Integrand size = 13, antiderivative size = 565 \[ \int \frac {1}{x^2 \left (b+a x^8\right )} \, dx=-\frac {1}{b x}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \arctan \left (\frac {-\sqrt [8]{b}+\sqrt {2} \sqrt [8]{b}-\sqrt {2 \left (2-\sqrt {2}\right )} \sqrt [8]{a} x}{\sqrt [8]{b}}\right )}{8 b^{9/8}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \arctan \left (\frac {-\sqrt [8]{b}+\sqrt {2} \sqrt [8]{b}+\sqrt {2 \left (2-\sqrt {2}\right )} \sqrt [8]{a} x}{\sqrt [8]{b}}\right )}{8 b^{9/8}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \arctan \left (\frac {\sqrt [8]{b}+\sqrt {2} \sqrt [8]{b}-\sqrt {2 \left (2+\sqrt {2}\right )} \sqrt [8]{a} x}{\sqrt [8]{b}}\right )}{8 b^{9/8}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \arctan \left (\frac {\sqrt [8]{b}+\sqrt {2} \sqrt [8]{b}+\sqrt {2 \left (2+\sqrt {2}\right )} \sqrt [8]{a} x}{\sqrt [8]{b}}\right )}{8 b^{9/8}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \log \left (\sqrt [4]{b}-\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a} x^2\right )}{16 b^{9/8}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \log \left (\sqrt [4]{b}+\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a} x^2\right )}{16 b^{9/8}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \log \left (\sqrt [4]{b}-\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a} x^2\right )}{16 b^{9/8}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \log \left (\sqrt [4]{b}+\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a} x^2\right )}{16 b^{9/8}} \] Output:
-1/b/x+1/8*(2+2^(1/2))^(1/2)*a^(1/8)*arctan((-b^(1/8)+b^(1/8)*2^(1/2)-(4-2 *2^(1/2))^(1/2)*a^(1/8)*x)/b^(1/8))/b^(9/8)-1/8*(2+2^(1/2))^(1/2)*a^(1/8)* arctan((-b^(1/8)+b^(1/8)*2^(1/2)+(4-2*2^(1/2))^(1/2)*a^(1/8)*x)/b^(1/8))/b ^(9/8)+1/8*(2-2^(1/2))^(1/2)*a^(1/8)*arctan((b^(1/8)+b^(1/8)*2^(1/2)-(4+2* 2^(1/2))^(1/2)*a^(1/8)*x)/b^(1/8))/b^(9/8)-1/8*(2-2^(1/2))^(1/2)*a^(1/8)*a rctan((b^(1/8)+b^(1/8)*2^(1/2)+(4+2*2^(1/2))^(1/2)*a^(1/8)*x)/b^(1/8))/b^( 9/8)-1/16*(2-2^(1/2))^(1/2)*a^(1/8)*ln(b^(1/4)-(2-2^(1/2))^(1/2)*a^(1/8)*b ^(1/8)*x+a^(1/4)*x^2)/b^(9/8)+1/16*(2-2^(1/2))^(1/2)*a^(1/8)*ln(b^(1/4)+(2 -2^(1/2))^(1/2)*a^(1/8)*b^(1/8)*x+a^(1/4)*x^2)/b^(9/8)-1/16*(2+2^(1/2))^(1 /2)*a^(1/8)*ln(b^(1/4)-(2+2^(1/2))^(1/2)*a^(1/8)*b^(1/8)*x+a^(1/4)*x^2)/b^ (9/8)+1/16*(2+2^(1/2))^(1/2)*a^(1/8)*ln(b^(1/4)+(2+2^(1/2))^(1/2)*a^(1/8)* b^(1/8)*x+a^(1/4)*x^2)/b^(9/8)
Time = 0.19 (sec) , antiderivative size = 377, normalized size of antiderivative = 0.67 \[ \int \frac {1}{x^2 \left (b+a x^8\right )} \, dx=-\frac {8 \sqrt [8]{b}+2 \sqrt [8]{a} x \arctan \left (\frac {\sqrt [8]{a} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{b}}-\tan \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )+2 \sqrt [8]{a} x \arctan \left (\frac {\sqrt [8]{a} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{b}}+\tan \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )+\sqrt [8]{a} x \cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{b}+\sqrt [4]{a} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )-\sqrt [8]{a} x \cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{b}+\sqrt [4]{a} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )-2 \sqrt [8]{a} x \arctan \left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{a} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{b}}\right ) \sin \left (\frac {\pi }{8}\right )+2 \sqrt [8]{a} x \arctan \left (\cot \left (\frac {\pi }{8}\right )+\frac {\sqrt [8]{a} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{b}}\right ) \sin \left (\frac {\pi }{8}\right )+\sqrt [8]{a} x \log \left (\sqrt [4]{b}+\sqrt [4]{a} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )-\sqrt [8]{a} x \log \left (\sqrt [4]{b}+\sqrt [4]{a} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )}{8 b^{9/8} x} \] Input:
Integrate[1/(x^2*(b + a*x^8)),x]
Output:
-1/8*(8*b^(1/8) + 2*a^(1/8)*x*ArcTan[(a^(1/8)*x*Sec[Pi/8])/b^(1/8) - Tan[P i/8]]*Cos[Pi/8] + 2*a^(1/8)*x*ArcTan[(a^(1/8)*x*Sec[Pi/8])/b^(1/8) + Tan[P i/8]]*Cos[Pi/8] + a^(1/8)*x*Cos[Pi/8]*Log[b^(1/4) + a^(1/4)*x^2 - 2*a^(1/8 )*b^(1/8)*x*Cos[Pi/8]] - a^(1/8)*x*Cos[Pi/8]*Log[b^(1/4) + a^(1/4)*x^2 + 2 *a^(1/8)*b^(1/8)*x*Cos[Pi/8]] - 2*a^(1/8)*x*ArcTan[Cot[Pi/8] - (a^(1/8)*x* Csc[Pi/8])/b^(1/8)]*Sin[Pi/8] + 2*a^(1/8)*x*ArcTan[Cot[Pi/8] + (a^(1/8)*x* Csc[Pi/8])/b^(1/8)]*Sin[Pi/8] + a^(1/8)*x*Log[b^(1/4) + a^(1/4)*x^2 - 2*a^ (1/8)*b^(1/8)*x*Sin[Pi/8]]*Sin[Pi/8] - a^(1/8)*x*Log[b^(1/4) + a^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]]*Sin[Pi/8])/(b^(9/8)*x)
Time = 1.15 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.58, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {847, 830, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (a x^8+b\right )} \, dx\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle -\frac {a \int \frac {x^6}{a x^8+b}dx}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 830 |
| \(\displaystyle -\frac {a \left (\frac {\int \frac {x^2}{\sqrt {b}-\sqrt {-a} x^4}dx}{2 \sqrt {-a}}-\frac {\int \frac {x^2}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt {-a}}\right )}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle -\frac {a \left (\frac {\int \frac {x^2}{\sqrt {b}-\sqrt {-a} x^4}dx}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{-a} x^2+\sqrt [4]{b}}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{-a}}-\frac {\int \frac {\sqrt [4]{b}-\sqrt [4]{-a} x^2}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\int \frac {1}{\sqrt [4]{b}-\sqrt [4]{-a} x^2}dx}{2 \sqrt [4]{-a}}-\frac {\int \frac {1}{\sqrt [4]{-a} x^2+\sqrt [4]{b}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{-a} x^2+\sqrt [4]{b}}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{-a}}-\frac {\int \frac {\sqrt [4]{b}-\sqrt [4]{-a} x^2}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\int \frac {1}{\sqrt [4]{b}-\sqrt [4]{-a} x^2}dx}{2 \sqrt [4]{-a}}-\frac {\arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{-a} x^2+\sqrt [4]{b}}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{-a}}-\frac {\int \frac {\sqrt [4]{b}-\sqrt [4]{-a} x^2}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}-\frac {\arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt [4]{-a} x^2+\sqrt [4]{b}}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{-a}}-\frac {\int \frac {\sqrt [4]{b}-\sqrt [4]{-a} x^2}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}-\frac {\arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}}dx}{2 \sqrt [4]{-a}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt [4]{-a}}-\frac {\int \frac {\sqrt [4]{b}-\sqrt [4]{-a} x^2}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}-\frac {\arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}-\frac {\int \frac {\sqrt [4]{b}-\sqrt [4]{-a} x^2}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}-\frac {\arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}-\frac {\int \frac {\sqrt [4]{b}-\sqrt [4]{-a} x^2}{\sqrt {-a} x^4+\sqrt {b}}dx}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}-\frac {\arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}-\frac {-\frac {\int -\frac {\frac {\sqrt {2} \sqrt [8]{b}}{\sqrt [8]{-a}}-2 x}{x^2-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} x+\frac {\sqrt [8]{b}}{\sqrt [8]{-a}}\right )}{x^2+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}-\frac {\arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}-\frac {\frac {\int \frac {\frac {\sqrt {2} \sqrt [8]{b}}{\sqrt [8]{-a}}-2 x}{x^2-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} x+\frac {\sqrt [8]{b}}{\sqrt [8]{-a}}\right )}{x^2+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}-\frac {\arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}-\frac {\frac {\int \frac {\frac {\sqrt {2} \sqrt [8]{b}}{\sqrt [8]{-a}}-2 x}{x^2-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}}dx}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}+\frac {\int \frac {\sqrt {2} x+\frac {\sqrt [8]{b}}{\sqrt [8]{-a}}}{x^2+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}}dx}{2 \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{b}-\frac {1}{b x}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}-\frac {\arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{2 (-a)^{3/8} \sqrt [8]{b}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2+\sqrt [4]{b}\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2+\sqrt [4]{b}\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{b}}}{2 \sqrt [4]{-a}}}{2 \sqrt {-a}}\right )}{b}-\frac {1}{b x}\) |
Input:
Int[1/(x^2*(b + a*x^8)),x]
Output:
-(1/(b*x)) - (a*((-1/2*ArcTan[((-a)^(1/8)*x)/b^(1/8)]/((-a)^(3/8)*b^(1/8)) + ArcTanh[((-a)^(1/8)*x)/b^(1/8)]/(2*(-a)^(3/8)*b^(1/8)))/(2*Sqrt[-a]) - ((-(ArcTan[1 - (Sqrt[2]*(-a)^(1/8)*x)/b^(1/8)]/(Sqrt[2]*(-a)^(1/8)*b^(1/8) )) + ArcTan[1 + (Sqrt[2]*(-a)^(1/8)*x)/b^(1/8)]/(Sqrt[2]*(-a)^(1/8)*b^(1/8 )))/(2*(-a)^(1/4)) - (-1/2*Log[b^(1/4) - Sqrt[2]*(-a)^(1/8)*b^(1/8)*x + (- a)^(1/4)*x^2]/(Sqrt[2]*(-a)^(1/8)*b^(1/8)) + Log[b^(1/4) + Sqrt[2]*(-a)^(1 /8)*b^(1/8)*x + (-a)^(1/4)*x^2]/(2*Sqrt[2]*(-a)^(1/8)*b^(1/8)))/(2*(-a)^(1 /4)))/(2*Sqrt[-a])))/b
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[s/(2*b) Int[x^(m - n/2)/( r + s*x^(n/2)), x], x] - Simp[s/(2*b) Int[x^(m - n/2)/(r - s*x^(n/2)), x] , x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LeQ[n/2, m] && L tQ[m, n] && !GtQ[a/b, 0]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.06
| method | result | size |
| default | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} a +b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}}{8 b}-\frac {1}{b x}\) | \(36\) |
| risch | \(-\frac {1}{b x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{9} \textit {\_Z}^{8}+a \right )}{\sum }\textit {\_R} \ln \left (\left (9 \textit {\_R}^{8} b^{9}+8 a \right ) x +b^{8} \textit {\_R}^{7}\right )\right )}{8}\) | \(50\) |
Input:
int(1/x^2/(a*x^8+b),x,method=_RETURNVERBOSE)
Output:
-1/8/b*sum(1/_R*ln(x-_R),_R=RootOf(_Z^8*a+b))-1/b/x
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.48 \[ \int \frac {1}{x^2 \left (b+a x^8\right )} \, dx=-\frac {-\left (i - 1\right ) \, \sqrt {2} b x \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} b^{8} \left (-\frac {a}{b^{9}}\right )^{\frac {7}{8}} + a x\right ) + \left (i + 1\right ) \, \sqrt {2} b x \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} b^{8} \left (-\frac {a}{b^{9}}\right )^{\frac {7}{8}} + a x\right ) - \left (i + 1\right ) \, \sqrt {2} b x \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} b^{8} \left (-\frac {a}{b^{9}}\right )^{\frac {7}{8}} + a x\right ) + \left (i - 1\right ) \, \sqrt {2} b x \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} b^{8} \left (-\frac {a}{b^{9}}\right )^{\frac {7}{8}} + a x\right ) + 2 \, b x \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (b^{8} \left (-\frac {a}{b^{9}}\right )^{\frac {7}{8}} + a x\right ) - 2 i \, b x \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (i \, b^{8} \left (-\frac {a}{b^{9}}\right )^{\frac {7}{8}} + a x\right ) + 2 i \, b x \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (-i \, b^{8} \left (-\frac {a}{b^{9}}\right )^{\frac {7}{8}} + a x\right ) - 2 \, b x \left (-\frac {a}{b^{9}}\right )^{\frac {1}{8}} \log \left (-b^{8} \left (-\frac {a}{b^{9}}\right )^{\frac {7}{8}} + a x\right ) + 16}{16 \, b x} \] Input:
integrate(1/x^2/(a*x^8+b),x, algorithm="fricas")
Output:
-1/16*(-(I - 1)*sqrt(2)*b*x*(-a/b^9)^(1/8)*log((1/2*I + 1/2)*sqrt(2)*b^8*( -a/b^9)^(7/8) + a*x) + (I + 1)*sqrt(2)*b*x*(-a/b^9)^(1/8)*log(-(1/2*I - 1/ 2)*sqrt(2)*b^8*(-a/b^9)^(7/8) + a*x) - (I + 1)*sqrt(2)*b*x*(-a/b^9)^(1/8)* log((1/2*I - 1/2)*sqrt(2)*b^8*(-a/b^9)^(7/8) + a*x) + (I - 1)*sqrt(2)*b*x* (-a/b^9)^(1/8)*log(-(1/2*I + 1/2)*sqrt(2)*b^8*(-a/b^9)^(7/8) + a*x) + 2*b* x*(-a/b^9)^(1/8)*log(b^8*(-a/b^9)^(7/8) + a*x) - 2*I*b*x*(-a/b^9)^(1/8)*lo g(I*b^8*(-a/b^9)^(7/8) + a*x) + 2*I*b*x*(-a/b^9)^(1/8)*log(-I*b^8*(-a/b^9) ^(7/8) + a*x) - 2*b*x*(-a/b^9)^(1/8)*log(-b^8*(-a/b^9)^(7/8) + a*x) + 16)/ (b*x)
Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.05 \[ \int \frac {1}{x^2 \left (b+a x^8\right )} \, dx=\operatorname {RootSum} {\left (16777216 t^{8} b^{9} + a, \left ( t \mapsto t \log {\left (- \frac {2097152 t^{7} b^{8}}{a} + x \right )} \right )\right )} - \frac {1}{b x} \] Input:
integrate(1/x**2/(a*x**8+b),x)
Output:
RootSum(16777216*_t**8*b**9 + a, Lambda(_t, _t*log(-2097152*_t**7*b**8/a + x))) - 1/(b*x)
\[ \int \frac {1}{x^2 \left (b+a x^8\right )} \, dx=\int { \frac {1}{{\left (a x^{8} + b\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(a*x^8+b),x, algorithm="maxima")
Output:
-a*integrate(x^6/(a*x^8 + b), x)/b - 1/(b*x)
Time = 0.20 (sec) , antiderivative size = 453, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^2 \left (b+a x^8\right )} \, dx=-\frac {a \left (\frac {b}{a}\right )^{\frac {7}{8}} \arctan \left (\frac {2 \, x + \sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}\right )}{4 \, b^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {a \left (\frac {b}{a}\right )^{\frac {7}{8}} \arctan \left (\frac {2 \, x - \sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}\right )}{4 \, b^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {a \left (\frac {b}{a}\right )^{\frac {7}{8}} \arctan \left (\frac {2 \, x + \sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}\right )}{4 \, b^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {a \left (\frac {b}{a}\right )^{\frac {7}{8}} \arctan \left (\frac {2 \, x - \sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}\right )}{4 \, b^{2} \sqrt {2 \, \sqrt {2} + 4}} + \frac {a \left (\frac {b}{a}\right )^{\frac {7}{8}} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}} + \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}{8 \, b^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {a \left (\frac {b}{a}\right )^{\frac {7}{8}} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}} + \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}{8 \, b^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {a \left (\frac {b}{a}\right )^{\frac {7}{8}} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}} + \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}{8 \, b^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {a \left (\frac {b}{a}\right )^{\frac {7}{8}} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}} + \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}{8 \, b^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {1}{b x} \] Input:
integrate(1/x^2/(a*x^8+b),x, algorithm="giac")
Output:
-1/4*a*(b/a)^(7/8)*arctan((2*x + sqrt(-sqrt(2) + 2)*(b/a)^(1/8))/(sqrt(sqr t(2) + 2)*(b/a)^(1/8)))/(b^2*sqrt(-2*sqrt(2) + 4)) - 1/4*a*(b/a)^(7/8)*arc tan((2*x - sqrt(-sqrt(2) + 2)*(b/a)^(1/8))/(sqrt(sqrt(2) + 2)*(b/a)^(1/8)) )/(b^2*sqrt(-2*sqrt(2) + 4)) - 1/4*a*(b/a)^(7/8)*arctan((2*x + sqrt(sqrt(2 ) + 2)*(b/a)^(1/8))/(sqrt(-sqrt(2) + 2)*(b/a)^(1/8)))/(b^2*sqrt(2*sqrt(2) + 4)) - 1/4*a*(b/a)^(7/8)*arctan((2*x - sqrt(sqrt(2) + 2)*(b/a)^(1/8))/(sq rt(-sqrt(2) + 2)*(b/a)^(1/8)))/(b^2*sqrt(2*sqrt(2) + 4)) + 1/8*a*(b/a)^(7/ 8)*log(x^2 + x*sqrt(sqrt(2) + 2)*(b/a)^(1/8) + (b/a)^(1/4))/(b^2*sqrt(-2*s qrt(2) + 4)) - 1/8*a*(b/a)^(7/8)*log(x^2 - x*sqrt(sqrt(2) + 2)*(b/a)^(1/8) + (b/a)^(1/4))/(b^2*sqrt(-2*sqrt(2) + 4)) + 1/8*a*(b/a)^(7/8)*log(x^2 + x *sqrt(-sqrt(2) + 2)*(b/a)^(1/8) + (b/a)^(1/4))/(b^2*sqrt(2*sqrt(2) + 4)) - 1/8*a*(b/a)^(7/8)*log(x^2 - x*sqrt(-sqrt(2) + 2)*(b/a)^(1/8) + (b/a)^(1/4 ))/(b^2*sqrt(2*sqrt(2) + 4)) - 1/(b*x)
Time = 0.18 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.21 \[ \int \frac {1}{x^2 \left (b+a x^8\right )} \, dx=-\frac {1}{b\,x}-\frac {{\left (-a\right )}^{1/8}\,\mathrm {atan}\left (\frac {{\left (-a\right )}^{1/8}\,x}{b^{1/8}}\right )}{4\,b^{9/8}}-\frac {{\left (-a\right )}^{1/8}\,\mathrm {atan}\left (\frac {{\left (-a\right )}^{1/8}\,x\,1{}\mathrm {i}}{b^{1/8}}\right )\,1{}\mathrm {i}}{4\,b^{9/8}}+\frac {\sqrt {2}\,{\left (-a\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-a\right )}^{1/8}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{b^{1/8}}\right )\,\left (-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )}{b^{9/8}}+\frac {\sqrt {2}\,{\left (-a\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-a\right )}^{1/8}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{b^{1/8}}\right )\,\left (-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )}{b^{9/8}} \] Input:
int(1/(x^2*(b + a*x^8)),x)
Output:
- 1/(b*x) - ((-a)^(1/8)*atan(((-a)^(1/8)*x)/b^(1/8)))/(4*b^(9/8)) - ((-a)^ (1/8)*atan(((-a)^(1/8)*x*1i)/b^(1/8))*1i)/(4*b^(9/8)) - (2^(1/2)*(-a)^(1/8 )*atan((2^(1/2)*(-a)^(1/8)*x*(1/2 - 1i/2))/b^(1/8))*(1/8 - 1i/8))/b^(9/8) - (2^(1/2)*(-a)^(1/8)*atan((2^(1/2)*(-a)^(1/8)*x*(1/2 + 1i/2))/b^(1/8))*(1 /8 + 1i/8))/b^(9/8)
Time = 0.22 (sec) , antiderivative size = 380, normalized size of antiderivative = 0.67 \[ \int \frac {1}{x^2 \left (b+a x^8\right )} \, dx=\frac {2 b^{\frac {7}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}-2 a^{\frac {1}{4}} x}{b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}}\right ) x -2 b^{\frac {7}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}+2 a^{\frac {1}{4}} x}{b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}}\right ) x +2 b^{\frac {7}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}-2 a^{\frac {1}{4}} x}{b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}}\right ) x -2 b^{\frac {7}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}+2 a^{\frac {1}{4}} x}{b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}}\right ) x -b^{\frac {7}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (-b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}\, x +a^{\frac {1}{4}} x^{2}+b^{\frac {1}{4}}\right ) x +b^{\frac {7}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}\, x +a^{\frac {1}{4}} x^{2}+b^{\frac {1}{4}}\right ) x -b^{\frac {7}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}\, \mathrm {log}\left (-b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}\, x +a^{\frac {1}{4}} x^{2}+b^{\frac {1}{4}}\right ) x +b^{\frac {7}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}\, \mathrm {log}\left (b^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}\, x +a^{\frac {1}{4}} x^{2}+b^{\frac {1}{4}}\right ) x -16 b}{16 b^{2} x} \] Input:
int(1/x^2/(a*x^8+b),x)
Output:
(2*b**(7/8)*a**(1/8)*sqrt(sqrt(2) + 2)*atan((b**(1/8)*a**(1/8)*sqrt( - sqr t(2) + 2) - 2*a**(1/4)*x)/(b**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)))*x - 2*b** (7/8)*a**(1/8)*sqrt(sqrt(2) + 2)*atan((b**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) + 2*a**(1/4)*x)/(b**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)))*x + 2*b**(7/8)* a**(1/8)*sqrt( - sqrt(2) + 2)*atan((b**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2) - 2*a**(1/4)*x)/(b**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2)))*x - 2*b**(7/8)*a** (1/8)*sqrt( - sqrt(2) + 2)*atan((b**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2) + 2*a **(1/4)*x)/(b**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2)))*x - b**(7/8)*a**(1/8) *sqrt( - sqrt(2) + 2)*log( - b**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2)*x + a* *(1/4)*x**2 + b**(1/4))*x + b**(7/8)*a**(1/8)*sqrt( - sqrt(2) + 2)*log(b** (1/8)*a**(1/8)*sqrt( - sqrt(2) + 2)*x + a**(1/4)*x**2 + b**(1/4))*x - b**( 7/8)*a**(1/8)*sqrt(sqrt(2) + 2)*log( - b**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2) *x + a**(1/4)*x**2 + b**(1/4))*x + b**(7/8)*a**(1/8)*sqrt(sqrt(2) + 2)*log (b**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)*x + a**(1/4)*x**2 + b**(1/4))*x - 16* b)/(16*b**2*x)