Integrand size = 18, antiderivative size = 257 \[ \int (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {8 b d \sqrt {d x}}{5 c}-\frac {2 b d^{3/2} \arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{5 c^{5/4}}+\frac {\sqrt {2} b d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{5 c^{5/4}}-\frac {\sqrt {2} b d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{5 c^{5/4}}+\frac {2 (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d}-\frac {2 b d^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{5 c^{5/4}}-\frac {\sqrt {2} b d^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d} \left (1+\sqrt {c} x\right )}\right )}{5 c^{5/4}} \] Output:
8/5*b*d*(d*x)^(1/2)/c-2/5*b*d^(3/2)*arctan(c^(1/4)*(d*x)^(1/2)/d^(1/2))/c^ (5/4)-1/5*2^(1/2)*b*d^(3/2)*arctan(-1+2^(1/2)*c^(1/4)*(d*x)^(1/2)/d^(1/2)) /c^(5/4)-1/5*2^(1/2)*b*d^(3/2)*arctan(1+2^(1/2)*c^(1/4)*(d*x)^(1/2)/d^(1/2 ))/c^(5/4)+2/5*(d*x)^(5/2)*(a+b*arctanh(c*x^2))/d-2/5*b*d^(3/2)*arctanh(c^ (1/4)*(d*x)^(1/2)/d^(1/2))/c^(5/4)-1/5*2^(1/2)*b*d^(3/2)*arctanh(2^(1/2)*c ^(1/4)*(d*x)^(1/2)/d^(1/2)/(1+c^(1/2)*x))/c^(5/4)
Time = 0.08 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.93 \[ \int (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {(d x)^{3/2} \left (16 b \sqrt [4]{c} \sqrt {x}+4 a c^{5/4} x^{5/2}+2 \sqrt {2} b \arctan \left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )-2 \sqrt {2} b \arctan \left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )-4 b \arctan \left (\sqrt [4]{c} \sqrt {x}\right )+4 b c^{5/4} x^{5/2} \text {arctanh}\left (c x^2\right )+2 b \log \left (1-\sqrt [4]{c} \sqrt {x}\right )-2 b \log \left (1+\sqrt [4]{c} \sqrt {x}\right )+\sqrt {2} b \log \left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )-\sqrt {2} b \log \left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )\right )}{10 c^{5/4} x^{3/2}} \] Input:
Integrate[(d*x)^(3/2)*(a + b*ArcTanh[c*x^2]),x]
Output:
((d*x)^(3/2)*(16*b*c^(1/4)*Sqrt[x] + 4*a*c^(5/4)*x^(5/2) + 2*Sqrt[2]*b*Arc Tan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]] - 2*Sqrt[2]*b*ArcTan[1 + Sqrt[2]*c^(1/4)* Sqrt[x]] - 4*b*ArcTan[c^(1/4)*Sqrt[x]] + 4*b*c^(5/4)*x^(5/2)*ArcTanh[c*x^2 ] + 2*b*Log[1 - c^(1/4)*Sqrt[x]] - 2*b*Log[1 + c^(1/4)*Sqrt[x]] + Sqrt[2]* b*Log[1 - Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - Sqrt[2]*b*Log[1 + Sqrt[2] *c^(1/4)*Sqrt[x] + Sqrt[c]*x]))/(10*c^(5/4)*x^(3/2))
Time = 0.63 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.36, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6464, 843, 851, 758, 755, 756, 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.
| \(\displaystyle \int (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx\) |
| \(\Big \downarrow \) 6464 |
| \(\displaystyle \frac {2 (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d}-\frac {4 b c \int \frac {(d x)^{7/2}}{1-c^2 x^4}dx}{5 d^2}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {2 (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d}-\frac {4 b c \left (\frac {d^4 \int \frac {1}{\sqrt {d x} \left (1-c^2 x^4\right )}dx}{c^2}-\frac {2 d^3 \sqrt {d x}}{c^2}\right )}{5 d^2}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {2 (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d}-\frac {4 b c \left (\frac {2 d^3 \int \frac {1}{1-c^2 x^4}d\sqrt {d x}}{c^2}-\frac {2 d^3 \sqrt {d x}}{c^2}\right )}{5 d^2}\) |
| \(\Big \downarrow \) 758 |
| \(\displaystyle \frac {2 (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d}-\frac {4 b c \left (\frac {2 d^3 \left (\frac {1}{2} d^2 \int \frac {1}{d^2-c d^2 x^2}d\sqrt {d x}+\frac {1}{2} d^2 \int \frac {1}{c x^2 d^2+d^2}d\sqrt {d x}\right )}{c^2}-\frac {2 d^3 \sqrt {d x}}{c^2}\right )}{5 d^2}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {2 (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d}-\frac {4 b c \left (\frac {2 d^3 \left (\frac {1}{2} d^2 \int \frac {1}{d^2-c d^2 x^2}d\sqrt {d x}+\frac {1}{2} d^2 \left (\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}+\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}\right )\right )}{c^2}-\frac {2 d^3 \sqrt {d x}}{c^2}\right )}{5 d^2}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {2 (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d}-\frac {4 b c \left (\frac {2 d^3 \left (\frac {1}{2} d^2 \left (\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}+\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}\right )+\frac {1}{2} d^2 \left (\frac {\int \frac {1}{d-\sqrt {c} d x}d\sqrt {d x}}{2 d}+\frac {\int \frac {1}{\sqrt {c} x d+d}d\sqrt {d x}}{2 d}\right )\right )}{c^2}-\frac {2 d^3 \sqrt {d x}}{c^2}\right )}{5 d^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d}-\frac {4 b c \left (\frac {2 d^3 \left (\frac {1}{2} d^2 \left (\frac {\int \frac {1}{d-\sqrt {c} d x}d\sqrt {d x}}{2 d}+\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}\right )+\frac {1}{2} d^2 \left (\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}+\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}\right )\right )}{c^2}-\frac {2 d^3 \sqrt {d x}}{c^2}\right )}{5 d^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d}-\frac {4 b c \left (\frac {2 d^3 \left (\frac {1}{2} d^2 \left (\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}+\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}\right )+\frac {1}{2} d^2 \left (\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}\right )\right )}{c^2}-\frac {2 d^3 \sqrt {d x}}{c^2}\right )}{5 d^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d}-\frac {4 b c \left (\frac {2 d^3 \left (\frac {1}{2} d^2 \left (\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}+\frac {\frac {\int \frac {1}{x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {c}}}{2 d}\right )+\frac {1}{2} d^2 \left (\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}\right )\right )}{c^2}-\frac {2 d^3 \sqrt {d x}}{c^2}\right )}{5 d^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d}-\frac {4 b c \left (\frac {2 d^3 \left (\frac {1}{2} d^2 \left (\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}+\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 d}\right )+\frac {1}{2} d^2 \left (\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}\right )\right )}{c^2}-\frac {2 d^3 \sqrt {d x}}{c^2}\right )}{5 d^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d}-\frac {4 b c \left (\frac {2 d^3 \left (\frac {1}{2} d^2 \left (\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 d}\right )+\frac {1}{2} d^2 \left (\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}\right )\right )}{c^2}-\frac {2 d^3 \sqrt {d x}}{c^2}\right )}{5 d^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d}-\frac {4 b c \left (\frac {2 d^3 \left (\frac {1}{2} d^2 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt [4]{c} \sqrt {d x}}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 d}\right )+\frac {1}{2} d^2 \left (\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}\right )\right )}{c^2}-\frac {2 d^3 \sqrt {d x}}{c^2}\right )}{5 d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d}-\frac {4 b c \left (\frac {2 d^3 \left (\frac {1}{2} d^2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt [4]{c} \sqrt {d x}}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 d}\right )+\frac {1}{2} d^2 \left (\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}\right )\right )}{c^2}-\frac {2 d^3 \sqrt {d x}}{c^2}\right )}{5 d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d}-\frac {4 b c \left (\frac {2 d^3 \left (\frac {1}{2} d^2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt [4]{c} \sqrt {d x}}{x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}+\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {c} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 d}\right )+\frac {1}{2} d^2 \left (\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}\right )\right )}{c^2}-\frac {2 d^3 \sqrt {d x}}{c^2}\right )}{5 d^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 (d x)^{5/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d}-\frac {4 b c \left (\frac {2 d^3 \left (\frac {1}{2} d^2 \left (\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}\right )+\frac {1}{2} d^2 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 d}+\frac {\frac {\log \left (\sqrt {c} d x+\sqrt {2} \sqrt [4]{c} \sqrt {d} \sqrt {d x}+d\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\log \left (\sqrt {c} d x-\sqrt {2} \sqrt [4]{c} \sqrt {d} \sqrt {d x}+d\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 d}\right )\right )}{c^2}-\frac {2 d^3 \sqrt {d x}}{c^2}\right )}{5 d^2}\) |
Input:
Int[(d*x)^(3/2)*(a + b*ArcTanh[c*x^2]),x]
Output:
(2*(d*x)^(5/2)*(a + b*ArcTanh[c*x^2]))/(5*d) - (4*b*c*((-2*d^3*Sqrt[d*x])/ c^2 + (2*d^3*((d^2*(ArcTan[(c^(1/4)*Sqrt[d*x])/Sqrt[d]]/(2*c^(1/4)*d^(3/2) ) + ArcTanh[(c^(1/4)*Sqrt[d*x])/Sqrt[d]]/(2*c^(1/4)*d^(3/2))))/2 + (d^2*(( -(ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]]/(Sqrt[2]*c^(1/4)*Sqrt[d] )) + ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]]/(Sqrt[2]*c^(1/4)*Sqrt [d]))/(2*d) + (-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*d)))/2))/c^2))/(5*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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) 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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b , 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^(n/2)), x], x] + Simp[r/(2*a) Int[1/(r + s*x^(n/2)), x], x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 1] && !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 = 0.38 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.96
| method | result | size |
| derivativedivides | \(\frac {\frac {2 \left (d x \right )^{\frac {5}{2}} a}{5}+2 b \left (\frac {\left (d x \right )^{\frac {5}{2}} \operatorname {arctanh}\left (c \,x^{2}\right )}{5}-\frac {4 c \,d^{2} \left (-\frac {\sqrt {d x}}{c^{2}}+\frac {\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )\right )}{8 c^{2}}+\frac {\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c^{2}}\right )}{5}\right )}{d}\) | \(248\) |
| default | \(\frac {\frac {2 \left (d x \right )^{\frac {5}{2}} a}{5}+2 b \left (\frac {\left (d x \right )^{\frac {5}{2}} \operatorname {arctanh}\left (c \,x^{2}\right )}{5}-\frac {4 c \,d^{2} \left (-\frac {\sqrt {d x}}{c^{2}}+\frac {\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )\right )}{8 c^{2}}+\frac {\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c^{2}}\right )}{5}\right )}{d}\) | \(248\) |
| parts | \(\frac {2 a \left (d x \right )^{\frac {5}{2}}}{5 d}+\frac {2 b \left (\frac {\left (d x \right )^{\frac {5}{2}} \operatorname {arctanh}\left (c \,x^{2}\right )}{5}-\frac {4 c \,d^{2} \left (-\frac {\sqrt {d x}}{c^{2}}+\frac {\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )\right )}{8 c^{2}}+\frac {\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c^{2}}\right )}{5}\right )}{d}\) | \(250\) |
Input:
int((d*x)^(3/2)*(a+b*arctanh(c*x^2)),x,method=_RETURNVERBOSE)
Output:
2/d*(1/5*(d*x)^(5/2)*a+b*(1/5*(d*x)^(5/2)*arctanh(c*x^2)-4/5*c*d^2*(-1/c^2 *(d*x)^(1/2)+1/8/c^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)))+2*arctan((d*x)^(1/2)/(d^2/c)^(1/4)))+1/16/c^2*(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))) ))
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.46 \[ \int (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=-\frac {\left (\frac {b^{4} d^{6}}{c^{5}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b d + \left (\frac {b^{4} d^{6}}{c^{5}}\right )^{\frac {1}{4}} c\right ) + i \, \left (\frac {b^{4} d^{6}}{c^{5}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b d + i \, \left (\frac {b^{4} d^{6}}{c^{5}}\right )^{\frac {1}{4}} c\right ) - i \, \left (\frac {b^{4} d^{6}}{c^{5}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b d - i \, \left (\frac {b^{4} d^{6}}{c^{5}}\right )^{\frac {1}{4}} c\right ) - \left (\frac {b^{4} d^{6}}{c^{5}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b d - \left (\frac {b^{4} d^{6}}{c^{5}}\right )^{\frac {1}{4}} c\right ) + \left (-\frac {b^{4} d^{6}}{c^{5}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b d + \left (-\frac {b^{4} d^{6}}{c^{5}}\right )^{\frac {1}{4}} c\right ) + i \, \left (-\frac {b^{4} d^{6}}{c^{5}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b d + i \, \left (-\frac {b^{4} d^{6}}{c^{5}}\right )^{\frac {1}{4}} c\right ) - i \, \left (-\frac {b^{4} d^{6}}{c^{5}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b d - i \, \left (-\frac {b^{4} d^{6}}{c^{5}}\right )^{\frac {1}{4}} c\right ) - \left (-\frac {b^{4} d^{6}}{c^{5}}\right )^{\frac {1}{4}} c \log \left (\sqrt {d x} b d - \left (-\frac {b^{4} d^{6}}{c^{5}}\right )^{\frac {1}{4}} c\right ) - {\left (b c d x^{2} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a c d x^{2} + 8 \, b d\right )} \sqrt {d x}}{5 \, c} \] Input:
integrate((d*x)^(3/2)*(a+b*arctanh(c*x^2)),x, algorithm="fricas")
Output:
-1/5*((b^4*d^6/c^5)^(1/4)*c*log(sqrt(d*x)*b*d + (b^4*d^6/c^5)^(1/4)*c) + I *(b^4*d^6/c^5)^(1/4)*c*log(sqrt(d*x)*b*d + I*(b^4*d^6/c^5)^(1/4)*c) - I*(b ^4*d^6/c^5)^(1/4)*c*log(sqrt(d*x)*b*d - I*(b^4*d^6/c^5)^(1/4)*c) - (b^4*d^ 6/c^5)^(1/4)*c*log(sqrt(d*x)*b*d - (b^4*d^6/c^5)^(1/4)*c) + (-b^4*d^6/c^5) ^(1/4)*c*log(sqrt(d*x)*b*d + (-b^4*d^6/c^5)^(1/4)*c) + I*(-b^4*d^6/c^5)^(1 /4)*c*log(sqrt(d*x)*b*d + I*(-b^4*d^6/c^5)^(1/4)*c) - I*(-b^4*d^6/c^5)^(1/ 4)*c*log(sqrt(d*x)*b*d - I*(-b^4*d^6/c^5)^(1/4)*c) - (-b^4*d^6/c^5)^(1/4)* c*log(sqrt(d*x)*b*d - (-b^4*d^6/c^5)^(1/4)*c) - (b*c*d*x^2*log(-(c*x^2 + 1 )/(c*x^2 - 1)) + 2*a*c*d*x^2 + 8*b*d)*sqrt(d*x))/c
\[ \int (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\int \left (d x\right )^{\frac {3}{2}} \left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )\, dx \] Input:
integrate((d*x)**(3/2)*(a+b*atanh(c*x**2)),x)
Output:
Integral((d*x)**(3/2)*(a + b*atanh(c*x**2)), x)
Time = 0.11 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.21 \[ \int (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {4 \, \left (d x\right )^{\frac {5}{2}} a + {\left (4 \, \left (d x\right )^{\frac {5}{2}} \operatorname {artanh}\left (c x^{2}\right ) + \frac {{\left (\frac {16 \, \sqrt {d x} d^{4}}{c^{2}} - \frac {\frac {2 \, \sqrt {2} d^{5} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} + 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d}} + \frac {2 \, \sqrt {2} d^{5} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} - 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d}} + \frac {\sqrt {2} d^{\frac {9}{2}} \log \left (\sqrt {c} d x + \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {1}{4}}} - \frac {\sqrt {2} d^{\frac {9}{2}} \log \left (\sqrt {c} d x - \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {1}{4}}}}{c^{2}} - \frac {2 \, {\left (\frac {2 \, d^{5} \arctan \left (\frac {\sqrt {d x} \sqrt {c}}{\sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d}} - \frac {d^{5} \log \left (\frac {\sqrt {d x} \sqrt {c} - \sqrt {\sqrt {c} d}}{\sqrt {d x} \sqrt {c} + \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d}}\right )}}{c^{2}}\right )} c}{d^{2}}\right )} b}{10 \, d} \] Input:
integrate((d*x)^(3/2)*(a+b*arctanh(c*x^2)),x, algorithm="maxima")
Output:
1/10*(4*(d*x)^(5/2)*a + (4*(d*x)^(5/2)*arctanh(c*x^2) + (16*sqrt(d*x)*d^4/ c^2 - (2*sqrt(2)*d^5*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) + 2*sqrt(2)*d^5*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(2)*d^(9/2)*log(sqrt(c)*d*x + sqrt(2)*sqrt(d*x)*c^(1 /4)*sqrt(d) + d)/c^(1/4) - sqrt(2)*d^(9/2)*log(sqrt(c)*d*x - sqrt(2)*sqrt( d*x)*c^(1/4)*sqrt(d) + d)/c^(1/4))/c^2 - 2*(2*d^5*arctan(sqrt(d*x)*sqrt(c) /sqrt(sqrt(c)*d))/sqrt(sqrt(c)*d) - d^5*log((sqrt(d*x)*sqrt(c) - sqrt(sqrt (c)*d))/(sqrt(d*x)*sqrt(c) + sqrt(sqrt(c)*d)))/sqrt(sqrt(c)*d))/c^2)*c/d^2 )*b)/d
\[ \int (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\int { \left (d x\right )^{\frac {3}{2}} {\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )} \,d x } \] Input:
integrate((d*x)^(3/2)*(a+b*arctanh(c*x^2)),x, algorithm="giac")
Output:
integrate((d*x)^(3/2)*(b*arctanh(c*x^2) + a), x)
Timed out. \[ \int (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\int {\left (d\,x\right )}^{3/2}\,\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right ) \,d x \] Input:
int((d*x)^(3/2)*(a + b*atanh(c*x^2)),x)
Output:
int((d*x)^(3/2)*(a + b*atanh(c*x^2)), x)
Time = 0.18 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.95 \[ \int (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {\sqrt {d}\, d \left (2 c^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} \sqrt {2}}\right ) b -2 c^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} \sqrt {2}}\right ) b -4 c^{\frac {3}{4}} \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}}}\right ) b -2 c^{\frac {3}{4}} \sqrt {2}\, \mathit {atanh} \left (c \,x^{2}\right ) b +4 \sqrt {x}\, \mathit {atanh} \left (c \,x^{2}\right ) b \,c^{2} x^{2}-c^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) b -c^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) b +2 c^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x +1\right ) b -c^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {c}\, x +1\right ) b -2 c^{\frac {3}{4}} \mathrm {log}\left (c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) b +2 c^{\frac {3}{4}} \mathrm {log}\left (-c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) b +4 \sqrt {x}\, a \,c^{2} x^{2}+16 \sqrt {x}\, b c \right )}{10 c^{2}} \] Input:
int((d*x)^(3/2)*(a+b*atanh(c*x^2)),x)
Output:
(sqrt(d)*d*(2*c**(3/4)*sqrt(2)*atan((c**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(c)) /(c**(1/4)*sqrt(2)))*b - 2*c**(3/4)*sqrt(2)*atan((c**(1/4)*sqrt(2) + 2*sqr t(x)*sqrt(c))/(c**(1/4)*sqrt(2)))*b - 4*c**(3/4)*atan((sqrt(x)*sqrt(c))/c* *(1/4))*b - 2*c**(3/4)*sqrt(2)*atanh(c*x**2)*b + 4*sqrt(x)*atanh(c*x**2)*b *c**2*x**2 - c**(3/4)*sqrt(2)*log(c**(1/4) + sqrt(x)*sqrt(c))*b - c**(3/4) *sqrt(2)*log( - c**(1/4) + sqrt(x)*sqrt(c))*b + 2*c**(3/4)*sqrt(2)*log( - sqrt(x)*c**(1/4)*sqrt(2) + sqrt(c)*x + 1)*b - c**(3/4)*sqrt(2)*log(sqrt(c) *x + 1)*b - 2*c**(3/4)*log(c**(1/4) + sqrt(x)*sqrt(c))*b + 2*c**(3/4)*log( - c**(1/4) + sqrt(x)*sqrt(c))*b + 4*sqrt(x)*a*c**2*x**2 + 16*sqrt(x)*b*c) )/(10*c**2)