Integrand size = 21, antiderivative size = 137 \[ \int \frac {1}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=-\frac {\left (-x^2+x^4\right )^{3/4}}{2 x \left (-1+x^2\right )}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{4 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{4 \sqrt [4]{2}}+\frac {1}{8} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
Time = 0.35 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=-\frac {\sqrt {x} \left (4 \sqrt {x}+2^{3/4} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+2^{3/4} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-\sqrt [4]{-1+x^2} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{-1+x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{8 \sqrt [4]{x^2 \left (-1+x^2\right )}} \]
-1/8*(Sqrt[x]*(4*Sqrt[x] + 2^(3/4)*(-1 + x^2)^(1/4)*ArcTan[(2^(1/4)*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)*RootSum[2 - 2*#1^4 + #1^8 & , (-Log[S qrt[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 = 0.56 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.78, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, 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.
\(\displaystyle \int \frac {1}{\sqrt [4]{x^4-x^2} \left (x^8-1\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^2-1} \int -\frac {1}{\sqrt {x} \sqrt [4]{x^2-1} \left (1-x^8\right )}dx}{\sqrt [4]{x^4-x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^2-1} \int \frac {1}{\sqrt {x} \sqrt [4]{x^2-1} \left (1-x^8\right )}dx}{\sqrt [4]{x^4-x^2}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \frac {1}{\sqrt [4]{x^2-1} \left (1-x^8\right )}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \left (\frac {1}{2 \left (1-x^4\right ) \sqrt [4]{x^2-1}}+\frac {1}{2 \left (x^4+1\right ) \sqrt [4]{x^2-1}}\right )d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2-1} \left (\frac {\arctan \left (\frac {\sqrt [4]{1-i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{8 \sqrt [4]{1-i}}+\frac {\arctan \left (\frac {\sqrt [4]{1+i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{8 \sqrt [4]{1+i}}+\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{8 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{1-i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{8 \sqrt [4]{1-i}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{1+i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{8 \sqrt [4]{1+i}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{8 \sqrt [4]{2}}+\frac {\sqrt {x}}{4 \sqrt [4]{x^2-1}}\right )}{\sqrt [4]{x^4-x^2}}\) |
(-2*Sqrt[x]*(-1 + x^2)^(1/4)*(Sqrt[x]/(4*(-1 + x^2)^(1/4)) + ArcTan[((1 - I)^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)]/(8*(1 - I)^(1/4)) + ArcTan[((1 + I)^(1 /4)*Sqrt[x])/(-1 + x^2)^(1/4)]/(8*(1 + I)^(1/4)) + ArcTan[(2^(1/4)*Sqrt[x] )/(-1 + x^2)^(1/4)]/(8*2^(1/4)) + ArcTanh[((1 - I)^(1/4)*Sqrt[x])/(-1 + x^ 2)^(1/4)]/(8*(1 - I)^(1/4)) + ArcTanh[((1 + I)^(1/4)*Sqrt[x])/(-1 + x^2)^( 1/4)]/(8*(1 + I)^(1/4)) + ArcTanh[(2^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)]/(8*2 ^(1/4))))/(-x^2 + x^4)^(1/4)
3.20.50.3.1 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]
Timed out.
\[\int \frac {1}{\left (x^{4}-x^{2}\right )^{\frac {1}{4}} \left (x^{8}-1\right )}d x\]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 3.83 (sec) , antiderivative size = 1469, normalized size of antiderivative = 10.72 \[ \int \frac {1}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\text {Too large to display} \]
-1/32*(sqrt(2)*(x^3 - x)*sqrt(sqrt(2)*sqrt(-I + 1))*log(-(2*sqrt(2)*sqrt(- I + 1)*((7*I + 1)*x^4 + (I - 7)*x^2)*(x^4 - x^2)^(1/4) + 4*(x^4 - x^2)^(3/ 4)*((3*I + 4)*x^2 + 4*I - 3) + (2*sqrt(-I + 1)*sqrt(x^4 - x^2)*((7*I + 1)* x^3 + (I - 7)*x) - sqrt(2)*(-(10*I + 5)*x^5 - (2*I - 14)*x^3 + (4*I - 3)*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)*((7*I + 1)*x^4 + (I - 7)*x^2)* (x^4 - x^2)^(1/4) + 4*(x^4 - x^2)^(3/4)*((3*I + 4)*x^2 + 4*I - 3) + (2*sqr t(-I + 1)*sqrt(x^4 - x^2)*(-(7*I + 1)*x^3 - (I - 7)*x) - sqrt(2)*((10*I + 5)*x^5 + (2*I - 14)*x^3 - (4*I - 3)*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)*(x^4 - x^2)^(1/4)*(-(7*I - 1)*x^4 - (I + 7)*x^2) + 4*(x^4 - x^2)^(3/4 )*(-(3*I - 4)*x^2 - 4*I - 3) + (2*sqrt(I + 1)*sqrt(x^4 - x^2)*(-(7*I - 1)* x^3 - (I + 7)*x) - sqrt(2)*((10*I - 5)*x^5 + (2*I + 14)*x^3 - (4*I + 3)*x) )*sqrt(sqrt(2)*sqrt(I + 1)))/(x^5 + x)) - sqrt(2)*(x^3 - x)*sqrt(sqrt(2)*s qrt(I + 1))*log(-(2*sqrt(2)*sqrt(I + 1)*(x^4 - x^2)^(1/4)*(-(7*I - 1)*x^4 - (I + 7)*x^2) + 4*(x^4 - x^2)^(3/4)*(-(3*I - 4)*x^2 - 4*I - 3) + (2*sqrt( I + 1)*sqrt(x^4 - x^2)*((7*I - 1)*x^3 + (I + 7)*x) - sqrt(2)*(-(10*I - 5)* x^5 - (2*I + 14)*x^3 + (4*I + 3)*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 )*(x^4 - x^2)^(1/4)*((7*I - 1)*x^4 + (I + 7)*x^2) + 4*(x^4 - x^2)^(3/4)...
Not integrable
Time = 0.87 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.23 \[ \int \frac {1}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\int \frac {1}{\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}\, dx \]
Not integrable
Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.15 \[ \int \frac {1}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} - 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}}} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.33 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.78 \[ \int \frac {1}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\frac {1}{8} \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}{16} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} - {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + 8 i \, \left (-\frac {1}{33554432} i + \frac {1}{33554432}\right )^{\frac {1}{4}} \log \left (i \, \left (-37778931862957161709568 i + 37778931862957161709568\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 524288\right ) - 8 i \, \left (-\frac {1}{33554432} i + \frac {1}{33554432}\right )^{\frac {1}{4}} \log \left (-i \, \left (-37778931862957161709568 i + 37778931862957161709568\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 524288\right ) + i \, \left (\frac {1}{8192} i + \frac {1}{8192}\right )^{\frac {1}{4}} \log \left (\left (549755813888 i + 549755813888\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1024 i\right ) - \left (\frac {1}{8192} i + \frac {1}{8192}\right )^{\frac {1}{4}} \log \left (i \, \left (549755813888 i + 549755813888\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1024 i\right ) + \left (\frac {1}{8192} i + \frac {1}{8192}\right )^{\frac {1}{4}} \log \left (-i \, \left (549755813888 i + 549755813888\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1024 i\right ) - i \, \left (\frac {1}{8192} i + \frac {1}{8192}\right )^{\frac {1}{4}} \log \left (-\left (549755813888 i + 549755813888\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1024 i\right ) + \left (-\frac {1}{8192} i + \frac {1}{8192}\right )^{\frac {1}{4}} \log \left (\left (-549755813888 i + 549755813888\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 1024\right ) - \left (-\frac {1}{8192} i + \frac {1}{8192}\right )^{\frac {1}{4}} \log \left (-\left (-549755813888 i + 549755813888\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 1024\right ) - \frac {1}{2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}} \]
1/8*2^(3/4)*arctan(1/2*2^(3/4)*(-1/x^2 + 1)^(1/4)) - 1/16*2^(3/4)*log(2^(1 /4) + (-1/x^2 + 1)^(1/4)) + 1/16*2^(3/4)*log(2^(1/4) - (-1/x^2 + 1)^(1/4)) + 8*I*(-1/33554432*I + 1/33554432)^(1/4)*log(I*(-37778931862957161709568* I + 37778931862957161709568)^(1/4)*(-1/x^2 + 1)^(1/4) - 524288) - 8*I*(-1/ 33554432*I + 1/33554432)^(1/4)*log(-I*(-37778931862957161709568*I + 377789 31862957161709568)^(1/4)*(-1/x^2 + 1)^(1/4) - 524288) + I*(1/8192*I + 1/81 92)^(1/4)*log((549755813888*I + 549755813888)^(1/4)*(-1/x^2 + 1)^(1/4) + 1 024*I) - (1/8192*I + 1/8192)^(1/4)*log(I*(549755813888*I + 549755813888)^( 1/4)*(-1/x^2 + 1)^(1/4) + 1024*I) + (1/8192*I + 1/8192)^(1/4)*log(-I*(5497 55813888*I + 549755813888)^(1/4)*(-1/x^2 + 1)^(1/4) + 1024*I) - I*(1/8192* I + 1/8192)^(1/4)*log(-(549755813888*I + 549755813888)^(1/4)*(-1/x^2 + 1)^ (1/4) + 1024*I) + (-1/8192*I + 1/8192)^(1/4)*log((-549755813888*I + 549755 813888)^(1/4)*(-1/x^2 + 1)^(1/4) - 1024) - (-1/8192*I + 1/8192)^(1/4)*log( -(-549755813888*I + 549755813888)^(1/4)*(-1/x^2 + 1)^(1/4) - 1024) - 1/2/( -1/x^2 + 1)^(1/4)
Not integrable
Time = 5.80 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.15 \[ \int \frac {1}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx=\int \frac {1}{\left (x^8-1\right )\,{\left (x^4-x^2\right )}^{1/4}} \,d x \]