Integrand size = 18, antiderivative size = 374 \[ \int x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\frac {47 b^3 \sqrt {x}}{70 c^7}+\frac {23 b^3 x^{3/2}}{420 c^5}+\frac {b^3 x^{5/2}}{140 c^3}-\frac {47 b^3 \text {arctanh}\left (c \sqrt {x}\right )}{70 c^8}+\frac {71 b^2 x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{140 c^6}+\frac {9 b^2 x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{70 c^4}+\frac {b^2 x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{28 c^2}+\frac {44 b \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{35 c^8}+\frac {3 b \sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{4 c^7}+\frac {b x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{4 c^5}+\frac {3 b x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{20 c^3}+\frac {3 b x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{28 c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{4 c^8}+\frac {1}{4} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {88 b^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{35 c^8}-\frac {44 b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{35 c^8} \]
23/420*b^3*x^(3/2)/c^5+1/140*b^3*x^(5/2)/c^3-47/70*b^3*arctanh(c*x^(1/2))/ c^8+71/140*b^2*x*(a+b*arctanh(c*x^(1/2)))/c^6+9/70*b^2*x^2*(a+b*arctanh(c* x^(1/2)))/c^4+1/28*b^2*x^3*(a+b*arctanh(c*x^(1/2)))/c^2+44/35*b*(a+b*arcta nh(c*x^(1/2)))^2/c^8+1/4*b*x^(3/2)*(a+b*arctanh(c*x^(1/2)))^2/c^5+3/20*b*x ^(5/2)*(a+b*arctanh(c*x^(1/2)))^2/c^3+3/28*b*x^(7/2)*(a+b*arctanh(c*x^(1/2 )))^2/c-1/4*(a+b*arctanh(c*x^(1/2)))^3/c^8+1/4*x^4*(a+b*arctanh(c*x^(1/2)) )^3-88/35*b^2*(a+b*arctanh(c*x^(1/2)))*ln(2/(1-c*x^(1/2)))/c^8-44/35*b^3*p olylog(2,1-2/(1-c*x^(1/2)))/c^8+47/70*b^3*x^(1/2)/c^7+3/4*b*(a+b*arctanh(c *x^(1/2)))^2*x^(1/2)/c^7
Time = 0.79 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.12 \[ \int x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\frac {-564 a b^2+630 a^2 b c \sqrt {x}+564 b^3 c \sqrt {x}+426 a b^2 c^2 x+210 a^2 b c^3 x^{3/2}+46 b^3 c^3 x^{3/2}+108 a b^2 c^4 x^2+126 a^2 b c^5 x^{5/2}+6 b^3 c^5 x^{5/2}+30 a b^2 c^6 x^3+90 a^2 b c^7 x^{7/2}+210 a^3 c^8 x^4+6 b^2 \left (b \left (-176+105 c \sqrt {x}+35 c^3 x^{3/2}+21 c^5 x^{5/2}+15 c^7 x^{7/2}\right )+105 a \left (-1+c^8 x^4\right )\right ) \text {arctanh}\left (c \sqrt {x}\right )^2+210 b^3 \left (-1+c^8 x^4\right ) \text {arctanh}\left (c \sqrt {x}\right )^3+6 b \text {arctanh}\left (c \sqrt {x}\right ) \left (105 a^2 c^8 x^4+b^2 \left (-94+71 c^2 x+18 c^4 x^2+5 c^6 x^3\right )+2 a b c \sqrt {x} \left (105+35 c^2 x+21 c^4 x^2+15 c^6 x^3\right )-352 b^2 \log \left (1+e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )+315 a^2 b \log \left (1-c \sqrt {x}\right )-315 a^2 b \log \left (1+c \sqrt {x}\right )+1056 a b^2 \log \left (1-c^2 x\right )+1056 b^3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )}{840 c^8} \]
(-564*a*b^2 + 630*a^2*b*c*Sqrt[x] + 564*b^3*c*Sqrt[x] + 426*a*b^2*c^2*x + 210*a^2*b*c^3*x^(3/2) + 46*b^3*c^3*x^(3/2) + 108*a*b^2*c^4*x^2 + 126*a^2*b *c^5*x^(5/2) + 6*b^3*c^5*x^(5/2) + 30*a*b^2*c^6*x^3 + 90*a^2*b*c^7*x^(7/2) + 210*a^3*c^8*x^4 + 6*b^2*(b*(-176 + 105*c*Sqrt[x] + 35*c^3*x^(3/2) + 21* c^5*x^(5/2) + 15*c^7*x^(7/2)) + 105*a*(-1 + c^8*x^4))*ArcTanh[c*Sqrt[x]]^2 + 210*b^3*(-1 + c^8*x^4)*ArcTanh[c*Sqrt[x]]^3 + 6*b*ArcTanh[c*Sqrt[x]]*(1 05*a^2*c^8*x^4 + b^2*(-94 + 71*c^2*x + 18*c^4*x^2 + 5*c^6*x^3) + 2*a*b*c*S qrt[x]*(105 + 35*c^2*x + 21*c^4*x^2 + 15*c^6*x^3) - 352*b^2*Log[1 + E^(-2* ArcTanh[c*Sqrt[x]])]) + 315*a^2*b*Log[1 - c*Sqrt[x]] - 315*a^2*b*Log[1 + c *Sqrt[x]] + 1056*a*b^2*Log[1 - c^2*x] + 1056*b^3*PolyLog[2, -E^(-2*ArcTanh [c*Sqrt[x]])])/(840*c^8)
Leaf count is larger than twice the leaf count of optimal. \(949\) vs. \(2(374)=748\).
Time = 5.46 (sec) , antiderivative size = 949, normalized size of antiderivative = 2.54, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.222, Rules used = {6454, 6452, 6542, 6452, 6542, 6452, 254, 2009, 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.
| \(\displaystyle \int x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx\) |
| \(\Big \downarrow \) 6454 |
| \(\displaystyle 2 \int x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3d\sqrt {x}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \int \frac {x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\int \frac {x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2d\sqrt {x}}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\int \frac {x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{7} x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{7} b c \int \frac {x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\frac {\int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{7} x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{7} b c \left (\frac {\int \frac {x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )d\sqrt {x}}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\frac {\int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \int \frac {x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{7} x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{7} b c \left (\frac {\int \frac {x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \int \frac {x^3}{1-c^2 x}d\sqrt {x}}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\frac {\int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \int \frac {x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{7} x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{7} b c \left (\frac {\int \frac {x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \int \left (-\frac {x^2}{c^2}-\frac {x}{c^4}+\frac {1}{c^6 \left (1-c^2 x\right )}-\frac {1}{c^6}\right )d\sqrt {x}}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\frac {\int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \int \frac {x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{7} x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{7} b c \left (\frac {\int \frac {x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^7}-\frac {\sqrt {x}}{c^6}-\frac {x^{3/2}}{3 c^4}-\frac {x^{5/2}}{5 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\frac {\frac {\int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \left (\frac {\int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )d\sqrt {x}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{7} x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{7} b c \left (\frac {\frac {\int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^7}-\frac {\sqrt {x}}{c^6}-\frac {x^{3/2}}{3 c^4}-\frac {x^{5/2}}{5 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\frac {\frac {\int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \left (\frac {\int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \int \frac {x^2}{1-c^2 x}d\sqrt {x}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{7} x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{7} b c \left (\frac {\frac {\int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \int \frac {x^2}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^7}-\frac {\sqrt {x}}{c^6}-\frac {x^{3/2}}{3 c^4}-\frac {x^{5/2}}{5 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\frac {\frac {\int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \left (\frac {\int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \int \left (-\frac {x}{c^2}+\frac {1}{c^4 \left (1-c^2 x\right )}-\frac {1}{c^4}\right )d\sqrt {x}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{7} x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{7} b c \left (\frac {\frac {\int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \int \left (-\frac {x}{c^2}+\frac {1}{c^4 \left (1-c^2 x\right )}-\frac {1}{c^4}\right )d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^7}-\frac {\sqrt {x}}{c^6}-\frac {x^{3/2}}{3 c^4}-\frac {x^{5/2}}{5 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\frac {\frac {\int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{5} x^{5/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{5} b c \left (\frac {\int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{7} x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{7} b c \left (\frac {\frac {\int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{4} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}-\frac {\sqrt {x}}{c^4}-\frac {x^{3/2}}{3 c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^7}-\frac {\sqrt {x}}{c^6}-\frac {x^{3/2}}{3 c^4}-\frac {x^{5/2}}{5 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\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}}{c^2}-\frac {\frac {1}{7} x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{7} b c \left (\frac {\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}}{c^2}-\frac {\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^7}-\frac {\sqrt {x}}{c^6}-\frac {x^{3/2}}{3 c^4}-\frac {x^{5/2}}{5 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6436 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\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}}{c^2}-\frac {\frac {1}{7} x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{7} b c \left (\frac {\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}}{c^2}-\frac {\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^7}-\frac {\sqrt {x}}{c^6}-\frac {x^{3/2}}{3 c^4}-\frac {x^{5/2}}{5 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\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}}{c^2}-\frac {\frac {1}{7} x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{7} b c \left (\frac {\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}}{c^2}-\frac {\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^7}-\frac {\sqrt {x}}{c^6}-\frac {x^{3/2}}{3 c^4}-\frac {x^{5/2}}{5 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\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}}{c^2}-\frac {\frac {1}{7} x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{7} b c \left (\frac {\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}}{c^2}-\frac {\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^7}-\frac {\sqrt {x}}{c^6}-\frac {x^{3/2}}{3 c^4}-\frac {x^{5/2}}{5 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\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}}{c^2}-\frac {\frac {1}{7} x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{7} b c \left (\frac {\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}}{c^2}-\frac {\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^7}-\frac {\sqrt {x}}{c^6}-\frac {x^{3/2}}{3 c^4}-\frac {x^{5/2}}{5 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\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}}{c^2}-\frac {\frac {1}{7} x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{7} b c \left (\frac {\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}}{c^2}-\frac {\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^7}-\frac {\sqrt {x}}{c^6}-\frac {x^{3/2}}{3 c^4}-\frac {x^{5/2}}{5 c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\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 {x^{3/2}}{3 c^2}-\frac {\sqrt {x}}{c^4}+\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}\right )}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{7} x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{7} b c \left (\frac {\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 {x^{3/2}}{3 c^2}-\frac {\sqrt {x}}{c^4}+\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}\right )}{c^2}}{c^2}-\frac {\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (-\frac {x^{5/2}}{5 c^2}-\frac {x^{3/2}}{3 c^4}-\frac {\sqrt {x}}{c^6}+\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^7}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\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 {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\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 {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\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 {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\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 {x^{3/2}}{3 c^2}-\frac {\sqrt {x}}{c^4}+\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}\right )}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{7} x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{7} b c \left (\frac {\frac {\frac {\frac {\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\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 {x^{3/2}}{3 c^2}-\frac {\sqrt {x}}{c^4}+\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}\right )}{c^2}}{c^2}-\frac {\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (-\frac {x^{5/2}}{5 c^2}-\frac {x^{3/2}}{3 c^4}-\frac {\sqrt {x}}{c^6}+\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^7}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\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 {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c}+\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}}{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 {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c}+\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}}{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 {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c}+\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}}{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 {x^{3/2}}{3 c^2}-\frac {\sqrt {x}}{c^4}+\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}\right )}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{7} x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{7} b c \left (\frac {\frac {\frac {\frac {\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c}+\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}}{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 {x^{3/2}}{3 c^2}-\frac {\sqrt {x}}{c^4}+\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}\right )}{c^2}}{c^2}-\frac {\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (-\frac {x^{5/2}}{5 c^2}-\frac {x^{3/2}}{3 c^4}-\frac {\sqrt {x}}{c^6}+\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^7}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{8} b c \left (\frac {\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 {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\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 {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\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 {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\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 {x^{3/2}}{3 c^2}-\frac {\sqrt {x}}{c^4}+\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}\right )}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{7} x^{7/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{7} b c \left (\frac {\frac {\frac {\frac {\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\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 {x^{3/2}}{3 c^2}-\frac {\sqrt {x}}{c^4}+\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^5}\right )}{c^2}}{c^2}-\frac {\frac {1}{6} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (-\frac {x^{5/2}}{5 c^2}-\frac {x^{3/2}}{3 c^4}-\frac {\sqrt {x}}{c^6}+\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^7}\right )}{c^2}\right )}{c^2}\right )\right )\) |
2*((x^4*(a + b*ArcTanh[c*Sqrt[x]])^3)/8 - (3*b*c*(-(((x^(7/2)*(a + b*ArcTa nh[c*Sqrt[x]])^2)/7 - (2*b*c*(-(((x^3*(a + b*ArcTanh[c*Sqrt[x]]))/6 - (b*c *(-(Sqrt[x]/c^6) - x^(3/2)/(3*c^4) - x^(5/2)/(5*c^2) + ArcTanh[c*Sqrt[x]]/ c^7))/6)/c^2) + (-(((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*A rcTanh[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*Sq rt[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))/7)/c^2) + (-(((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*A rcTanh[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*Sq rt[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*Sqrt[x]])^2)/ 3 - (2*b*c*(-(((x*(a + b*ArcTanh[c*Sqrt[x]]))/2 - (b*c*(-(Sqrt[x]/c^2) + A rcTanh[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*ArcTanh[c*Sqrt [x]])^3/(3*b*c^3) - (Sqrt[x]*(a + b*ArcTanh[c*Sqrt[x]])^2 - 2*b*c*(-1/2...
3.3.2.3.1 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 = 3.41 (sec) , antiderivative size = 1341, normalized size of antiderivative = 3.59
\[\text {Expression too large to display}\]
2/c^8*(-11/30*b^3-3/16*I*b^3*Pi*arctanh(c*x^(1/2))^2+9/140*b^3*arctanh(c*x ^(1/2))*c^4*x^2+1/56*b^3*arctanh(c*x^(1/2))*c^6*x^3+3/56*b^3*arctanh(c*x^( 1/2))^2*c^7*x^(7/2)+3/40*b^3*arctanh(c*x^(1/2))^2*c^5*x^(5/2)+1/8*b^3*arct anh(c*x^(1/2))^2*c^3*x^(3/2)+3/8*b^3*arctanh(c*x^(1/2))^2*c*x^(1/2)+1/8*b^ 3*c^8*x^4*arctanh(c*x^(1/2))^3+71/280*b^3*arctanh(c*x^(1/2))*c^2*x+3/32*I* b^3*Pi*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1))^3*arctanh(c*x^(1/2))^2+3/32*I*b^3 *Pi*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1)/(1-(1+c*x^(1/2))^2/(c^2*x-1)))^3*arct anh(c*x^(1/2))^2+3/16*I*b^3*Pi*csgn(I/(1-(1+c*x^(1/2))^2/(c^2*x-1)))^2*arc tanh(c*x^(1/2))^2-3/16*I*b^3*Pi*csgn(I/(1-(1+c*x^(1/2))^2/(c^2*x-1)))^3*ar ctanh(c*x^(1/2))^2-3/32*I*b^3*Pi*csgn(I/(1-(1+c*x^(1/2))^2/(c^2*x-1)))*csg n(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)))*arctanh(c*x^(1/2))^2+3/32*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))^2-3/32*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*a rctanh(c*x^(1/2))^2+3/16*I*b^3*Pi*csgn(I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))*c sgn(I*(1+c*x^(1/2))^2/(c^2*x-1))^2*arctanh(c*x^(1/2))^2+3/32*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))*arc tanh(c*x^(1/2))^2-47/140*b^3*arctanh(c*x^(1/2))-1/8*b^3*arctanh(c*x^(1/2)) ^3+22/35*b^3*arctanh(c*x^(1/2))^2-44/35*b^3*dilog(1-I*(1+c*x^(1/2))/(-c...
\[ \int x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{3} x^{3} \,d x } \]
integral(b^3*x^3*arctanh(c*sqrt(x))^3 + 3*a*b^2*x^3*arctanh(c*sqrt(x))^2 + 3*a^2*b*x^3*arctanh(c*sqrt(x)) + a^3*x^3, x)
\[ \int x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\int x^{3} \left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )^{3}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 1972 vs. \(2 (297) = 594\).
Time = 0.85 (sec) , antiderivative size = 1972, normalized size of antiderivative = 5.27 \[ \int x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\text {Too large to display} \]
1/4*a^3*x^4 - 1/26880*a*b^2*c*((315*c^7*x^4 + 500*c^5*x^3 + 1002*c^3*x^2 + 3684*c*x - 12*(105*c^7*x^4 + 120*c^6*x^(7/2) + 140*c^5*x^3 + 168*c^4*x^(5 /2) + 210*c^3*x^2 + 280*c^2*x^(3/2) + 420*c*x + 840*sqrt(x))*log(c*sqrt(x) + 1))/c^8 - 6396*log(c*sqrt(x) + 1)/c^9 - 6396*log(c*sqrt(x) - 1)/c^9) - 1/2240*(840*x^4*log(c*sqrt(x) + 1) - c*((105*c^7*x^4 - 120*c^6*x^(7/2) + 1 40*c^5*x^3 - 168*c^4*x^(5/2) + 210*c^3*x^2 - 280*c^2*x^(3/2) + 420*c*x - 8 40*sqrt(x))/c^8 + 840*log(c*sqrt(x) + 1)/c^9))*a*b^2*log(-c*sqrt(x) + 1) + 1/2240*(840*x^4*log(c*sqrt(x) + 1) - c*((105*c^7*x^4 - 120*c^6*x^(7/2) + 140*c^5*x^3 - 168*c^4*x^(5/2) + 210*c^3*x^2 - 280*c^2*x^(3/2) + 420*c*x - 840*sqrt(x))/c^8 + 840*log(c*sqrt(x) + 1)/c^9))*a^2*b - 1/2240*(840*x^4*lo g(-c*sqrt(x) + 1) - c*((105*c^7*x^4 + 120*c^6*x^(7/2) + 140*c^5*x^3 + 168* c^4*x^(5/2) + 210*c^3*x^2 + 280*c^2*x^(3/2) + 420*c*x + 840*sqrt(x))/c^8 + 840*log(c*sqrt(x) - 1)/c^9))*a^2*b + 1/1881600*(11025*(32*log(-c*sqrt(x) + 1)^2 - 8*log(-c*sqrt(x) + 1) + 1)*(c*sqrt(x) - 1)^8 + 57600*(49*log(-c*s qrt(x) + 1)^2 - 14*log(-c*sqrt(x) + 1) + 2)*(c*sqrt(x) - 1)^7 + 548800*(18 *log(-c*sqrt(x) + 1)^2 - 6*log(-c*sqrt(x) + 1) + 1)*(c*sqrt(x) - 1)^6 + 79 0272*(25*log(-c*sqrt(x) + 1)^2 - 10*log(-c*sqrt(x) + 1) + 2)*(c*sqrt(x) - 1)^5 + 3087000*(8*log(-c*sqrt(x) + 1)^2 - 4*log(-c*sqrt(x) + 1) + 1)*(c*sq rt(x) - 1)^4 + 2195200*(9*log(-c*sqrt(x) + 1)^2 - 6*log(-c*sqrt(x) + 1) + 2)*(c*sqrt(x) - 1)^3 + 4939200*(2*log(-c*sqrt(x) + 1)^2 - 2*log(-c*sqrt...
\[ \int x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{3} x^{3} \,d x } \]
Timed out. \[ \int x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\int x^3\,{\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}^3 \,d x \]