Integrand size = 30, antiderivative size = 105 \[ \int \frac {\sqrt [4]{-1+x^4} \left (2+x^4\right )}{x^2 \left (2+2 x^4+x^8\right )} \, dx=-\frac {\sqrt [4]{-1+x^4}}{x}+\frac {1}{8} \text {RootSum}\left [5-6 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-5 \log (x)+5 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )+3 \log (x) \text {$\#$1}^4-3 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
Time = 0.25 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{-1+x^4} \left (2+x^4\right )}{x^2 \left (2+2 x^4+x^8\right )} \, dx=-\frac {\sqrt [4]{-1+x^4}}{x}+\frac {1}{8} \text {RootSum}\left [5-6 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-5 \log (x)+5 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )+3 \log (x) \text {$\#$1}^4-3 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
-((-1 + x^4)^(1/4)/x) + RootSum[5 - 6*#1^4 + 2*#1^8 & , (-5*Log[x] + 5*Log [(-1 + x^4)^(1/4) - x*#1] + 3*Log[x]*#1^4 - 3*Log[(-1 + x^4)^(1/4) - x*#1] *#1^4)/(-3*#1^3 + 2*#1^7) & ]/8
Result contains complex when optimal does not.
Time = 1.36 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.81, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, 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.
| \(\displaystyle \int \frac {\sqrt [4]{x^4-1} \left (x^4+2\right )}{x^2 \left (x^8+2 x^4+2\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt [4]{x^4-1}}{x^2}+\frac {\left (-x^4-1\right ) \sqrt [4]{x^4-1} x^2}{x^8+2 x^4+2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (\frac {1}{12}+\frac {i}{12}\right ) \sqrt [4]{x^4-1} x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,\left (-\frac {1}{2}+\frac {i}{2}\right ) x^4\right )}{\sqrt [4]{1-x^4}}-\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) \sqrt [4]{x^4-1} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\left (-\frac {1}{2}-\frac {i}{2}\right ) x^4,x^4\right )}{\sqrt [4]{1-x^4}}-\left (\frac {1}{4}+\frac {i}{2}\right ) \left (\frac {3}{5}-\frac {i}{5}\right )^{3/4} \arctan \left (\frac {x}{\sqrt [4]{\frac {3}{5}-\frac {i}{5}} \sqrt [4]{x^4-1}}\right )-\left (\frac {1}{4}-\frac {i}{2}\right ) \left (\frac {3}{5}+\frac {i}{5}\right )^{3/4} \arctan \left (\frac {x}{\sqrt [4]{\frac {3}{5}+\frac {i}{5}} \sqrt [4]{x^4-1}}\right )+\left (\frac {1}{4}+\frac {i}{2}\right ) \left (\frac {3}{5}-\frac {i}{5}\right )^{3/4} \text {arctanh}\left (\frac {x}{\sqrt [4]{\frac {3}{5}-\frac {i}{5}} \sqrt [4]{x^4-1}}\right )+\left (\frac {1}{4}-\frac {i}{2}\right ) \left (\frac {3}{5}+\frac {i}{5}\right )^{3/4} \text {arctanh}\left (\frac {x}{\sqrt [4]{\frac {3}{5}+\frac {i}{5}} \sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{x^4-1}}{x}\) |
-((-1 + x^4)^(1/4)/x) - ((1/12 + I/12)*x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, -1/4, 1, 7/4, x^4, (-1/2 + I/2)*x^4])/(1 - x^4)^(1/4) - ((1/12 - I/12)*x^3 *(-1 + x^4)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, (-1/2 - I/2)*x^4, x^4])/(1 - x^4)^(1/4) - (1/4 + I/2)*(3/5 - I/5)^(3/4)*ArcTan[x/((3/5 - I/5)^(1/4)*(- 1 + x^4)^(1/4))] - (1/4 - I/2)*(3/5 + I/5)^(3/4)*ArcTan[x/((3/5 + I/5)^(1/ 4)*(-1 + x^4)^(1/4))] + (1/4 + I/2)*(3/5 - I/5)^(3/4)*ArcTanh[x/((3/5 - I/ 5)^(1/4)*(-1 + x^4)^(1/4))] + (1/4 - I/2)*(3/5 + I/5)^(3/4)*ArcTanh[x/((3/ 5 + I/5)^(1/4)*(-1 + x^4)^(1/4))]
3.16.16.3.1 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 1.
Time = 13.19 (sec) , antiderivative size = 4181, normalized size of antiderivative = 39.82
\[\text {output too large to display}\]
-(x^4-1)^(1/4)/x+(-8192*RootOf(33554432*_Z^8-24576*_Z^4+5)^5*ln(-(-16384*R ootOf(33554432*_Z^8-24576*_Z^4+5)^5*x^12+524288*(x^12-3*x^8+3*x^4-1)^(1/4) *RootOf(33554432*_Z^8-24576*_Z^4+5)^6*x^9+32768*RootOf(33554432*_Z^8-24576 *_Z^4+5)^5*x^8-4*RootOf(33554432*_Z^8-24576*_Z^4+5)*x^12-1048576*(x^12-3*x ^8+3*x^4-1)^(1/4)*RootOf(33554432*_Z^8-24576*_Z^4+5)^6*x^5-192*RootOf(3355 4432*_Z^8-24576*_Z^4+5)^2*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9+512*(x^12-3*x^8+3 *x^4-1)^(1/2)*RootOf(33554432*_Z^8-24576*_Z^4+5)^3*x^6-12288*(x^12-3*x^8+3 *x^4-1)^(3/4)*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*x^3-16384*RootOf(335544 32*_Z^8-24576*_Z^4+5)^5*x^4+12*RootOf(33554432*_Z^8-24576*_Z^4+5)*x^8+5242 88*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(33554432*_Z^8-24576*_Z^4+5)^6*x+384*R ootOf(33554432*_Z^8-24576*_Z^4+5)^2*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5-512*(x^ 12-3*x^8+3*x^4-1)^(1/2)*RootOf(33554432*_Z^8-24576*_Z^4+5)^3*x^2+5*(x^12-3 *x^8+3*x^4-1)^(3/4)*x^3-12*RootOf(33554432*_Z^8-24576*_Z^4+5)*x^4-192*Root Of(33554432*_Z^8-24576*_Z^4+5)^2*(x^12-3*x^8+3*x^4-1)^(1/4)*x+4*RootOf(335 54432*_Z^8-24576*_Z^4+5))/(4096*x^4*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-x ^4+1)/(8192*x*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3*x-1)^2/(1+x)^2/(-1+x) ^2/(8192*x*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3*x+1)^2)+3*RootOf(3355443 2*_Z^8-24576*_Z^4+5)*ln(-(-16384*RootOf(33554432*_Z^8-24576*_Z^4+5)^5*x^12 +524288*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(33554432*_Z^8-24576*_Z^4+5)^6*x^ 9+32768*RootOf(33554432*_Z^8-24576*_Z^4+5)^5*x^8-4*RootOf(33554432*_Z^8...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 5.76 (sec) , antiderivative size = 1027, normalized size of antiderivative = 9.78 \[ \int \frac {\sqrt [4]{-1+x^4} \left (2+x^4\right )}{x^2 \left (2+2 x^4+x^8\right )} \, dx=\text {Too large to display} \]
-1/32*(sqrt(2)*sqrt(-sqrt(-I + 3)*sqrt(2))*x*log(-(2*sqrt(-I + 3)*sqrt(2)* ((63*I + 16)*x^7 + (47*I + 79)*x^3)*(x^4 - 1)^(1/4) + 4*(-(63*I + 16)*x^5 - (47*I + 79)*x)*(x^4 - 1)^(3/4) + (2*sqrt(2)*((63*I + 16)*x^6 + (47*I + 7 9)*x^2)*sqrt(x^4 - 1) - sqrt(-I + 3)*((104*I + 13)*x^8 + (50*I + 120)*x^4 - 44*I - 38))*sqrt(-sqrt(-I + 3)*sqrt(2)))/(x^8 + 2*x^4 + 2)) - sqrt(2)*sq rt(-sqrt(-I + 3)*sqrt(2))*x*log(-(2*sqrt(-I + 3)*sqrt(2)*((63*I + 16)*x^7 + (47*I + 79)*x^3)*(x^4 - 1)^(1/4) + 4*(-(63*I + 16)*x^5 - (47*I + 79)*x)* (x^4 - 1)^(3/4) + (2*sqrt(2)*(-(63*I + 16)*x^6 - (47*I + 79)*x^2)*sqrt(x^4 - 1) - sqrt(-I + 3)*(-(104*I + 13)*x^8 - (50*I + 120)*x^4 + 44*I + 38))*s qrt(-sqrt(-I + 3)*sqrt(2)))/(x^8 + 2*x^4 + 2)) + sqrt(2)*sqrt(-sqrt(I + 3) *sqrt(2))*x*log(-(2*sqrt(I + 3)*sqrt(2)*(-(63*I - 16)*x^7 - (47*I - 79)*x^ 3)*(x^4 - 1)^(1/4) + 4*((63*I - 16)*x^5 + (47*I - 79)*x)*(x^4 - 1)^(3/4) + (2*sqrt(2)*(-(63*I - 16)*x^6 - (47*I - 79)*x^2)*sqrt(x^4 - 1) - sqrt(I + 3)*(-(104*I - 13)*x^8 - (50*I - 120)*x^4 + 44*I - 38))*sqrt(-sqrt(I + 3)*s qrt(2)))/(x^8 + 2*x^4 + 2)) - sqrt(2)*sqrt(-sqrt(I + 3)*sqrt(2))*x*log(-(2 *sqrt(I + 3)*sqrt(2)*(-(63*I - 16)*x^7 - (47*I - 79)*x^3)*(x^4 - 1)^(1/4) + 4*((63*I - 16)*x^5 + (47*I - 79)*x)*(x^4 - 1)^(3/4) + (2*sqrt(2)*((63*I - 16)*x^6 + (47*I - 79)*x^2)*sqrt(x^4 - 1) - sqrt(I + 3)*((104*I - 13)*x^8 + (50*I - 120)*x^4 - 44*I + 38))*sqrt(-sqrt(I + 3)*sqrt(2)))/(x^8 + 2*x^4 + 2)) + sqrt(2)*sqrt(sqrt(I + 3)*sqrt(2))*x*log(-(2*sqrt(I + 3)*sqrt(2...
Timed out. \[ \int \frac {\sqrt [4]{-1+x^4} \left (2+x^4\right )}{x^2 \left (2+2 x^4+x^8\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.29 \[ \int \frac {\sqrt [4]{-1+x^4} \left (2+x^4\right )}{x^2 \left (2+2 x^4+x^8\right )} \, dx=\int { \frac {{\left (x^{4} + 2\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{{\left (x^{8} + 2 \, x^{4} + 2\right )} x^{2}} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.36 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.98 \[ \int \frac {\sqrt [4]{-1+x^4} \left (2+x^4\right )}{x^2 \left (2+2 x^4+x^8\right )} \, dx=\text {Too large to display} \]
1/1152921504606846976*I*(8*I + 8)^(65/4)*(-I + 2)^(1/4)*log(I*(-4994214705 84397634687058932589151711107044531621558329071816032093522024492452282368 00000*I + 3457533257891983624756561841001819538433385218918480739727957145 2628447849477465702400000)^(1/4) - (4427218577690292387840*I - 13281655733 070877163520)*(x^4 - 1)^(1/4)/x) - 1/1152921504606846976*I*(8*I + 8)^(65/4 )*(-I + 2)^(1/4)*log(I*(-4994214705843976346870589325891517111070445316215 5832907181603209352202449245228236800000*I + 34575332578919836247565618410 018195384333852189184807397279571452628447849477465702400000)^(1/4) + (442 7218577690292387840*I - 13281655733070877163520)*(x^4 - 1)^(1/4)/x) - 1/42 94967296*(8*I + 8)^(33/4)*(-I + 2)^(1/4)*log(-I*(-375722955104843938688593 78145757959465275804876800000*I + 2601158919956611883228726179321704886057 5557222400000)^(1/4) + (12369505812480*I + 4123168604160)*(x^4 - 1)^(1/4)/ x) + 1/4294967296*(8*I + 8)^(33/4)*(-I + 2)^(1/4)*log(-I*(-375722955104843 93868859378145757959465275804876800000*I + 2601158919956611883228726179321 7048860575557222400000)^(1/4) - (12369505812480*I + 4123168604160)*(x^4 - 1)^(1/4)/x) - (x^4 - 1)^(1/4)/x - 1/32768*(8*I + 8)^(33/4)*(I + 2)^(1/4)*l og(I*(37572295510484393868859378145757959465275804876800000*I + 2601158919 9566118832287261793217048860575557222400000)^(1/4) - (12369505812480*I - 4 123168604160)*(x^4 - 1)^(1/4)/x)/(sqrt(sqrt(2) + 2) + I*sqrt(-sqrt(2) + 2) )^17 + 1/32768*(8*I + 8)^(33/4)*(I + 2)^(1/4)*log(I*(375722955104843938...
Not integrable
Time = 6.69 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.29 \[ \int \frac {\sqrt [4]{-1+x^4} \left (2+x^4\right )}{x^2 \left (2+2 x^4+x^8\right )} \, dx=\int \frac {{\left (x^4-1\right )}^{1/4}\,\left (x^4+2\right )}{x^2\,\left (x^8+2\,x^4+2\right )} \,d x \]