Integrand size = 33, antiderivative size = 432 \[ \int \frac {\left (-1+a x^8\right ) \left (1+a x^8\right )^{3/4}}{1+x^8+a^2 x^{16}} \, dx=\frac {\left (1+\sqrt [4]{-1}\right ) \arctan \left (\frac {(-1)^{7/8} \sqrt {2+\sqrt {2}} \sqrt [8]{-1+2 a} x \sqrt [4]{1+a x^8}}{(-1)^{3/4} \sqrt [4]{-1+2 a} x^2+\sqrt {1+a x^8}}\right )}{8 \sqrt [8]{-1+2 a}}-\frac {i \left (-i \sqrt {2}+\sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \arctan \left (\frac {(-1)^{7/8} \left (-2+\sqrt {2}\right ) \sqrt [8]{-1+2 a} x \sqrt [4]{1+a x^8}}{(-1)^{3/4} \sqrt {2-\sqrt {2}} \sqrt [4]{-1+2 a} x^2+\sqrt {2-\sqrt {2}} \sqrt {1+a x^8}}\right )}{16 \sqrt [8]{-1+2 a}}+\frac {\left (\sqrt {2}+i \sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \text {arctanh}\left (\frac {(-1)^{7/8} \sqrt [4]{-1+2 a} x^2-\sqrt [8]{-1} \sqrt {1+a x^8}}{\sqrt {2-\sqrt {2}} \sqrt [8]{-1+2 a} x \sqrt [4]{1+a x^8}}\right )}{16 \sqrt [8]{-1+2 a}}+\frac {\left (1+\sqrt [4]{-1}\right ) \text {arctanh}\left (\frac {(-1)^{7/8} \sqrt [4]{-1+2 a} x^2-\sqrt [8]{-1} \sqrt {1+a x^8}}{\sqrt {2+\sqrt {2}} \sqrt [8]{-1+2 a} x \sqrt [4]{1+a x^8}}\right )}{8 \sqrt [8]{-1+2 a}} \]
1/8*(1+(-1)^(1/4))*arctan((-1)^(7/8)*(2+2^(1/2))^(1/2)*(-1+2*a)^(1/8)*x*(a *x^8+1)^(1/4)/((-1)^(3/4)*(-1+2*a)^(1/4)*x^2+(a*x^8+1)^(1/2)))/(-1+2*a)^(1 /8)-1/16*I*(-I*2^(1/2)+2-2^(1/2))*arctan((-1)^(7/8)*(-2+2^(1/2))*(-1+2*a)^ (1/8)*x*(a*x^8+1)^(1/4)/((-1)^(3/4)*(2-2^(1/2))^(1/2)*(-1+2*a)^(1/4)*x^2+( 2-2^(1/2))^(1/2)*(a*x^8+1)^(1/2)))/(-1+2*a)^(1/8)+1/16*(2^(1/2)+I*(2-2^(1/ 2)))*arctanh(((-1)^(7/8)*(-1+2*a)^(1/4)*x^2-(-1)^(1/8)*(a*x^8+1)^(1/2))/(2 -2^(1/2))^(1/2)/(-1+2*a)^(1/8)/x/(a*x^8+1)^(1/4))/(-1+2*a)^(1/8)+1/8*(1+(- 1)^(1/4))*arctanh(((-1)^(7/8)*(-1+2*a)^(1/4)*x^2-(-1)^(1/8)*(a*x^8+1)^(1/2 ))/(2+2^(1/2))^(1/2)/(-1+2*a)^(1/8)/x/(a*x^8+1)^(1/4))/(-1+2*a)^(1/8)
Time = 8.87 (sec) , antiderivative size = 385, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-1+a x^8\right ) \left (1+a x^8\right )^{3/4}}{1+x^8+a^2 x^{16}} \, dx=\frac {-2 \left (i+(-1)^{3/4}\right ) \text {arctanh}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\left (-i+(-1)^{3/4}\right ) \sqrt [4]{-1+2 a} x^2+\left (1+(-1)^{3/4}\right ) \sqrt {1+a x^8}\right )}{\sqrt [8]{-1+2 a} x \sqrt [4]{1+a x^8}}\right )+\sqrt {2} \left (i \left (i+\sqrt {3+2 \sqrt {2}}\right ) \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\left (\left (-i+(-1)^{3/4}\right ) \sqrt [4]{-1+2 a} x^2\right )+\left (1+(-1)^{3/4}\right ) \sqrt {1+a x^8}\right )}{\sqrt [8]{-1+2 a} x \sqrt [4]{1+a x^8}}\right )+\left (1+i \sqrt {3-2 \sqrt {2}}\right ) \arctan \left (\frac {2 \sqrt [8]{-1+2 a} x \sqrt [4]{1+a x^8}}{\left ((1-i)+\sqrt {2}\right ) \sqrt [4]{-1+2 a} x^2-\left ((1+i)+\sqrt {2}\right ) \sqrt {1+a x^8}}\right )+\left (-1-i \sqrt {3-2 \sqrt {2}}\right ) \text {arctanh}\left (\frac {\left ((1-i)+\sqrt {2}\right ) \sqrt [4]{-1+2 a} x^2+\left ((1+i)+\sqrt {2}\right ) \sqrt {1+a x^8}}{2 \sqrt [8]{-1+2 a} x \sqrt [4]{1+a x^8}}\right )\right )}{16 \sqrt [8]{-1+2 a}} \]
(-2*(I + (-1)^(3/4))*ArcTanh[((1/2 + I/2)*((-I + (-1)^(3/4))*(-1 + 2*a)^(1 /4)*x^2 + (1 + (-1)^(3/4))*Sqrt[1 + a*x^8]))/((-1 + 2*a)^(1/8)*x*(1 + a*x^ 8)^(1/4))] + Sqrt[2]*(I*(I + Sqrt[3 + 2*Sqrt[2]])*ArcTan[((1/2 + I/2)*(-(( -I + (-1)^(3/4))*(-1 + 2*a)^(1/4)*x^2) + (1 + (-1)^(3/4))*Sqrt[1 + a*x^8]) )/((-1 + 2*a)^(1/8)*x*(1 + a*x^8)^(1/4))] + (1 + I*Sqrt[3 - 2*Sqrt[2]])*Ar cTan[(2*(-1 + 2*a)^(1/8)*x*(1 + a*x^8)^(1/4))/(((1 - I) + Sqrt[2])*(-1 + 2 *a)^(1/4)*x^2 - ((1 + I) + Sqrt[2])*Sqrt[1 + a*x^8])] + (-1 - I*Sqrt[3 - 2 *Sqrt[2]])*ArcTanh[(((1 - I) + Sqrt[2])*(-1 + 2*a)^(1/4)*x^2 + ((1 + I) + Sqrt[2])*Sqrt[1 + a*x^8])/(2*(-1 + 2*a)^(1/8)*x*(1 + a*x^8)^(1/4))]))/(16* (-1 + 2*a)^(1/8))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.56 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.37, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, 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.
\(\displaystyle \int \frac {\left (a x^8-1\right ) \left (a x^8+1\right )^{3/4}}{a^2 x^{16}+x^8+1} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {\left (a-\frac {a (2 a+1)}{\sqrt {1-4 a^2}}\right ) \left (a x^8+1\right )^{3/4}}{2 a^2 x^8-\sqrt {1-4 a^2}+1}+\frac {\left (\frac {(2 a+1) a}{\sqrt {1-4 a^2}}+a\right ) \left (a x^8+1\right )^{3/4}}{2 a^2 x^8+\sqrt {1-4 a^2}+1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a \left (1-\frac {2 a+1}{\sqrt {1-4 a^2}}\right ) x \operatorname {AppellF1}\left (\frac {1}{8},-\frac {3}{4},1,\frac {9}{8},-a x^8,-\frac {2 a^2 x^8}{1-\sqrt {1-4 a^2}}\right )}{1-\sqrt {1-4 a^2}}+\frac {a \left (\frac {2 a+1}{\sqrt {1-4 a^2}}+1\right ) x \operatorname {AppellF1}\left (\frac {1}{8},-\frac {3}{4},1,\frac {9}{8},-a x^8,-\frac {2 a^2 x^8}{\sqrt {1-4 a^2}+1}\right )}{\sqrt {1-4 a^2}+1}\) |
(a*(1 - (1 + 2*a)/Sqrt[1 - 4*a^2])*x*AppellF1[1/8, -3/4, 1, 9/8, -(a*x^8), (-2*a^2*x^8)/(1 - Sqrt[1 - 4*a^2])])/(1 - Sqrt[1 - 4*a^2]) + (a*(1 + (1 + 2*a)/Sqrt[1 - 4*a^2])*x*AppellF1[1/8, -3/4, 1, 9/8, -(a*x^8), (-2*a^2*x^8 )/(1 + Sqrt[1 - 4*a^2])])/(1 + Sqrt[1 - 4*a^2])
3.31.28.3.1 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 = 0.38 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.09
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 a +1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (a \,x^{8}+1\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right )}{8}\) | \(38\) |
Timed out. \[ \int \frac {\left (-1+a x^8\right ) \left (1+a x^8\right )^{3/4}}{1+x^8+a^2 x^{16}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (-1+a x^8\right ) \left (1+a x^8\right )^{3/4}}{1+x^8+a^2 x^{16}} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-1+a x^8\right ) \left (1+a x^8\right )^{3/4}}{1+x^8+a^2 x^{16}} \, dx=\int { \frac {{\left (a x^{8} + 1\right )}^{\frac {3}{4}} {\left (a x^{8} - 1\right )}}{a^{2} x^{16} + x^{8} + 1} \,d x } \]
\[ \int \frac {\left (-1+a x^8\right ) \left (1+a x^8\right )^{3/4}}{1+x^8+a^2 x^{16}} \, dx=\int { \frac {{\left (a x^{8} + 1\right )}^{\frac {3}{4}} {\left (a x^{8} - 1\right )}}{a^{2} x^{16} + x^{8} + 1} \,d x } \]
Timed out. \[ \int \frac {\left (-1+a x^8\right ) \left (1+a x^8\right )^{3/4}}{1+x^8+a^2 x^{16}} \, dx=\int \frac {\left (a\,x^8-1\right )\,{\left (a\,x^8+1\right )}^{3/4}}{a^2\,x^{16}+x^8+1} \,d x \]