Integrand size = 16, antiderivative size = 234 \[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\frac {b^3 \sqrt {x}}{2 c^3}-\frac {b^3 \text {arctanh}\left (c \sqrt {x}\right )}{2 c^4}+\frac {b^2 x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{2 c^2}+\frac {2 b \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{c^4}+\frac {3 b \sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 c^3}+\frac {b x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{2 c^4}+\frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {4 b^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c^4}-\frac {2 b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{c^4} \]
-1/2*b^3*arctanh(c*x^(1/2))/c^4+1/2*b^2*x*(a+b*arctanh(c*x^(1/2)))/c^2+2*b *(a+b*arctanh(c*x^(1/2)))^2/c^4+1/2*b*x^(3/2)*(a+b*arctanh(c*x^(1/2)))^2/c -1/2*(a+b*arctanh(c*x^(1/2)))^3/c^4+1/2*x^2*(a+b*arctanh(c*x^(1/2)))^3-4*b ^2*(a+b*arctanh(c*x^(1/2)))*ln(2/(1-c*x^(1/2)))/c^4-2*b^3*polylog(2,1-2/(1 -c*x^(1/2)))/c^4+1/2*b^3*x^(1/2)/c^3+3/2*b*(a+b*arctanh(c*x^(1/2)))^2*x^(1 /2)/c^3
Time = 0.38 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.22 \[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\frac {-2 a b^2+6 a^2 b c \sqrt {x}+2 b^3 c \sqrt {x}+2 a b^2 c^2 x+2 a^2 b c^3 x^{3/2}+2 a^3 c^4 x^2+2 b^2 \left (b \left (-4+3 c \sqrt {x}+c^3 x^{3/2}\right )+3 a \left (-1+c^4 x^2\right )\right ) \text {arctanh}\left (c \sqrt {x}\right )^2+2 b^3 \left (-1+c^4 x^2\right ) \text {arctanh}\left (c \sqrt {x}\right )^3+2 b \text {arctanh}\left (c \sqrt {x}\right ) \left (3 a^2 c^4 x^2+b^2 \left (-1+c^2 x\right )+2 a b c \sqrt {x} \left (3+c^2 x\right )-8 b^2 \log \left (1+e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )+3 a^2 b \log \left (1-c \sqrt {x}\right )-3 a^2 b \log \left (1+c \sqrt {x}\right )+8 a b^2 \log \left (1-c^2 x\right )+8 b^3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )}{4 c^4} \]
(-2*a*b^2 + 6*a^2*b*c*Sqrt[x] + 2*b^3*c*Sqrt[x] + 2*a*b^2*c^2*x + 2*a^2*b* c^3*x^(3/2) + 2*a^3*c^4*x^2 + 2*b^2*(b*(-4 + 3*c*Sqrt[x] + c^3*x^(3/2)) + 3*a*(-1 + c^4*x^2))*ArcTanh[c*Sqrt[x]]^2 + 2*b^3*(-1 + c^4*x^2)*ArcTanh[c* Sqrt[x]]^3 + 2*b*ArcTanh[c*Sqrt[x]]*(3*a^2*c^4*x^2 + b^2*(-1 + c^2*x) + 2* a*b*c*Sqrt[x]*(3 + c^2*x) - 8*b^2*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) + 3* a^2*b*Log[1 - c*Sqrt[x]] - 3*a^2*b*Log[1 + c*Sqrt[x]] + 8*a*b^2*Log[1 - c^ 2*x] + 8*b^3*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])])/(4*c^4)
Time = 2.10 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.52, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {6454, 6452, 6542, 6452, 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.
| \(\displaystyle \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx\) |
| \(\Big \downarrow \) 6454 |
| \(\displaystyle 2 \int x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3d\sqrt {x}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{4} b c \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{4} b c \left (\frac {\int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2d\sqrt {x}}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{4} b c \left (\frac {\int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \int \frac {x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{4} b c \left (\frac {\frac {\int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int \sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )d\sqrt {x}}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6436 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{4} b c \left (\frac {\frac {\int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-2 b c \int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int \sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )d\sqrt {x}}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{4} b c \left (\frac {\frac {\int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-2 b c \int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \int \frac {x}{1-c^2 x}d\sqrt {x}}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{4} b c \left (\frac {\frac {\int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-2 b c \int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\int \frac {1}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\sqrt {x}}{c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{4} b c \left (\frac {\frac {\int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-2 b c \int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{4} b c \left (\frac {\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{3 b c^3}-\frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-2 b c \int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\int \frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{4} b c \left (\frac {\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{3 b c^3}-\frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-2 b c \left (\frac {\int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c \sqrt {x}}d\sqrt {x}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\frac {\int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c \sqrt {x}}d\sqrt {x}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{4} b c \left (\frac {\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{3 b c^3}-\frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-2 b c \left (\frac {\frac {\log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{c}-b \int \frac {\log \left (\frac {2}{1-c \sqrt {x}}\right )}{1-c^2 x}d\sqrt {x}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\frac {\frac {\log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{c}-b \int \frac {\log \left (\frac {2}{1-c \sqrt {x}}\right )}{1-c^2 x}d\sqrt {x}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{4} b c \left (\frac {\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{3 b c^3}-\frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-2 b c \left (\frac {\frac {b \int \frac {\log \left (\frac {2}{1-c \sqrt {x}}\right )}{1-\frac {2}{1-c \sqrt {x}}}d\frac {1}{1-c \sqrt {x}}}{c}+\frac {\log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{c}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\frac {\frac {b \int \frac {\log \left (\frac {2}{1-c \sqrt {x}}\right )}{1-\frac {2}{1-c \sqrt {x}}}d\frac {1}{1-c \sqrt {x}}}{c}+\frac {\log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{c}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {3}{4} b c \left (\frac {\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{3 b c^3}-\frac {\sqrt {x} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-2 b c \left (\frac {\frac {\log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{c}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{2 c}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {2}{3} b c \left (\frac {\frac {\frac {\log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{c}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{2 c}}{c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )}{c^2}\right )}{c^2}\right )\right )\) |
2*((x^2*(a + b*ArcTanh[c*Sqrt[x]])^3)/4 - (3*b*c*(-(((x^(3/2)*(a + b*ArcTa nh[c*Sqrt[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*ArcTanh[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))/4)
3.3.4.3.1 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[((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 = 30.44 (sec) , antiderivative size = 1145, normalized size of antiderivative = 4.89
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1145\) |
| default | \(\text {Expression too large to display}\) | \(1145\) |
| parts | \(\text {Expression too large to display}\) | \(1147\) |
2/c^4*(1/4*a^3*c^4*x^2+b^3*(-1/4+arctanh(c*x^(1/2))^2+1/4*c*x^(1/2)+1/4*ar ctanh(c*x^(1/2))^2*c^3*x^(3/2)+3/4*arctanh(c*x^(1/2))^2*c*x^(1/2)-1/4*arct anh(c*x^(1/2))^3+3/8*arctanh(c*x^(1/2))^2*ln(c*x^(1/2)-1)-3/8*arctanh(c*x^ (1/2))^2*ln(1+c*x^(1/2))-2*arctanh(c*x^(1/2))*ln(1+I*(1+c*x^(1/2))/(-c^2*x +1)^(1/2))-2*arctanh(c*x^(1/2))*ln(1-I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))+3/4 *arctanh(c*x^(1/2))^2*ln((1+c*x^(1/2))/(-c^2*x+1)^(1/2))+3/16*I*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+3/8*I*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 -3/16*I*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+3/16*I*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))*arcta nh(c*x^(1/2))^2+3/16*I*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*arctanh(c*x^(1/2))^2+3/16*I*Pi*csgn(I*(1+c*x^(1/2))^2/(c ^2*x-1))^3*arctanh(c*x^(1/2))^2+3/8*I*Pi*csgn(I/(1-(1+c*x^(1/2))^2/(c^2*x- 1)))^2*arctanh(c*x^(1/2))^2-3/8*I*Pi*csgn(I/(1-(1+c*x^(1/2))^2/(c^2*x-1))) ^3*arctanh(c*x^(1/2))^2-1/4*arctanh(c*x^(1/2))*(1+c*x^(1/2))^2+1/2*(1+c*x^ (1/2))*arctanh(c*x^(1/2))-3/8*I*Pi*arctanh(c*x^(1/2))^2+1/4*c^4*x^2*arctan h(c*x^(1/2))^3+1/2*(c*x^(1/2)-1)*(1+c*x^(1/2))*arctanh(c*x^(1/2))-2*dilog( 1-I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))-2*dilog(1+I*(1+c*x^(1/2))/(-c^2*x+1...
\[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{3} x \,d x } \]
integral(b^3*x*arctanh(c*sqrt(x))^3 + 3*a*b^2*x*arctanh(c*sqrt(x))^2 + 3*a ^2*b*x*arctanh(c*sqrt(x)) + a^3*x, x)
\[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\int x \left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )^{3}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 1184 vs. \(2 (191) = 382\).
Time = 0.71 (sec) , antiderivative size = 1184, normalized size of antiderivative = 5.06 \[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\text {Too large to display} \]
1/2*a^3*x^2 - 1/32*a*b^2*c*((3*c^3*x^2 + 10*c*x - 2*(3*c^3*x^2 + 4*c^2*x^( 3/2) + 6*c*x + 12*sqrt(x))*log(c*sqrt(x) + 1))/c^4 - 14*log(c*sqrt(x) + 1) /c^5 - 14*log(c*sqrt(x) - 1)/c^5) - 1/16*(12*x^2*log(c*sqrt(x) + 1) - c*(( 3*c^3*x^2 - 4*c^2*x^(3/2) + 6*c*x - 12*sqrt(x))/c^4 + 12*log(c*sqrt(x) + 1 )/c^5))*a*b^2*log(-c*sqrt(x) + 1) + 1/16*(12*x^2*log(c*sqrt(x) + 1) - c*(( 3*c^3*x^2 - 4*c^2*x^(3/2) + 6*c*x - 12*sqrt(x))/c^4 + 12*log(c*sqrt(x) + 1 )/c^5))*a^2*b - 1/16*(12*x^2*log(-c*sqrt(x) + 1) - c*((3*c^3*x^2 + 4*c^2*x ^(3/2) + 6*c*x + 12*sqrt(x))/c^4 + 12*log(c*sqrt(x) - 1)/c^5))*a^2*b + 1/1 92*(9*(8*log(-c*sqrt(x) + 1)^2 - 4*log(-c*sqrt(x) + 1) + 1)*(c*sqrt(x) - 1 )^4 + 32*(9*log(-c*sqrt(x) + 1)^2 - 6*log(-c*sqrt(x) + 1) + 2)*(c*sqrt(x) - 1)^3 + 216*(2*log(-c*sqrt(x) + 1)^2 - 2*log(-c*sqrt(x) + 1) + 1)*(c*sqrt (x) - 1)^2 + 288*(log(-c*sqrt(x) + 1)^2 - 2*log(-c*sqrt(x) + 1) + 2)*(c*sq rt(x) - 1))*a*b^2/c^4 - 1/4608*(9*(32*log(-c*sqrt(x) + 1)^3 - 24*log(-c*sq rt(x) + 1)^2 + 12*log(-c*sqrt(x) + 1) - 3)*(c*sqrt(x) - 1)^4 + 128*(9*log( -c*sqrt(x) + 1)^3 - 9*log(-c*sqrt(x) + 1)^2 + 6*log(-c*sqrt(x) + 1) - 2)*( c*sqrt(x) - 1)^3 + 432*(4*log(-c*sqrt(x) + 1)^3 - 6*log(-c*sqrt(x) + 1)^2 + 6*log(-c*sqrt(x) + 1) - 3)*(c*sqrt(x) - 1)^2 + 1152*(log(-c*sqrt(x) + 1) ^3 - 3*log(-c*sqrt(x) + 1)^2 + 6*log(-c*sqrt(x) + 1) - 6)*(c*sqrt(x) - 1)) *b^3/c^4 + 2*(log(c*sqrt(x) + 1)*log(-1/2*c*sqrt(x) + 1/2) + dilog(1/2*c*s qrt(x) + 1/2))*b^3/c^4 - 319/384*b^3*log(c*sqrt(x) - 1)/c^4 + 1/16*(25*...
\[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{3} x \,d x } \]
Timed out. \[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3 \, dx=\int x\,{\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}^3 \,d x \]