Integrand size = 18, antiderivative size = 257 \[ \int (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {8 b d (d x)^{3/2}}{21 c}+\frac {2 b d^{5/2} \arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 c^{7/4}}+\frac {\sqrt {2} b d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 c^{7/4}}-\frac {\sqrt {2} b d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 c^{7/4}}+\frac {2 (d x)^{7/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{7 d}-\frac {2 b d^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 c^{7/4}}+\frac {\sqrt {2} b d^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d} \left (1+\sqrt {c} x\right )}\right )}{7 c^{7/4}} \] Output:
8/21*b*d*(d*x)^(3/2)/c+2/7*b*d^(5/2)*arctan(c^(1/4)*(d*x)^(1/2)/d^(1/2))/c ^(7/4)-1/7*2^(1/2)*b*d^(5/2)*arctan(-1+2^(1/2)*c^(1/4)*(d*x)^(1/2)/d^(1/2) )/c^(7/4)-1/7*2^(1/2)*b*d^(5/2)*arctan(1+2^(1/2)*c^(1/4)*(d*x)^(1/2)/d^(1/ 2))/c^(7/4)+2/7*(d*x)^(7/2)*(a+b*arctanh(c*x^2))/d-2/7*b*d^(5/2)*arctanh(c ^(1/4)*(d*x)^(1/2)/d^(1/2))/c^(7/4)+1/7*2^(1/2)*b*d^(5/2)*arctanh(2^(1/2)* c^(1/4)*(d*x)^(1/2)/d^(1/2)/(1+c^(1/2)*x))/c^(7/4)
Time = 0.12 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.94 \[ \int (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {(d x)^{5/2} \left (16 b c^{3/4} x^{3/2}+12 a c^{7/4} x^{7/2}+6 \sqrt {2} b \arctan \left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )-6 \sqrt {2} b \arctan \left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )+12 b \arctan \left (\sqrt [4]{c} \sqrt {x}\right )+12 b c^{7/4} x^{7/2} \text {arctanh}\left (c x^2\right )+6 b \log \left (1-\sqrt [4]{c} \sqrt {x}\right )-6 b \log \left (1+\sqrt [4]{c} \sqrt {x}\right )-3 \sqrt {2} b \log \left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )+3 \sqrt {2} b \log \left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )\right )}{42 c^{7/4} x^{5/2}} \] Input:
Integrate[(d*x)^(5/2)*(a + b*ArcTanh[c*x^2]),x]
Output:
((d*x)^(5/2)*(16*b*c^(3/4)*x^(3/2) + 12*a*c^(7/4)*x^(7/2) + 6*Sqrt[2]*b*Ar cTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]] - 6*Sqrt[2]*b*ArcTan[1 + Sqrt[2]*c^(1/4) *Sqrt[x]] + 12*b*ArcTan[c^(1/4)*Sqrt[x]] + 12*b*c^(7/4)*x^(7/2)*ArcTanh[c* x^2] + 6*b*Log[1 - c^(1/4)*Sqrt[x]] - 6*b*Log[1 + c^(1/4)*Sqrt[x]] - 3*Sqr t[2]*b*Log[1 - Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 3*Sqrt[2]*b*Log[1 + Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x]))/(42*c^(7/4)*x^(5/2))
Time = 0.72 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.39, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {6464, 843, 851, 27, 829, 826, 827, 218, 221, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx\) |
| \(\Big \downarrow \) 6464 |
| \(\displaystyle \frac {2 (d x)^{7/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{7 d}-\frac {4 b c \int \frac {(d x)^{9/2}}{1-c^2 x^4}dx}{7 d^2}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {2 (d x)^{7/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{7 d}-\frac {4 b c \left (\frac {d^4 \int \frac {\sqrt {d x}}{1-c^2 x^4}dx}{c^2}-\frac {2 d^3 (d x)^{3/2}}{3 c^2}\right )}{7 d^2}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {2 (d x)^{7/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{7 d}-\frac {4 b c \left (\frac {2 d^3 \int \frac {d^5 x}{d^4-c^2 d^4 x^4}d\sqrt {d x}}{c^2}-\frac {2 d^3 (d x)^{3/2}}{3 c^2}\right )}{7 d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (d x)^{7/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{7 d}-\frac {4 b c \left (\frac {2 d^7 \int \frac {d x}{d^4-c^2 d^4 x^4}d\sqrt {d x}}{c^2}-\frac {2 d^3 (d x)^{3/2}}{3 c^2}\right )}{7 d^2}\) |
| \(\Big \downarrow \) 829 |
| \(\displaystyle \frac {2 (d x)^{7/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{7 d}-\frac {4 b c \left (\frac {2 d^7 \left (\frac {\int \frac {d x}{d^2-c d^2 x^2}d\sqrt {d x}}{2 d^2}+\frac {\int \frac {d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d^2}\right )}{c^2}-\frac {2 d^3 (d x)^{3/2}}{3 c^2}\right )}{7 d^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {2 (d x)^{7/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{7 d}-\frac {4 b c \left (\frac {2 d^7 \left (\frac {\int \frac {d x}{d^2-c d^2 x^2}d\sqrt {d x}}{2 d^2}+\frac {\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 d^2}\right )}{c^2}-\frac {2 d^3 (d x)^{3/2}}{3 c^2}\right )}{7 d^2}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {2 (d x)^{7/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{7 d}-\frac {4 b c \left (\frac {2 d^7 \left (\frac {\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 d^2}+\frac {\frac {\int \frac {1}{d-\sqrt {c} d x}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {1}{\sqrt {c} x d+d}d\sqrt {d x}}{2 \sqrt {c}}}{2 d^2}\right )}{c^2}-\frac {2 d^3 (d x)^{3/2}}{3 c^2}\right )}{7 d^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 (d x)^{7/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{7 d}-\frac {4 b c \left (\frac {2 d^7 \left (\frac {\frac {\int \frac {1}{d-\sqrt {c} d x}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 d^2}+\frac {\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 d^2}\right )}{c^2}-\frac {2 d^3 (d x)^{3/2}}{3 c^2}\right )}{7 d^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 (d x)^{7/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{7 d}-\frac {4 b c \left (\frac {2 d^7 \left (\frac {\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 d^2}+\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 d^2}\right )}{c^2}-\frac {2 d^3 (d x)^{3/2}}{3 c^2}\right )}{7 d^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 (d x)^{7/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{7 d}-\frac {4 b c \left (\frac {2 d^7 \left (\frac {\frac {\frac {\int \frac {1}{x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {c}}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 d^2}+\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 d^2}\right )}{c^2}-\frac {2 d^3 (d x)^{3/2}}{3 c^2}\right )}{7 d^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 (d x)^{7/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{7 d}-\frac {4 b c \left (\frac {2 d^7 \left (\frac {\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 d^2}+\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 d^2}\right )}{c^2}-\frac {2 d^3 (d x)^{3/2}}{3 c^2}\right )}{7 d^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 (d x)^{7/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{7 d}-\frac {4 b c \left (\frac {2 d^7 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 d^2}+\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 d^2}\right )}{c^2}-\frac {2 d^3 (d x)^{3/2}}{3 c^2}\right )}{7 d^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 (d x)^{7/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{7 d}-\frac {4 b c \left (\frac {2 d^7 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt [4]{c} \sqrt {d x}}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}}{2 d^2}+\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 d^2}\right )}{c^2}-\frac {2 d^3 (d x)^{3/2}}{3 c^2}\right )}{7 d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (d x)^{7/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{7 d}-\frac {4 b c \left (\frac {2 d^7 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt [4]{c} \sqrt {d x}}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}}{2 d^2}+\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 d^2}\right )}{c^2}-\frac {2 d^3 (d x)^{3/2}}{3 c^2}\right )}{7 d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (d x)^{7/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{7 d}-\frac {4 b c \left (\frac {2 d^7 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt [4]{c} \sqrt {d x}}{x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}+\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {c} \sqrt {d}}}{2 \sqrt {c}}}{2 d^2}+\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 d^2}\right )}{c^2}-\frac {2 d^3 (d x)^{3/2}}{3 c^2}\right )}{7 d^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 (d x)^{7/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{7 d}-\frac {4 b c \left (\frac {2 d^7 \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 d^2}+\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\frac {\log \left (\sqrt {c} d x+\sqrt {2} \sqrt [4]{c} \sqrt {d} \sqrt {d x}+d\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\log \left (\sqrt {c} d x-\sqrt {2} \sqrt [4]{c} \sqrt {d} \sqrt {d x}+d\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}}{2 d^2}\right )}{c^2}-\frac {2 d^3 (d x)^{3/2}}{3 c^2}\right )}{7 d^2}\) |
Input:
Int[(d*x)^(5/2)*(a + b*ArcTanh[c*x^2]),x]
Output:
(2*(d*x)^(7/2)*(a + b*ArcTanh[c*x^2]))/(7*d) - (4*b*c*((-2*d^3*(d*x)^(3/2) )/(3*c^2) + (2*d^7*((-1/2*ArcTan[(c^(1/4)*Sqrt[d*x])/Sqrt[d]]/(c^(3/4)*Sqr t[d]) + ArcTanh[(c^(1/4)*Sqrt[d*x])/Sqrt[d]]/(2*c^(3/4)*Sqrt[d]))/(2*d^2) + ((-(ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]]/(Sqrt[2]*c^(1/4)*Sqr t[d])) + ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]]/(Sqrt[2]*c^(1/4)* Sqrt[d]))/(2*Sqrt[c]) - (-1/2*Log[d + Sqrt[c]*d*x - Sqrt[2]*c^(1/4)*Sqrt[d ]*Sqrt[d*x]]/(Sqrt[2]*c^(1/4)*Sqrt[d]) + Log[d + Sqrt[c]*d*x + Sqrt[2]*c^( 1/4)*Sqrt[d]*Sqrt[d*x]]/(2*Sqrt[2]*c^(1/4)*Sqrt[d]))/(2*Sqrt[c]))/(2*d^2)) )/c^2))/(7*d^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt [-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[x^m/(r + s*x^ (n/2)), x], x] + Simp[r/(2*a) Int[x^m/(r - s*x^(n/2)), x], x]] /; FreeQ[{ a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LtQ[m, n/2] && !GtQ[a/b, 0]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))*((d_)*(x_))^(m_), x_Symbol] : > Simp[(d*x)^(m + 1)*((a + b*ArcTanh[c*x^n])/(d*(m + 1))), x] - Simp[b*c*(n /(d^n*(m + 1))) Int[(d*x)^(m + n)/(1 - c^2*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IntegerQ[n] && NeQ[m, -1]
Time = 4.39 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00
| method | result | size |
| derivativedivides | \(\frac {\frac {2 a \left (d x \right )^{\frac {7}{2}}}{7}+2 b \left (\frac {\left (d x \right )^{\frac {7}{2}} \operatorname {arctanh}\left (c \,x^{2}\right )}{7}-\frac {4 c \,d^{2} \left (-\frac {\left (d x \right )^{\frac {3}{2}}}{3 c^{2}}+\frac {d^{2} \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c^{3} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {d^{2} \left (2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )\right )}{8 c^{3} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{7}\right )}{d}\) | \(256\) |
| default | \(\frac {\frac {2 a \left (d x \right )^{\frac {7}{2}}}{7}+2 b \left (\frac {\left (d x \right )^{\frac {7}{2}} \operatorname {arctanh}\left (c \,x^{2}\right )}{7}-\frac {4 c \,d^{2} \left (-\frac {\left (d x \right )^{\frac {3}{2}}}{3 c^{2}}+\frac {d^{2} \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c^{3} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {d^{2} \left (2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )\right )}{8 c^{3} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{7}\right )}{d}\) | \(256\) |
| parts | \(\frac {2 a \left (d x \right )^{\frac {7}{2}}}{7 d}+\frac {2 b \left (\frac {\left (d x \right )^{\frac {7}{2}} \operatorname {arctanh}\left (c \,x^{2}\right )}{7}-\frac {4 c \,d^{2} \left (-\frac {\left (d x \right )^{\frac {3}{2}}}{3 c^{2}}+\frac {d^{2} \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c^{3} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {d^{2} \left (2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )\right )}{8 c^{3} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{7}\right )}{d}\) | \(258\) |
Input:
int((d*x)^(5/2)*(a+b*arctanh(c*x^2)),x,method=_RETURNVERBOSE)
Output:
2/d*(1/7*a*(d*x)^(7/2)+b*(1/7*(d*x)^(7/2)*arctanh(c*x^2)-4/7*c*d^2*(-1/3*( d*x)^(3/2)/c^2+1/16*d^2/c^3/(d^2/c)^(1/4)*2^(1/2)*(ln((d*x-(d^2/c)^(1/4)*( d*x)^(1/2)*2^(1/2)+(d^2/c)^(1/2))/(d*x+(d^2/c)^(1/4)*(d*x)^(1/2)*2^(1/2)+( d^2/c)^(1/2)))+2*arctan(2^(1/2)/(d^2/c)^(1/4)*(d*x)^(1/2)+1)+2*arctan(2^(1 /2)/(d^2/c)^(1/4)*(d*x)^(1/2)-1))-1/8*d^2/c^3/(d^2/c)^(1/4)*(2*arctan((d*x )^(1/2)/(d^2/c)^(1/4))-ln(((d*x)^(1/2)+(d^2/c)^(1/4))/((d*x)^(1/2)-(d^2/c) ^(1/4)))))))
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.69 \[ \int (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=-\frac {3 \, \left (\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b^{3} d^{7} + \left (\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {3}{4}} c^{5}\right ) - 3 i \, \left (\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b^{3} d^{7} + i \, \left (\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {3}{4}} c^{5}\right ) + 3 i \, \left (\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b^{3} d^{7} - i \, \left (\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {3}{4}} c^{5}\right ) - 3 \, \left (\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b^{3} d^{7} - \left (\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {3}{4}} c^{5}\right ) + 3 \, \left (-\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b^{3} d^{7} + \left (-\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {3}{4}} c^{5}\right ) - 3 i \, \left (-\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b^{3} d^{7} + i \, \left (-\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {3}{4}} c^{5}\right ) + 3 i \, \left (-\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b^{3} d^{7} - i \, \left (-\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {3}{4}} c^{5}\right ) - 3 \, \left (-\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b^{3} d^{7} - \left (-\frac {b^{4} d^{10}}{c^{7}}\right )^{\frac {3}{4}} c^{5}\right ) - {\left (3 \, b c d^{2} x^{3} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 6 \, a c d^{2} x^{3} + 8 \, b d^{2} x\right )} \sqrt {d x}}{21 \, c} \] Input:
integrate((d*x)^(5/2)*(a+b*arctanh(c*x^2)),x, algorithm="fricas")
Output:
-1/21*(3*(b^4*d^10/c^7)^(1/4)*c*log(sqrt(d*x)*b^3*d^7 + (b^4*d^10/c^7)^(3/ 4)*c^5) - 3*I*(b^4*d^10/c^7)^(1/4)*c*log(sqrt(d*x)*b^3*d^7 + I*(b^4*d^10/c ^7)^(3/4)*c^5) + 3*I*(b^4*d^10/c^7)^(1/4)*c*log(sqrt(d*x)*b^3*d^7 - I*(b^4 *d^10/c^7)^(3/4)*c^5) - 3*(b^4*d^10/c^7)^(1/4)*c*log(sqrt(d*x)*b^3*d^7 - ( b^4*d^10/c^7)^(3/4)*c^5) + 3*(-b^4*d^10/c^7)^(1/4)*c*log(sqrt(d*x)*b^3*d^7 + (-b^4*d^10/c^7)^(3/4)*c^5) - 3*I*(-b^4*d^10/c^7)^(1/4)*c*log(sqrt(d*x)* b^3*d^7 + I*(-b^4*d^10/c^7)^(3/4)*c^5) + 3*I*(-b^4*d^10/c^7)^(1/4)*c*log(s qrt(d*x)*b^3*d^7 - I*(-b^4*d^10/c^7)^(3/4)*c^5) - 3*(-b^4*d^10/c^7)^(1/4)* c*log(sqrt(d*x)*b^3*d^7 - (-b^4*d^10/c^7)^(3/4)*c^5) - (3*b*c*d^2*x^3*log( -(c*x^2 + 1)/(c*x^2 - 1)) + 6*a*c*d^2*x^3 + 8*b*d^2*x)*sqrt(d*x))/c
Timed out. \[ \int (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\text {Timed out} \] Input:
integrate((d*x)**(5/2)*(a+b*atanh(c*x**2)),x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.23 \[ \int (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {12 \, \left (d x\right )^{\frac {7}{2}} a + {\left (12 \, \left (d x\right )^{\frac {7}{2}} \operatorname {artanh}\left (c x^{2}\right ) - \frac {{\left (\frac {3 \, d^{6} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} + 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} - 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {c} d x + \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {3}{4}} \sqrt {d}} + \frac {\sqrt {2} \log \left (\sqrt {c} d x - \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {3}{4}} \sqrt {d}}\right )}}{c^{2}} - \frac {6 \, d^{6} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {d x} \sqrt {c}}{\sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} + \frac {\log \left (\frac {\sqrt {d x} \sqrt {c} - \sqrt {\sqrt {c} d}}{\sqrt {d x} \sqrt {c} + \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}}\right )}}{c^{2}} - \frac {16 \, \left (d x\right )^{\frac {3}{2}} d^{4}}{c^{2}}\right )} c}{d^{2}}\right )} b}{42 \, d} \] Input:
integrate((d*x)^(5/2)*(a+b*arctanh(c*x^2)),x, algorithm="maxima")
Output:
1/42*(12*(d*x)^(7/2)*a + (12*(d*x)^(7/2)*arctanh(c*x^2) - (3*d^6*(2*sqrt(2 )*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*sqrt(d) + 2*sqrt(d*x)*sqrt(c))/sqrt( sqrt(c)*d))/(sqrt(sqrt(c)*d)*sqrt(c)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqr t(2)*c^(1/4)*sqrt(d) - 2*sqrt(d*x)*sqrt(c))/sqrt(sqrt(c)*d))/(sqrt(sqrt(c) *d)*sqrt(c)) - sqrt(2)*log(sqrt(c)*d*x + sqrt(2)*sqrt(d*x)*c^(1/4)*sqrt(d) + d)/(c^(3/4)*sqrt(d)) + sqrt(2)*log(sqrt(c)*d*x - sqrt(2)*sqrt(d*x)*c^(1 /4)*sqrt(d) + d)/(c^(3/4)*sqrt(d)))/c^2 - 6*d^6*(2*arctan(sqrt(d*x)*sqrt(c )/sqrt(sqrt(c)*d))/(sqrt(sqrt(c)*d)*sqrt(c)) + log((sqrt(d*x)*sqrt(c) - sq rt(sqrt(c)*d))/(sqrt(d*x)*sqrt(c) + sqrt(sqrt(c)*d)))/(sqrt(sqrt(c)*d)*sqr t(c)))/c^2 - 16*(d*x)^(3/2)*d^4/c^2)*c/d^2)*b)/d
\[ \int (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\int { \left (d x\right )^{\frac {5}{2}} {\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )} \,d x } \] Input:
integrate((d*x)^(5/2)*(a+b*arctanh(c*x^2)),x, algorithm="giac")
Output:
integrate((d*x)^(5/2)*(b*arctanh(c*x^2) + a), x)
Timed out. \[ \int (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\int {\left (d\,x\right )}^{5/2}\,\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right ) \,d x \] Input:
int((d*x)^(5/2)*(a + b*atanh(c*x^2)),x)
Output:
int((d*x)^(5/2)*(a + b*atanh(c*x^2)), x)
Time = 0.18 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.96 \[ \int (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {\sqrt {d}\, d^{2} \left (6 c^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} \sqrt {2}}\right ) b -6 c^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} \sqrt {2}}\right ) b +12 c^{\frac {1}{4}} \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}}}\right ) b +6 c^{\frac {1}{4}} \sqrt {2}\, \mathit {atanh} \left (c \,x^{2}\right ) b +12 \sqrt {x}\, \mathit {atanh} \left (c \,x^{2}\right ) b \,c^{2} x^{3}+3 c^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) b +3 c^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) b -6 c^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x +1\right ) b +3 c^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {c}\, x +1\right ) b -6 c^{\frac {1}{4}} \mathrm {log}\left (c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) b +6 c^{\frac {1}{4}} \mathrm {log}\left (-c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) b +12 \sqrt {x}\, a \,c^{2} x^{3}+16 \sqrt {x}\, b c x \right )}{42 c^{2}} \] Input:
int((d*x)^(5/2)*(a+b*atanh(c*x^2)),x)
Output:
(sqrt(d)*d**2*(6*c**(1/4)*sqrt(2)*atan((c**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt( c))/(c**(1/4)*sqrt(2)))*b - 6*c**(1/4)*sqrt(2)*atan((c**(1/4)*sqrt(2) + 2* sqrt(x)*sqrt(c))/(c**(1/4)*sqrt(2)))*b + 12*c**(1/4)*atan((sqrt(x)*sqrt(c) )/c**(1/4))*b + 6*c**(1/4)*sqrt(2)*atanh(c*x**2)*b + 12*sqrt(x)*atanh(c*x* *2)*b*c**2*x**3 + 3*c**(1/4)*sqrt(2)*log(c**(1/4) + sqrt(x)*sqrt(c))*b + 3 *c**(1/4)*sqrt(2)*log( - c**(1/4) + sqrt(x)*sqrt(c))*b - 6*c**(1/4)*sqrt(2 )*log( - sqrt(x)*c**(1/4)*sqrt(2) + sqrt(c)*x + 1)*b + 3*c**(1/4)*sqrt(2)* log(sqrt(c)*x + 1)*b - 6*c**(1/4)*log(c**(1/4) + sqrt(x)*sqrt(c))*b + 6*c* *(1/4)*log( - c**(1/4) + sqrt(x)*sqrt(c))*b + 12*sqrt(x)*a*c**2*x**3 + 16* sqrt(x)*b*c*x))/(42*c**2)