Integrand size = 26, antiderivative size = 272 \[ \int \frac {1+x^8}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=-\frac {\left (-x^2+x^4\right )^{3/4}}{x \left (-1+x^2\right )}+\arctan \left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{2 \sqrt [4]{2}}+\text {arctanh}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{2 \sqrt [4]{2}}+\frac {5}{8} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-1+\text {$\#$1}^4}\&\right ]+\frac {1}{8} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {2 \log (x)-2 \log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right )+3 \log (x) \text {$\#$1}^4-3 \log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}+\text {$\#$1}^5}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.79 \[ \int \frac {1+x^8}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=-\frac {\sqrt {x} \left (4 \sqrt {x}-4 \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+2^{3/4} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-4 \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+2^{3/4} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-\sqrt [4]{-1+x^2} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{-1+x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{4 \sqrt [4]{x^2 \left (-1+x^2\right )}} \]
-1/4*(Sqrt[x]*(4*Sqrt[x] - 4*(-1 + x^2)^(1/4)*ArcTan[Sqrt[x]/(-1 + x^2)^(1 /4)] + 2^(3/4)*(-1 + x^2)^(1/4)*ArcTan[(2^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)] - 4*(-1 + x^2)^(1/4)*ArcTanh[Sqrt[x]/(-1 + x^2)^(1/4)] + 2^(3/4)*(-1 + x^ 2)^(1/4)*ArcTanh[(2^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)] - (-1 + x^2)^(1/4)*Ro otSum[2 - 2*#1^4 + #1^8 & , (-Log[Sqrt[x]] + Log[(-1 + x^2)^(1/4) - Sqrt[x ]*#1])/#1 & ]))/(x^2*(-1 + x^2))^(1/4)
Result contains complex when optimal does not.
Time = 0.64 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2467, 25, 2035, 7276, 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.
| \(\displaystyle \int \frac {x^8+1}{\sqrt [4]{x^4-x^2} \left (x^8-1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^2-1} \int -\frac {x^8+1}{\sqrt {x} \sqrt [4]{x^2-1} \left (1-x^8\right )}dx}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^2-1} \int \frac {x^8+1}{\sqrt {x} \sqrt [4]{x^2-1} \left (1-x^8\right )}dx}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \frac {x^8+1}{\sqrt [4]{x^2-1} \left (1-x^8\right )}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \left (\frac {2}{\sqrt [4]{x^2-1} \left (1-x^8\right )}-\frac {1}{\sqrt [4]{x^2-1}}\right )d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2-1} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )+\frac {\arctan \left (\frac {\sqrt [4]{1-i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{1-i}}+\frac {\arctan \left (\frac {\sqrt [4]{1+i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{1+i}}+\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{2}}-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt [4]{1-i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{1-i}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{1+i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{1+i}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{2}}+\frac {\sqrt {x}}{2 \sqrt [4]{x^2-1}}\right )}{\sqrt [4]{x^4-x^2}}\) |
(-2*Sqrt[x]*(-1 + x^2)^(1/4)*(Sqrt[x]/(2*(-1 + x^2)^(1/4)) - ArcTan[Sqrt[x ]/(-1 + x^2)^(1/4)]/2 + ArcTan[((1 - I)^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)]/( 4*(1 - I)^(1/4)) + ArcTan[((1 + I)^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)]/(4*(1 + I)^(1/4)) + ArcTan[(2^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)]/(4*2^(1/4)) - Arc Tanh[Sqrt[x]/(-1 + x^2)^(1/4)]/2 + ArcTanh[((1 - I)^(1/4)*Sqrt[x])/(-1 + x ^2)^(1/4)]/(4*(1 - I)^(1/4)) + ArcTanh[((1 + I)^(1/4)*Sqrt[x])/(-1 + x^2)^ (1/4)]/(4*(1 + I)^(1/4)) + ArcTanh[(2^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)]/(4* 2^(1/4))))/(-x^2 + x^4)^(1/4)
3.29.2.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.00 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.74
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\left (-\frac {\arctan \left (\frac {2^{\frac {3}{4}} \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{2 x}\right )}{2}+\frac {\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}\right )}{4}\right ) 2^{\frac {3}{4}}+\ln \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{4}}-x}{x}\right )-\ln \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{4}}+x}{x}\right )-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right )}{2}+2 \arctan \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x}\right )\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{4}}+2 x}{2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}\) | \(202\) |
-1/2/(x^4-x^2)^(1/4)*(((-1/2*arctan(1/2*2^(3/4)/x*(x^4-x^2)^(1/4))+1/4*ln( (-2^(1/4)*x-(x^4-x^2)^(1/4))/(2^(1/4)*x-(x^4-x^2)^(1/4))))*2^(3/4)+ln(((x^ 4-x^2)^(1/4)-x)/x)-ln(((x^4-x^2)^(1/4)+x)/x)-1/2*sum(ln((-_R*x+(x^4-x^2)^( 1/4))/x)/_R,_R=RootOf(_Z^8-2*_Z^4+2))+2*arctan(1/x*(x^4-x^2)^(1/4)))*(x^4- x^2)^(1/4)+2*x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 75.42 (sec) , antiderivative size = 1577, normalized size of antiderivative = 5.80 \[ \int \frac {1+x^8}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\text {Too large to display} \]
-1/16*(sqrt(2)*(x^3 - x)*sqrt(sqrt(2)*sqrt(-I + 1))*log(-(2*sqrt(2)*sqrt(- I + 1)*((379*I + 3)*x^4 + (3*I - 379)*x^2)*(x^4 - x^2)^(1/4) + 4*(x^4 - x^ 2)^(3/4)*((188*I + 191)*x^2 + 191*I - 188) + (2*sqrt(-I + 1)*sqrt(x^4 - x^ 2)*((379*I + 3)*x^3 + (3*I - 379)*x) - sqrt(2)*(-(567*I + 194)*x^5 - (6*I - 758)*x^3 + (191*I - 188)*x))*sqrt(sqrt(2)*sqrt(-I + 1)))/(x^5 + x)) - sq rt(2)*(x^3 - x)*sqrt(sqrt(2)*sqrt(-I + 1))*log(-(2*sqrt(2)*sqrt(-I + 1)*(( 379*I + 3)*x^4 + (3*I - 379)*x^2)*(x^4 - x^2)^(1/4) + 4*(x^4 - x^2)^(3/4)* ((188*I + 191)*x^2 + 191*I - 188) + (2*sqrt(-I + 1)*sqrt(x^4 - x^2)*(-(379 *I + 3)*x^3 - (3*I - 379)*x) - sqrt(2)*((567*I + 194)*x^5 + (6*I - 758)*x^ 3 - (191*I - 188)*x))*sqrt(sqrt(2)*sqrt(-I + 1)))/(x^5 + x)) + sqrt(2)*(x^ 3 - x)*sqrt(sqrt(2)*sqrt(I + 1))*log(-(2*sqrt(2)*sqrt(I + 1)*(-(379*I - 3) *x^4 - (3*I + 379)*x^2)*(x^4 - x^2)^(1/4) + 4*(x^4 - x^2)^(3/4)*(-(188*I - 191)*x^2 - 191*I - 188) + (2*sqrt(I + 1)*sqrt(x^4 - x^2)*(-(379*I - 3)*x^ 3 - (3*I + 379)*x) - sqrt(2)*((567*I - 194)*x^5 + (6*I + 758)*x^3 - (191*I + 188)*x))*sqrt(sqrt(2)*sqrt(I + 1)))/(x^5 + x)) - sqrt(2)*(x^3 - x)*sqrt (sqrt(2)*sqrt(I + 1))*log(-(2*sqrt(2)*sqrt(I + 1)*(-(379*I - 3)*x^4 - (3*I + 379)*x^2)*(x^4 - x^2)^(1/4) + 4*(x^4 - x^2)^(3/4)*(-(188*I - 191)*x^2 - 191*I - 188) + (2*sqrt(I + 1)*sqrt(x^4 - x^2)*((379*I - 3)*x^3 + (3*I + 3 79)*x) - sqrt(2)*(-(567*I - 194)*x^5 - (6*I + 758)*x^3 + (191*I + 188)*x)) *sqrt(sqrt(2)*sqrt(I + 1)))/(x^5 + x)) - sqrt(2)*(x^3 - x)*sqrt(-sqrt(2...
Not integrable
Time = 2.41 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.13 \[ \int \frac {1+x^8}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\int \frac {x^{8} + 1}{\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}\, dx \]
Integral((x**8 + 1)/((x**2*(x - 1)*(x + 1))**(1/4)*(x - 1)*(x + 1)*(x**2 + 1)*(x**4 + 1)), x)
Not integrable
Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.10 \[ \int \frac {1+x^8}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\int { \frac {x^{8} + 1}{{\left (x^{8} - 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}}} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.34 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.05 \[ \int \frac {1+x^8}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\frac {1}{4} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} - {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + 4 i \, \left (-\frac {1}{131072} i + \frac {1}{131072}\right )^{\frac {1}{4}} \log \left (i \, \left (-2251799813685248 i + 2251799813685248\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 8192\right ) - 4 i \, \left (-\frac {1}{131072} i + \frac {1}{131072}\right )^{\frac {1}{4}} \log \left (-i \, \left (-2251799813685248 i + 2251799813685248\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 8192\right ) + i \, \left (\frac {1}{512} i + \frac {1}{512}\right )^{\frac {1}{4}} \log \left (\left (134217728 i + 134217728\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 128 i\right ) - \left (\frac {1}{512} i + \frac {1}{512}\right )^{\frac {1}{4}} \log \left (i \, \left (134217728 i + 134217728\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 128 i\right ) + \left (\frac {1}{512} i + \frac {1}{512}\right )^{\frac {1}{4}} \log \left (-i \, \left (134217728 i + 134217728\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 128 i\right ) - i \, \left (\frac {1}{512} i + \frac {1}{512}\right )^{\frac {1}{4}} \log \left (-\left (134217728 i + 134217728\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 128 i\right ) + \left (-\frac {1}{512} i + \frac {1}{512}\right )^{\frac {1}{4}} \log \left (\left (-134217728 i + 134217728\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 128\right ) - \left (-\frac {1}{512} i + \frac {1}{512}\right )^{\frac {1}{4}} \log \left (-\left (-134217728 i + 134217728\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 128\right ) - \frac {1}{{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}} - \arctan \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{2} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{2} \, \log \left (-{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) \]
1/4*2^(3/4)*arctan(1/2*2^(3/4)*(-1/x^2 + 1)^(1/4)) - 1/8*2^(3/4)*log(2^(1/ 4) + (-1/x^2 + 1)^(1/4)) + 1/8*2^(3/4)*log(2^(1/4) - (-1/x^2 + 1)^(1/4)) + 4*I*(-1/131072*I + 1/131072)^(1/4)*log(I*(-2251799813685248*I + 225179981 3685248)^(1/4)*(-1/x^2 + 1)^(1/4) - 8192) - 4*I*(-1/131072*I + 1/131072)^( 1/4)*log(-I*(-2251799813685248*I + 2251799813685248)^(1/4)*(-1/x^2 + 1)^(1 /4) - 8192) + I*(1/512*I + 1/512)^(1/4)*log((134217728*I + 134217728)^(1/4 )*(-1/x^2 + 1)^(1/4) + 128*I) - (1/512*I + 1/512)^(1/4)*log(I*(134217728*I + 134217728)^(1/4)*(-1/x^2 + 1)^(1/4) + 128*I) + (1/512*I + 1/512)^(1/4)* log(-I*(134217728*I + 134217728)^(1/4)*(-1/x^2 + 1)^(1/4) + 128*I) - I*(1/ 512*I + 1/512)^(1/4)*log(-(134217728*I + 134217728)^(1/4)*(-1/x^2 + 1)^(1/ 4) + 128*I) + (-1/512*I + 1/512)^(1/4)*log((-134217728*I + 134217728)^(1/4 )*(-1/x^2 + 1)^(1/4) - 128) - (-1/512*I + 1/512)^(1/4)*log(-(-134217728*I + 134217728)^(1/4)*(-1/x^2 + 1)^(1/4) - 128) - 1/(-1/x^2 + 1)^(1/4) - arct an((-1/x^2 + 1)^(1/4)) + 1/2*log((-1/x^2 + 1)^(1/4) + 1) - 1/2*log(-(-1/x^ 2 + 1)^(1/4) + 1)
Not integrable
Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.10 \[ \int \frac {1+x^8}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\int \frac {x^8+1}{\left (x^8-1\right )\,{\left (x^4-x^2\right )}^{1/4}} \,d x \]