Integrand size = 30, antiderivative size = 80 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\frac {\sqrt [4]{2+x^4} \left (-1+2 x^4\right )}{5 x^5}+\frac {1}{8} \text {RootSum}\left [-1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{2+x^4}-x \text {$\#$1}\right )}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\frac {\sqrt [4]{2+x^4} \left (-1+2 x^4\right )}{5 x^5}+\frac {1}{8} \text {RootSum}\left [-1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{2+x^4}-x \text {$\#$1}\right )}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
((2 + x^4)^(1/4)*(-1 + 2*x^4))/(5*x^5) + RootSum[-1 - #1^4 + #1^8 & , (-Lo g[x] + Log[(2 + x^4)^(1/4) - x*#1])/(-#1^3 + 2*#1^7) & ]/8
Leaf count is larger than twice the leaf count of optimal. \(1034\) vs. \(2(80)=160\).
Time = 2.88 (sec) , antiderivative size = 1034, normalized size of antiderivative = 12.92, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1388, 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.
\(\displaystyle \int \frac {\sqrt [4]{x^4+2} \left (x^8-4\right )}{x^6 \left (x^8-2 x^4-4\right )} \, dx\) |
\(\Big \downarrow \) 1388 |
\(\displaystyle \int \frac {\left (x^4-2\right ) \left (x^4+2\right )^{5/4}}{x^6 \left (x^8-2 x^4-4\right )}dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {\left (x^4+2\right )^{5/4}}{2 x^6}-\frac {\left (x^4+2\right )^{5/4}}{2 x^2}+\frac {\left (x^4-3\right ) \left (x^4+2\right )^{5/4} x^2}{2 \left (x^8-2 x^4-4\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\operatorname {AppellF1}\left (\frac {3}{4},-\frac {5}{4},1,\frac {7}{4},-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right ) x^3}{2^{3/4} \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {\sqrt [4]{2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},-\frac {x^4}{2},\frac {x^4}{1-\sqrt {5}}\right ) x^3}{3 \sqrt {5} \left (1-\sqrt {5}\right )}+\frac {\operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {5}{4},\frac {7}{4},\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right ) x^3}{2^{3/4} \sqrt {5} \left (1+\sqrt {5}\right )}-\frac {\sqrt [4]{2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {x^4}{1+\sqrt {5}},-\frac {x^4}{2}\right ) x^3}{3 \sqrt {5} \left (1+\sqrt {5}\right )}-\frac {1}{2} \sqrt [4]{x^4+2} x^3-\frac {3}{4} \arctan \left (\frac {x}{\sqrt [4]{x^4+2}}\right )+\frac {\sqrt [4]{\frac {1}{2} \left (-29+13 \sqrt {5}\right )} \arctan \left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{x^4+2}}\right )}{\sqrt {5}}+\frac {2 \sqrt [4]{-2+\sqrt {5}} \arctan \left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{x^4+2}}\right )}{\sqrt {5}}+\frac {\sqrt [4]{29+13 \sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{x^4+2}}\right )}{2^{3/4} \sqrt {5}}-\sqrt {\frac {2}{5}} \sqrt [4]{2+\sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{x^4+2}}\right )-\frac {\sqrt [4]{29+13 \sqrt {5}} \arctan \left (\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{x^4+2}}+1\right )}{2^{3/4} \sqrt {5}}+\sqrt {\frac {2}{5}} \sqrt [4]{2+\sqrt {5}} \arctan \left (\frac {2^{3/4} x}{\sqrt [4]{1+\sqrt {5}} \sqrt [4]{x^4+2}}+1\right )+\frac {3}{4} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+2}}\right )-\frac {\sqrt [4]{\frac {1}{2} \left (-29+13 \sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{x^4+2}}\right )}{\sqrt {5}}-\frac {2 \sqrt [4]{-2+\sqrt {5}} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt [4]{x^4+2}}\right )}{\sqrt {5}}-\frac {\sqrt [4]{29+13 \sqrt {5}} \log \left (\frac {2 x^2}{\sqrt {x^4+2}}-\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{x^4+2}}+\sqrt {2 \left (1+\sqrt {5}\right )}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{2+\sqrt {5}} \log \left (\frac {2 x^2}{\sqrt {x^4+2}}-\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{x^4+2}}+\sqrt {2 \left (1+\sqrt {5}\right )}\right )}{\sqrt {10}}+\frac {\sqrt [4]{29+13 \sqrt {5}} \log \left (\frac {2 x^2}{\sqrt {x^4+2}}+\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{x^4+2}}+\sqrt {2 \left (1+\sqrt {5}\right )}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{2+\sqrt {5}} \log \left (\frac {2 x^2}{\sqrt {x^4+2}}+\frac {2 \sqrt [4]{2 \left (1+\sqrt {5}\right )} x}{\sqrt [4]{x^4+2}}+\sqrt {2 \left (1+\sqrt {5}\right )}\right )}{\sqrt {10}}+\frac {\left (x^4+2\right )^{5/4}}{2 x}-\frac {\sqrt [4]{x^4+2}}{2 x}-\frac {\left (x^4+2\right )^{5/4}}{10 x^5}\) |
-1/2*(2 + x^4)^(1/4)/x - (x^3*(2 + x^4)^(1/4))/2 - (2 + x^4)^(5/4)/(10*x^5 ) + (2 + x^4)^(5/4)/(2*x) - (x^3*AppellF1[3/4, -5/4, 1, 7/4, -1/2*x^4, x^4 /(1 - Sqrt[5])])/(2^(3/4)*Sqrt[5]*(1 - Sqrt[5])) + (2^(1/4)*x^3*AppellF1[3 /4, -1/4, 1, 7/4, -1/2*x^4, x^4/(1 - Sqrt[5])])/(3*Sqrt[5]*(1 - Sqrt[5])) + (x^3*AppellF1[3/4, 1, -5/4, 7/4, x^4/(1 + Sqrt[5]), -1/2*x^4])/(2^(3/4)* Sqrt[5]*(1 + Sqrt[5])) - (2^(1/4)*x^3*AppellF1[3/4, 1, -1/4, 7/4, x^4/(1 + Sqrt[5]), -1/2*x^4])/(3*Sqrt[5]*(1 + Sqrt[5])) - (3*ArcTan[x/(2 + x^4)^(1 /4)])/4 + (2*(-2 + Sqrt[5])^(1/4)*ArcTan[((2/(-1 + Sqrt[5]))^(1/4)*x)/(2 + x^4)^(1/4)])/Sqrt[5] + (((-29 + 13*Sqrt[5])/2)^(1/4)*ArcTan[((2/(-1 + Sqr t[5]))^(1/4)*x)/(2 + x^4)^(1/4)])/Sqrt[5] - Sqrt[2/5]*(2 + Sqrt[5])^(1/4)* ArcTan[1 - (2^(3/4)*x)/((1 + Sqrt[5])^(1/4)*(2 + x^4)^(1/4))] + ((29 + 13* Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/((1 + Sqrt[5])^(1/4)*(2 + x^4)^(1/4) )])/(2^(3/4)*Sqrt[5]) + Sqrt[2/5]*(2 + Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)* x)/((1 + Sqrt[5])^(1/4)*(2 + x^4)^(1/4))] - ((29 + 13*Sqrt[5])^(1/4)*ArcTa n[1 + (2^(3/4)*x)/((1 + Sqrt[5])^(1/4)*(2 + x^4)^(1/4))])/(2^(3/4)*Sqrt[5] ) + (3*ArcTanh[x/(2 + x^4)^(1/4)])/4 - (2*(-2 + Sqrt[5])^(1/4)*ArcTanh[((2 /(-1 + Sqrt[5]))^(1/4)*x)/(2 + x^4)^(1/4)])/Sqrt[5] - (((-29 + 13*Sqrt[5]) /2)^(1/4)*ArcTanh[((2/(-1 + Sqrt[5]))^(1/4)*x)/(2 + x^4)^(1/4)])/Sqrt[5] + ((2 + Sqrt[5])^(1/4)*Log[Sqrt[2*(1 + Sqrt[5])] + (2*x^2)/Sqrt[2 + x^4] - (2*(2*(1 + Sqrt[5]))^(1/4)*x)/(2 + x^4)^(1/4)])/Sqrt[10] - ((29 + 13*Sq...
3.11.67.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
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 = 32.28 (sec) , antiderivative size = 6505, normalized size of antiderivative = 81.31
\[\text {output too large to display}\]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 4.41 (sec) , antiderivative size = 1398, normalized size of antiderivative = 17.48 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\text {Too large to display} \]
-1/80*(sqrt(5)*x^5*sqrt(-sqrt(sqrt(5) - 2))*log((4*(x^5 + sqrt(5)*x - x)*( x^4 + 2)^(3/4) - 2*(3*x^7 + 2*x^3 + sqrt(5)*(x^7 + 2*x^3))*(x^4 + 2)^(1/4) *sqrt(sqrt(5) - 2) + (2*(sqrt(5)*x^6 + x^6 + 4*x^2)*sqrt(x^4 + 2) - (7*x^8 + 14*x^4 + sqrt(5)*(3*x^8 + 6*x^4 + 4) + 4)*sqrt(sqrt(5) - 2))*sqrt(-sqrt (sqrt(5) - 2)))/(x^8 - 2*x^4 - 4)) - sqrt(5)*x^5*sqrt(-sqrt(sqrt(5) - 2))* log((4*(x^5 + sqrt(5)*x - x)*(x^4 + 2)^(3/4) - 2*(3*x^7 + 2*x^3 + sqrt(5)* (x^7 + 2*x^3))*(x^4 + 2)^(1/4)*sqrt(sqrt(5) - 2) - (2*(sqrt(5)*x^6 + x^6 + 4*x^2)*sqrt(x^4 + 2) - (7*x^8 + 14*x^4 + sqrt(5)*(3*x^8 + 6*x^4 + 4) + 4) *sqrt(sqrt(5) - 2))*sqrt(-sqrt(sqrt(5) - 2)))/(x^8 - 2*x^4 - 4)) + sqrt(5) *x^5*sqrt(-sqrt(-sqrt(5) - 2))*log((4*(x^5 - sqrt(5)*x - x)*(x^4 + 2)^(3/4 ) - 2*(3*x^7 + 2*x^3 - sqrt(5)*(x^7 + 2*x^3))*(x^4 + 2)^(1/4)*sqrt(-sqrt(5 ) - 2) + (2*(sqrt(5)*x^6 - x^6 - 4*x^2)*sqrt(x^4 + 2) + (7*x^8 + 14*x^4 - sqrt(5)*(3*x^8 + 6*x^4 + 4) + 4)*sqrt(-sqrt(5) - 2))*sqrt(-sqrt(-sqrt(5) - 2)))/(x^8 - 2*x^4 - 4)) - sqrt(5)*x^5*sqrt(-sqrt(-sqrt(5) - 2))*log((4*(x ^5 - sqrt(5)*x - x)*(x^4 + 2)^(3/4) - 2*(3*x^7 + 2*x^3 - sqrt(5)*(x^7 + 2* x^3))*(x^4 + 2)^(1/4)*sqrt(-sqrt(5) - 2) - (2*(sqrt(5)*x^6 - x^6 - 4*x^2)* sqrt(x^4 + 2) + (7*x^8 + 14*x^4 - sqrt(5)*(3*x^8 + 6*x^4 + 4) + 4)*sqrt(-s qrt(5) - 2))*sqrt(-sqrt(-sqrt(5) - 2)))/(x^8 - 2*x^4 - 4)) + sqrt(5)*x^5*( sqrt(5) - 2)^(1/4)*log((4*(x^5 + sqrt(5)*x - x)*(x^4 + 2)^(3/4) + 2*(3*x^7 + 2*x^3 + sqrt(5)*(x^7 + 2*x^3))*(x^4 + 2)^(1/4)*sqrt(sqrt(5) - 2) + (...
Timed out. \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.38 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\int { \frac {{\left (x^{8} - 4\right )} {\left (x^{4} + 2\right )}^{\frac {1}{4}}}{{\left (x^{8} - 2 \, x^{4} - 4\right )} x^{6}} \,d x } \]
Not integrable
Time = 0.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.38 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\int { \frac {{\left (x^{8} - 4\right )} {\left (x^{4} + 2\right )}^{\frac {1}{4}}}{{\left (x^{8} - 2 \, x^{4} - 4\right )} x^{6}} \,d x } \]
Not integrable
Time = 5.98 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=-\int \frac {{\left (x^4+2\right )}^{1/4}\,\left (x^8-4\right )}{x^6\,\left (-x^8+2\,x^4+4\right )} \,d x \]