Integrand size = 40, antiderivative size = 482 \[ \int \frac {b-a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b-2 a x^4+x^8\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {3 \left (1+\sqrt [4]{-1}\right ) \arctan \left (\frac {(-1)^{7/8} \sqrt {2+\sqrt {2}} \sqrt [8]{a^2+b} x \sqrt [4]{b+a x^4}}{(-1)^{3/4} \sqrt [4]{a^2+b} x^2+\sqrt {b+a x^4}}\right )}{8 \sqrt [8]{a^2+b}}-\frac {3 i \left (-i \sqrt {2}+\sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \arctan \left (\frac {(-1)^{7/8} \left (-2+\sqrt {2}\right ) \sqrt [8]{a^2+b} x \sqrt [4]{b+a x^4}}{(-1)^{3/4} \sqrt {2-\sqrt {2}} \sqrt [4]{a^2+b} x^2+\sqrt {2-\sqrt {2}} \sqrt {b+a x^4}}\right )}{16 \sqrt [8]{a^2+b}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {3 \left (\sqrt {2}+i \sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \text {arctanh}\left (\frac {(-1)^{7/8} \sqrt [4]{a^2+b} x^2-\sqrt [8]{-1} \sqrt {b+a x^4}}{\sqrt {2-\sqrt {2}} \sqrt [8]{a^2+b} x \sqrt [4]{b+a x^4}}\right )}{16 \sqrt [8]{a^2+b}}+\frac {3 \left (1+\sqrt [4]{-1}\right ) \text {arctanh}\left (\frac {(-1)^{7/8} \sqrt [4]{a^2+b} x^2-\sqrt [8]{-1} \sqrt {b+a x^4}}{\sqrt {2+\sqrt {2}} \sqrt [8]{a^2+b} x \sqrt [4]{b+a x^4}}\right )}{8 \sqrt [8]{a^2+b}} \]
arctan(a^(1/4)*x/(a*x^4+b)^(1/4))/a^(1/4)+3/8*(1+(-1)^(1/4))*arctan((-1)^( 7/8)*(2+2^(1/2))^(1/2)*(a^2+b)^(1/8)*x*(a*x^4+b)^(1/4)/((-1)^(3/4)*(a^2+b) ^(1/4)*x^2+(a*x^4+b)^(1/2)))/(a^2+b)^(1/8)-3/16*I*(-I*2^(1/2)+2-2^(1/2))*a rctan((-1)^(7/8)*(-2+2^(1/2))*(a^2+b)^(1/8)*x*(a*x^4+b)^(1/4)/((-1)^(3/4)* (2-2^(1/2))^(1/2)*(a^2+b)^(1/4)*x^2+(2-2^(1/2))^(1/2)*(a*x^4+b)^(1/2)))/(a ^2+b)^(1/8)+arctanh(a^(1/4)*x/(a*x^4+b)^(1/4))/a^(1/4)+3/16*(2^(1/2)+I*(2- 2^(1/2)))*arctanh(((-1)^(7/8)*(a^2+b)^(1/4)*x^2-(-1)^(1/8)*(a*x^4+b)^(1/2) )/(2-2^(1/2))^(1/2)/(a^2+b)^(1/8)/x/(a*x^4+b)^(1/4))/(a^2+b)^(1/8)+3/8*(1+ (-1)^(1/4))*arctanh(((-1)^(7/8)*(a^2+b)^(1/4)*x^2-(-1)^(1/8)*(a*x^4+b)^(1/ 2))/(2+2^(1/2))^(1/2)/(a^2+b)^(1/8)/x/(a*x^4+b)^(1/4))/(a^2+b)^(1/8)
\[ \int \frac {b-a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b-2 a x^4+x^8\right )} \, dx=\int \frac {b-a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b-2 a x^4+x^8\right )} \, dx \]
Time = 1.27 (sec) , antiderivative size = 451, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {7279, 2009}
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.
| \(\displaystyle \int \frac {-a x^4+b+2 x^8}{\sqrt [4]{a x^4+b} \left (-2 a x^4-b+x^8\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {2}{\sqrt [4]{a x^4+b}}+\frac {3 \left (a x^4+b\right )^{3/4}}{-2 a x^4-b+x^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \left (-a \sqrt {a^2+b}+a^2+b\right )^{3/4} \arctan \left (\frac {x \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+b}} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt {a^2+b} \left (a-\sqrt {a^2+b}\right )^{3/4}}-\frac {3 \left (a \sqrt {a^2+b}+a^2+b\right )^{3/4} \arctan \left (\frac {x \sqrt [4]{a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+b}+a} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt {a^2+b} \left (\sqrt {a^2+b}+a\right )^{3/4}}+\frac {3 \left (-a \sqrt {a^2+b}+a^2+b\right )^{3/4} \text {arctanh}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+b}} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt {a^2+b} \left (a-\sqrt {a^2+b}\right )^{3/4}}-\frac {3 \left (a \sqrt {a^2+b}+a^2+b\right )^{3/4} \text {arctanh}\left (\frac {x \sqrt [4]{a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+b}+a} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt {a^2+b} \left (\sqrt {a^2+b}+a\right )^{3/4}}+\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}\) |
ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) + (3*(a^2 + b - a*Sqrt[a^2 + b])^(3/4)*ArcTan[((a^2 + b - a*Sqrt[a^2 + b])^(1/4)*x)/((a - Sqrt[a^2 + b ])^(1/4)*(b + a*x^4)^(1/4))])/(4*Sqrt[a^2 + b]*(a - Sqrt[a^2 + b])^(3/4)) - (3*(a^2 + b + a*Sqrt[a^2 + b])^(3/4)*ArcTan[((a^2 + b + a*Sqrt[a^2 + b]) ^(1/4)*x)/((a + Sqrt[a^2 + b])^(1/4)*(b + a*x^4)^(1/4))])/(4*Sqrt[a^2 + b] *(a + Sqrt[a^2 + b])^(3/4)) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/ 4) + (3*(a^2 + b - a*Sqrt[a^2 + b])^(3/4)*ArcTanh[((a^2 + b - a*Sqrt[a^2 + b])^(1/4)*x)/((a - Sqrt[a^2 + b])^(1/4)*(b + a*x^4)^(1/4))])/(4*Sqrt[a^2 + b]*(a - Sqrt[a^2 + b])^(3/4)) - (3*(a^2 + b + a*Sqrt[a^2 + b])^(3/4)*Arc Tanh[((a^2 + b + a*Sqrt[a^2 + b])^(1/4)*x)/((a + Sqrt[a^2 + b])^(1/4)*(b + a*x^4)^(1/4))])/(4*Sqrt[a^2 + b]*(a + Sqrt[a^2 + b])^(3/4))
3.31.68.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.22
| method | result | size |
| pseudoelliptic | \(\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-a^{2}-b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right ) a^{\frac {1}{4}}-8 \arctan \left (\frac {\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+4 \ln \left (\frac {a^{\frac {1}{4}} x +\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right )}{8 a^{\frac {1}{4}}}\) | \(107\) |
1/8*(3*sum(ln((-_R*x+(a*x^4+b)^(1/4))/x)/_R,_R=RootOf(_Z^8-a^2-b))*a^(1/4) -8*arctan(1/a^(1/4)/x*(a*x^4+b)^(1/4))+4*ln((a^(1/4)*x+(a*x^4+b)^(1/4))/(- a^(1/4)*x+(a*x^4+b)^(1/4))))/a^(1/4)
Time = 0.30 (sec) , antiderivative size = 415, normalized size of antiderivative = 0.86 \[ \int \frac {b-a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b-2 a x^4+x^8\right )} \, dx=-\frac {\left (3 i - 3\right ) \, \sqrt {2} \log \left (-\frac {\left (i + 1\right ) \, \sqrt {2} {\left (a^{2} + b\right )}^{\frac {1}{8}} x - 2 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{2 \, x}\right )}{16 \, {\left (a^{2} + b\right )}^{\frac {1}{8}}} + \frac {\left (3 i + 3\right ) \, \sqrt {2} \log \left (-\frac {-\left (i - 1\right ) \, \sqrt {2} {\left (a^{2} + b\right )}^{\frac {1}{8}} x - 2 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{2 \, x}\right )}{16 \, {\left (a^{2} + b\right )}^{\frac {1}{8}}} - \frac {\left (3 i + 3\right ) \, \sqrt {2} \log \left (-\frac {\left (i - 1\right ) \, \sqrt {2} {\left (a^{2} + b\right )}^{\frac {1}{8}} x - 2 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{2 \, x}\right )}{16 \, {\left (a^{2} + b\right )}^{\frac {1}{8}}} + \frac {\left (3 i - 3\right ) \, \sqrt {2} \log \left (-\frac {-\left (i + 1\right ) \, \sqrt {2} {\left (a^{2} + b\right )}^{\frac {1}{8}} x - 2 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{2 \, x}\right )}{16 \, {\left (a^{2} + b\right )}^{\frac {1}{8}}} - \frac {3 \, \log \left (\frac {{\left (a^{2} + b\right )}^{\frac {1}{8}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{8 \, {\left (a^{2} + b\right )}^{\frac {1}{8}}} + \frac {3 \, \log \left (-\frac {{\left (a^{2} + b\right )}^{\frac {1}{8}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{8 \, {\left (a^{2} + b\right )}^{\frac {1}{8}}} + \frac {3 i \, \log \left (\frac {i \, {\left (a^{2} + b\right )}^{\frac {1}{8}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{8 \, {\left (a^{2} + b\right )}^{\frac {1}{8}}} - \frac {3 i \, \log \left (\frac {-i \, {\left (a^{2} + b\right )}^{\frac {1}{8}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{8 \, {\left (a^{2} + b\right )}^{\frac {1}{8}}} + \frac {\log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} - \frac {\log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} - \frac {i \, \log \left (\frac {i \, a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} + \frac {i \, \log \left (\frac {-i \, a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} \]
-(3/16*I - 3/16)*sqrt(2)*log(-1/2*((I + 1)*sqrt(2)*(a^2 + b)^(1/8)*x - 2*( a*x^4 + b)^(1/4))/x)/(a^2 + b)^(1/8) + (3/16*I + 3/16)*sqrt(2)*log(-1/2*(- (I - 1)*sqrt(2)*(a^2 + b)^(1/8)*x - 2*(a*x^4 + b)^(1/4))/x)/(a^2 + b)^(1/8 ) - (3/16*I + 3/16)*sqrt(2)*log(-1/2*((I - 1)*sqrt(2)*(a^2 + b)^(1/8)*x - 2*(a*x^4 + b)^(1/4))/x)/(a^2 + b)^(1/8) + (3/16*I - 3/16)*sqrt(2)*log(-1/2 *(-(I + 1)*sqrt(2)*(a^2 + b)^(1/8)*x - 2*(a*x^4 + b)^(1/4))/x)/(a^2 + b)^( 1/8) - 3/8*log(((a^2 + b)^(1/8)*x + (a*x^4 + b)^(1/4))/x)/(a^2 + b)^(1/8) + 3/8*log(-((a^2 + b)^(1/8)*x - (a*x^4 + b)^(1/4))/x)/(a^2 + b)^(1/8) + 3/ 8*I*log((I*(a^2 + b)^(1/8)*x + (a*x^4 + b)^(1/4))/x)/(a^2 + b)^(1/8) - 3/8 *I*log((-I*(a^2 + b)^(1/8)*x + (a*x^4 + b)^(1/4))/x)/(a^2 + b)^(1/8) + 1/2 *log((a^(1/4)*x + (a*x^4 + b)^(1/4))/x)/a^(1/4) - 1/2*log(-(a^(1/4)*x - (a *x^4 + b)^(1/4))/x)/a^(1/4) - 1/2*I*log((I*a^(1/4)*x + (a*x^4 + b)^(1/4))/ x)/a^(1/4) + 1/2*I*log((-I*a^(1/4)*x + (a*x^4 + b)^(1/4))/x)/a^(1/4)
Timed out. \[ \int \frac {b-a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b-2 a x^4+x^8\right )} \, dx=\text {Timed out} \]
\[ \int \frac {b-a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b-2 a x^4+x^8\right )} \, dx=\int { \frac {2 \, x^{8} - a x^{4} + b}{{\left (x^{8} - 2 \, a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {b-a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b-2 a x^4+x^8\right )} \, dx=\int { \frac {2 \, x^{8} - a x^{4} + b}{{\left (x^{8} - 2 \, a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {b-a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b-2 a x^4+x^8\right )} \, dx=\int -\frac {2\,x^8-a\,x^4+b}{{\left (a\,x^4+b\right )}^{1/4}\,\left (-x^8+2\,a\,x^4+b\right )} \,d x \]