Integrand size = 35, antiderivative size = 197 \[ \int \frac {1-x^4+x^8}{x^2 \left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx=-\frac {\sqrt [4]{-1+x^4}}{x}+\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )-\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )+\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right ) \]
-(x^4-1)^(1/4)/x+1/10*(-10+10*5^(1/2))^(1/2)*arctan(1/2*(-2+2*5^(1/2))^(1/ 2)*x/(x^4-1)^(1/4))-1/10*(10+10*5^(1/2))^(1/2)*arctan(1/2*(2+2*5^(1/2))^(1 /2)*x/(x^4-1)^(1/4))-1/10*(-10+10*5^(1/2))^(1/2)*arctanh(1/2*(-2+2*5^(1/2) )^(1/2)*x/(x^4-1)^(1/4))+1/10*(10+10*5^(1/2))^(1/2)*arctanh(1/2*(2+2*5^(1/ 2))^(1/2)*x/(x^4-1)^(1/4))
Time = 0.78 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.94 \[ \int \frac {1-x^4+x^8}{x^2 \left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx=\frac {1}{10} \left (-\frac {10 \sqrt [4]{-1+x^4}}{x}+\sqrt {10 \left (-1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )-\sqrt {10 \left (1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )-\sqrt {10 \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )+\sqrt {10 \left (1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )\right ) \]
((-10*(-1 + x^4)^(1/4))/x + Sqrt[10*(-1 + Sqrt[5])]*ArcTan[(Sqrt[(-1 + Sqr t[5])/2]*x)/(-1 + x^4)^(1/4)] - Sqrt[10*(1 + Sqrt[5])]*ArcTan[(Sqrt[(1 + S qrt[5])/2]*x)/(-1 + x^4)^(1/4)] - Sqrt[10*(-1 + Sqrt[5])]*ArcTanh[(Sqrt[(- 1 + Sqrt[5])/2]*x)/(-1 + x^4)^(1/4)] + Sqrt[10*(1 + Sqrt[5])]*ArcTanh[(Sqr t[(1 + Sqrt[5])/2]*x)/(-1 + x^4)^(1/4)])/10
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.65 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.80, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, 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.
\(\displaystyle \int \frac {x^8-x^4+1}{x^2 \left (x^4-1\right )^{3/4} \left (x^8-x^4-1\right )} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {2 x^2 \sqrt [4]{x^4-1}}{x^8-x^4-1}-\frac {1}{x^2 \left (x^4-1\right )^{3/4}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 \sqrt [4]{x^4-1} x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,\frac {2 x^4}{1-\sqrt {5}}\right )}{3 \sqrt {5} \left (1-\sqrt {5}\right ) \sqrt [4]{1-x^4}}-\frac {4 \sqrt [4]{x^4-1} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {2 x^4}{1+\sqrt {5}},x^4\right )}{3 \sqrt {5} \left (1+\sqrt {5}\right ) \sqrt [4]{1-x^4}}-\frac {\sqrt [4]{x^4-1}}{x}\) |
-((-1 + x^4)^(1/4)/x) + (4*x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, -1/4, 1, 7/4 , x^4, (2*x^4)/(1 - Sqrt[5])])/(3*Sqrt[5]*(1 - Sqrt[5])*(1 - x^4)^(1/4)) - (4*x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, (2*x^4)/(1 + Sqrt[5]) , x^4])/(3*Sqrt[5]*(1 + Sqrt[5])*(1 - x^4)^(1/4))
3.25.34.3.1 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 = 11.40 (sec) , antiderivative size = 3552, normalized size of antiderivative = 18.03
\[\text {output too large to display}\]
-(x^4-1)^(1/4)/x+(1/10*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*ln ((-32000*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^12+1600*RootOf(6400*_Z^4-80*_Z^2- 1)^2*x^12+160*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^2+400*RootOf(6400*_Z^4- 80*_Z^2-1)^2-5)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^9+64000*RootOf(6400*_Z^4-8 0*_Z^2-1)^4*x^8-15*x^12-6*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^2+400*RootO f(6400*_Z^4-80*_Z^2-1)^2-5)*x^9+800*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(6400 *_Z^4-80*_Z^2-1)^2*x^6-3600*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^8+480*(x^12-3* x^8+3*x^4-1)^(3/4)*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf (6400*_Z^4-80*_Z^2-1)^2*x^3-320*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^2+400 *RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^5-32000* RootOf(6400*_Z^4-80*_Z^2-1)^4*x^4-20*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6+35*x^8 -8*(x^12-3*x^8+3*x^4-1)^(3/4)*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^ 2-5)*x^3+12*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^2+400*RootOf(6400*_Z^4-80 *_Z^2-1)^2-5)*x^5-800*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(6400*_Z^4-80*_Z^2- 1)^2*x^2+2400*x^4*RootOf(6400*_Z^4-80*_Z^2-1)^2+160*(x^12-3*x^8+3*x^4-1)^( 1/4)*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf(6400*_Z^4-80* _Z^2-1)^2*x+20*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2-25*x^4-6*(x^12-3*x^8+3*x^4-1 )^(1/4)*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*x-400*RootOf(6400 *_Z^4-80*_Z^2-1)^2+5)/(320*RootOf(6400*_Z^4-80*_Z^2-1)^3*x^2-12*RootOf(640 0*_Z^4-80*_Z^2-1)*x^2-1)/(320*RootOf(6400*_Z^4-80*_Z^2-1)^3*x^2-12*Root...
Leaf count of result is larger than twice the leaf count of optimal. 1271 vs. \(2 (137) = 274\).
Time = 9.48 (sec) , antiderivative size = 1271, normalized size of antiderivative = 6.45 \[ \int \frac {1-x^4+x^8}{x^2 \left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx=\text {Too large to display} \]
-1/40*(sqrt(10)*x*sqrt(-sqrt(5) + 1)*log((sqrt(10)*sqrt(x^4 - 1)*(5*x^2 + sqrt(5)*(2*x^6 - x^2))*sqrt(-sqrt(5) + 1) - sqrt(10)*(5*x^8 - 5*x^4 + sqrt (5)*(2*x^4 - 1))*sqrt(-sqrt(5) + 1) + 10*(2*x^5 + sqrt(5)*x - x)*(x^4 - 1) ^(3/4) + 10*(x^7 - 3*x^3 - sqrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^ 4 - 1)) - sqrt(10)*x*sqrt(-sqrt(5) + 1)*log(-(sqrt(10)*sqrt(x^4 - 1)*(5*x^ 2 + sqrt(5)*(2*x^6 - x^2))*sqrt(-sqrt(5) + 1) - sqrt(10)*(5*x^8 - 5*x^4 + sqrt(5)*(2*x^4 - 1))*sqrt(-sqrt(5) + 1) - 10*(2*x^5 + sqrt(5)*x - x)*(x^4 - 1)^(3/4) - 10*(x^7 - 3*x^3 - sqrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^4 - 1)) + sqrt(10)*x*sqrt(-sqrt(5) - 1)*log((sqrt(10)*sqrt(x^4 - 1)*(5 *x^2 - sqrt(5)*(2*x^6 - x^2))*sqrt(-sqrt(5) - 1) + sqrt(10)*(5*x^8 - 5*x^4 - sqrt(5)*(2*x^4 - 1))*sqrt(-sqrt(5) - 1) + 10*(2*x^5 - sqrt(5)*x - x)*(x ^4 - 1)^(3/4) - 10*(x^7 - 3*x^3 + sqrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x ^8 - x^4 - 1)) - sqrt(10)*x*sqrt(-sqrt(5) - 1)*log(-(sqrt(10)*sqrt(x^4 - 1 )*(5*x^2 - sqrt(5)*(2*x^6 - x^2))*sqrt(-sqrt(5) - 1) + sqrt(10)*(5*x^8 - 5 *x^4 - sqrt(5)*(2*x^4 - 1))*sqrt(-sqrt(5) - 1) - 10*(2*x^5 - sqrt(5)*x - x )*(x^4 - 1)^(3/4) + 10*(x^7 - 3*x^3 + sqrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4) )/(x^8 - x^4 - 1)) + sqrt(10)*x*sqrt(sqrt(5) - 1)*log((10*(2*x^5 + sqrt(5) *x - x)*(x^4 - 1)^(3/4) + (sqrt(10)*sqrt(x^4 - 1)*(5*x^2 + sqrt(5)*(2*x^6 - x^2)) + sqrt(10)*(5*x^8 - 5*x^4 + sqrt(5)*(2*x^4 - 1)))*sqrt(sqrt(5) - 1 ) - 10*(x^7 - 3*x^3 - sqrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^4 ...
Timed out. \[ \int \frac {1-x^4+x^8}{x^2 \left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx=\text {Timed out} \]
\[ \int \frac {1-x^4+x^8}{x^2 \left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx=\int { \frac {x^{8} - x^{4} + 1}{{\left (x^{8} - x^{4} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \]
\[ \int \frac {1-x^4+x^8}{x^2 \left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx=\int { \frac {x^{8} - x^{4} + 1}{{\left (x^{8} - x^{4} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \]
Timed out. \[ \int \frac {1-x^4+x^8}{x^2 \left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx=\int -\frac {x^8-x^4+1}{x^2\,{\left (x^4-1\right )}^{3/4}\,\left (-x^8+x^4+1\right )} \,d x \]