*Sqrt[x]]*(30*a^2*c^6*x^3 + 4*a*b*c*Sq rt[x]*(15 + 5*c^2*x + 3*c^4*x^2) + b^2*(-19 + 16*c^2*x + 3*c^4*x^2) - 92*b ^2*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) + 15*a^2*b*Log[1 - c*Sqrt[x]] - 15* a^2*b*Log[1 + c*Sqrt[x]] + 46*a*b^2*Log[1 - c^2*x] + 46*b^3*PolyLog[2, -E^ (-2*ArcTanh[c*Sqrt[x]])])/(30*c^6)
Leaf count is larger than twice the leaf count of optimal. \(611\) vs. \(2(304)=608\).
Time = 3.53 (sec) , antiderivative size = 611, normalized size of antiderivative = 2.01, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6454, 6452, 6542, 6452, 6542, 6452, 254, 2009, 6542, 6436, 6452, 262, 219, 6510, 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.
| \(\displaystyle \int x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx\) |
| \(\Big \downarrow \) 6454 |
| \(\displaystyle 2 \int x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3d\sqrt {x}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {1}{2} b c \int \frac {x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {1}{2} b c \left (\frac {\int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2d\sqrt {x}}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {1}{2} b c \left (\frac {\int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \int \frac {x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {1}{2} b c \left (\frac {\frac {\int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \left (\frac {\int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )d\sqrt {x}}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {1}{2} b c \left (\frac {\frac {\int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \left (\frac {\int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \int \frac {x^2}{1-c^2 x}d\sqrt {x}}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {1}{2} b c \left (\frac {\frac {\int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \left (\frac {\int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \int \left (-\frac {x}{c^2}+\frac {1}{c^4 \left (1-c^2 x\right )}-\frac {1}{c^4}\right )d\sqrt {x}}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {1}{2} b c \left (\frac {\frac {\int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \left (\frac {\int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {1}{2} b c \left (\frac {\frac {\frac {\int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int \sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )d\sqrt {x}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \left (\frac {\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int \sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6436 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {1}{2} b c \left (\frac {\frac {\frac {\int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-2 b c \int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int \sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )d\sqrt {x}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \left (\frac {\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int \sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {1}{2} b c \left (\frac {\frac {\frac {\int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-2 b c \int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \int \frac {x}{1-c^2 x}d\sqrt {x}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \left (\frac {\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \int \frac {x}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {1}{2} b c \left (\frac {\frac {\frac {\int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-2 b c \int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\int \frac {1}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\sqrt {x}}{c^2}\right )}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \left (\frac {\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\int \frac {1}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\sqrt {x}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {1}{2} b c \left (\frac {\frac {\frac {\int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-2 b c \int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \left (\frac {\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {1}{2} b c \left (\frac {\frac {\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{3 b c^3}-\frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-2 b c \int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \left (\frac {\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {1}{2} b c \left (\frac {\frac {\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{3 b c^3}-\frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-2 b c \left (\frac {\int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c \sqrt {x}}d\sqrt {x}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\frac {\int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c \sqrt {x}}d\sqrt {x}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \left (\frac {\frac {\frac {\int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c \sqrt {x}}d\sqrt {x}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {1}{2} b c \left (\frac {\frac {\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{3 b c^3}-\frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-2 b c \left (\frac {\frac {\log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{c}-b \int \frac {\log \left (\frac {2}{1-c \sqrt {x}}\right )}{1-c^2 x}d\sqrt {x}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\frac {\frac {\log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{c}-b \int \frac {\log \left (\frac {2}{1-c \sqrt {x}}\right )}{1-c^2 x}d\sqrt {x}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \left (\frac {\frac {\frac {\frac {\log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{c}-b \int \frac {\log \left (\frac {2}{1-c \sqrt {x}}\right )}{1-c^2 x}d\sqrt {x}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {1}{2} b c \left (\frac {\frac {\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{3 b c^3}-\frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-2 b c \left (\frac {\frac {b \int \frac {\log \left (\frac {2}{1-c \sqrt {x}}\right )}{1-\frac {2}{1-c \sqrt {x}}}d\frac {1}{1-c \sqrt {x}}}{c}+\frac {\log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{c}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\frac {\frac {b \int \frac {\log \left (\frac {2}{1-c \sqrt {x}}\right )}{1-\frac {2}{1-c \sqrt {x}}}d\frac {1}{1-c \sqrt {x}}}{c}+\frac {\log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{c}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \left (\frac {\frac {\frac {\frac {b \int \frac {\log \left (\frac {2}{1-c \sqrt {x}}\right )}{1-\frac {2}{1-c \sqrt {x}}}d\frac {1}{1-c \sqrt {x}}}{c}+\frac {\log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{c}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {1}{2} b c \left (\frac {\frac {\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{3 b c^3}-\frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-2 b c \left (\frac {\frac {\log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{c}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{2 c}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\frac {\frac {\log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{c}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{2 c}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \left (\frac {\frac {\frac {\frac {\log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{c}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{2 c}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
2*((x^3*(a + b*ArcTanh[c*Sqrt[x]])^3)/6 - (b*c*(-(((x^(5/2)*(a + b*ArcTanh [c*Sqrt[x]])^2)/5 - (2*b*c*(-(((x^2*(a + b*ArcTanh[c*Sqrt[x]]))/4 - (b*c*( -(Sqrt[x]/c^4) - x^(3/2)/(3*c^2) + ArcTanh[c*Sqrt[x]]/c^5))/4)/c^2) + (-(( (x*(a + b*ArcTanh[c*Sqrt[x]]))/2 - (b*c*(-(Sqrt[x]/c^2) + ArcTanh[c*Sqrt[x ]]/c^3))/2)/c^2) + (-1/2*(a + b*ArcTanh[c*Sqrt[x]])^2/(b*c^2) + (((a + b*A rcTanh[c*Sqrt[x]])*Log[2/(1 - c*Sqrt[x])])/c + (b*PolyLog[2, 1 - 2/(1 - c* Sqrt[x])])/(2*c))/c)/c^2)/c^2))/5)/c^2) + (-(((x^(3/2)*(a + b*ArcTanh[c*Sq rt[x]])^2)/3 - (2*b*c*(-(((x*(a + b*ArcTanh[c*Sqrt[x]]))/2 - (b*c*(-(Sqrt[ x]/c^2) + ArcTanh[c*Sqrt[x]]/c^3))/2)/c^2) + (-1/2*(a + b*ArcTanh[c*Sqrt[x ]])^2/(b*c^2) + (((a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 - c*Sqrt[x])])/c + ( b*PolyLog[2, 1 - 2/(1 - c*Sqrt[x])])/(2*c))/c)/c^2))/3)/c^2) + ((a + b*Arc Tanh[c*Sqrt[x]])^3/(3*b*c^3) - (Sqrt[x]*(a + b*ArcTanh[c*Sqrt[x]])^2 - 2*b *c*(-1/2*(a + b*ArcTanh[c*Sqrt[x]])^2/(b*c^2) + (((a + b*ArcTanh[c*Sqrt[x] ])*Log[2/(1 - c*Sqrt[x])])/c + (b*PolyLog[2, 1 - 2/(1 - c*Sqrt[x])])/(2*c) )/c))/c^2)/c^2)/c^2))/2)
3.3.3.3.1 Defintions of rubi rules used
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_Symbol] :> Simp[x*(a + b*ArcTanh[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTanh[c*x^n]) ^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
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_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*ArcTanh[c*x])^p, x ], x, x^n], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 1] && IntegerQ[Simpl ify[(m + 1)/n]]
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_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b , c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.76 (sec) , antiderivative size = 1264, normalized size of antiderivative = 4.16
\[\text {Expression too large to display}\]
2/c^6*(1/8*I*b^3*Pi*csgn(I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))^2*csgn(I*(1+c*x ^(1/2))^2/(c^2*x-1))*arctanh(c*x^(1/2))^2+1/4*I*b^3*Pi*csgn(I*(1+c*x^(1/2) )/(-c^2*x+1)^(1/2))*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1))^2*arctanh(c*x^(1/2)) ^2+1/8*I*b^3*Pi*csgn(I/(1-(1+c*x^(1/2))^2/(c^2*x-1)))*csgn(I*(1+c*x^(1/2)) ^2/(c^2*x-1)/(1-(1+c*x^(1/2))^2/(c^2*x-1)))^2*arctanh(c*x^(1/2))^2-1/8*I*b ^3*Pi*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1))*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1)/( 1-(1+c*x^(1/2))^2/(c^2*x-1)))^2*arctanh(c*x^(1/2))^2-1/3*b^3-1/8*I*b^3*Pi* csgn(I/(1-(1+c*x^(1/2))^2/(c^2*x-1)))*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1))*cs gn(I*(1+c*x^(1/2))^2/(c^2*x-1)/(1-(1+c*x^(1/2))^2/(c^2*x-1)))*arctanh(c*x^ (1/2))^2+1/6*c^6*x^3*a^3+3*a*b^2*(1/6*c^6*x^3*arctanh(c*x^(1/2))^2+1/15*ar ctanh(c*x^(1/2))*c^5*x^(5/2)+1/9*arctanh(c*x^(1/2))*c^3*x^(3/2)+1/3*arctan h(c*x^(1/2))*c*x^(1/2)+1/6*arctanh(c*x^(1/2))*ln(c*x^(1/2)-1)-1/6*arctanh( c*x^(1/2))*ln(1+c*x^(1/2))-1/12*ln(c*x^(1/2)-1)*ln(1/2*c*x^(1/2)+1/2)+1/24 *ln(c*x^(1/2)-1)^2+1/24*ln(1+c*x^(1/2))^2-1/12*(ln(1+c*x^(1/2))-ln(1/2*c*x ^(1/2)+1/2))*ln(-1/2*c*x^(1/2)+1/2)+1/60*c^4*x^2+4/45*c^2*x+23/90*ln(c*x^( 1/2)-1)+23/90*ln(1+c*x^(1/2)))+3*a^2*b*(1/6*c^6*x^3*arctanh(c*x^(1/2))+1/3 0*c^5*x^(5/2)+1/18*c^3*x^(3/2)+1/6*c*x^(1/2)+1/12*ln(c*x^(1/2)-1)-1/12*ln( 1+c*x^(1/2)))+1/4*I*b^3*Pi*csgn(I/(1-(1+c*x^(1/2))^2/(c^2*x-1)))^2*arctanh (c*x^(1/2))^2-1/4*I*b^3*Pi*csgn(I/(1-(1+c*x^(1/2))^2/(c^2*x-1)))^3*arctanh (c*x^(1/2))^2+1/8*I*b^3*Pi*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1))^3*arctanh(...
\[ \int x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{3} x^{2} \,d x } \]
integral(b^3*x^2*arctanh(c*sqrt(x))^3 + 3*a*b^2*x^2*arctanh(c*sqrt(x))^2 + 3*a^2*b*x^2*arctanh(c*sqrt(x)) + a^3*x^2, x)
\[ \int x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\int x^{2} \left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )^{3}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 1579 vs. \(2 (243) = 486\).
Time = 0.79 (sec) , antiderivative size = 1579, normalized size of antiderivative = 5.19 \[ \int x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\text {Too large to display} \]
1/3*a^3*x^3 - 1/720*a*b^2*c*((20*c^5*x^3 + 39*c^3*x^2 + 138*c*x - 6*(10*c^ 5*x^3 + 12*c^4*x^(5/2) + 15*c^3*x^2 + 20*c^2*x^(3/2) + 30*c*x + 60*sqrt(x) )*log(c*sqrt(x) + 1))/c^6 - 222*log(c*sqrt(x) + 1)/c^7 - 222*log(c*sqrt(x) - 1)/c^7) - 1/120*(60*x^3*log(c*sqrt(x) + 1) - c*((10*c^5*x^3 - 12*c^4*x^ (5/2) + 15*c^3*x^2 - 20*c^2*x^(3/2) + 30*c*x - 60*sqrt(x))/c^6 + 60*log(c* sqrt(x) + 1)/c^7))*a*b^2*log(-c*sqrt(x) + 1) + 1/120*(60*x^3*log(c*sqrt(x) + 1) - c*((10*c^5*x^3 - 12*c^4*x^(5/2) + 15*c^3*x^2 - 20*c^2*x^(3/2) + 30 *c*x - 60*sqrt(x))/c^6 + 60*log(c*sqrt(x) + 1)/c^7))*a^2*b - 1/120*(60*x^3 *log(-c*sqrt(x) + 1) - c*((10*c^5*x^3 + 12*c^4*x^(5/2) + 15*c^3*x^2 + 20*c ^2*x^(3/2) + 30*c*x + 60*sqrt(x))/c^6 + 60*log(c*sqrt(x) - 1)/c^7))*a^2*b + 1/7200*(100*(18*log(-c*sqrt(x) + 1)^2 - 6*log(-c*sqrt(x) + 1) + 1)*(c*sq rt(x) - 1)^6 + 432*(25*log(-c*sqrt(x) + 1)^2 - 10*log(-c*sqrt(x) + 1) + 2) *(c*sqrt(x) - 1)^5 + 3375*(8*log(-c*sqrt(x) + 1)^2 - 4*log(-c*sqrt(x) + 1) + 1)*(c*sqrt(x) - 1)^4 + 4000*(9*log(-c*sqrt(x) + 1)^2 - 6*log(-c*sqrt(x) + 1) + 2)*(c*sqrt(x) - 1)^3 + 13500*(2*log(-c*sqrt(x) + 1)^2 - 2*log(-c*s qrt(x) + 1) + 1)*(c*sqrt(x) - 1)^2 + 10800*(log(-c*sqrt(x) + 1)^2 - 2*log( -c*sqrt(x) + 1) + 2)*(c*sqrt(x) - 1))*a*b^2/c^6 - 1/864000*(1000*(36*log(- c*sqrt(x) + 1)^3 - 18*log(-c*sqrt(x) + 1)^2 + 6*log(-c*sqrt(x) + 1) - 1)*( c*sqrt(x) - 1)^6 + 1728*(125*log(-c*sqrt(x) + 1)^3 - 75*log(-c*sqrt(x) + 1 )^2 + 30*log(-c*sqrt(x) + 1) - 6)*(c*sqrt(x) - 1)^5 + 16875*(32*log(-c*...
\[ \int x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{3} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\int x^2\,{\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}^3 \,d x \]