Integrand size = 30, antiderivative size = 80 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\frac {\sqrt [4]{2+x^4} \left (-1+2 x^4\right )}{5 x^5}+\frac {1}{8} \text {RootSum}\left [-1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{2+x^4}-x \text {$\#$1}\right )}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \] Output:
Unintegrable
Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\frac {\sqrt [4]{2+x^4} \left (-1+2 x^4\right )}{5 x^5}+\frac {1}{8} \text {RootSum}\left [-1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{2+x^4}-x \text {$\#$1}\right )}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \] Input:
Integrate[((2 + x^4)^(1/4)*(-4 + x^8))/(x^6*(-4 - 2*x^4 + x^8)),x]
Output:
((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 = 2.83 (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}\) |
Input:
Int[((2 + x^4)^(1/4)*(-4 + x^8))/(x^6*(-4 - 2*x^4 + x^8)),x]
Output:
-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...
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]
Time = 830.87 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96
method | result | size |
pseudoelliptic | \(\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}-1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{4}+2\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (2 \textit {\_R}^{4}-1\right )}\right ) x^{5}+16 \left (x^{4}+2\right )^{\frac {1}{4}} x^{4}-8 \left (x^{4}+2\right )^{\frac {1}{4}}}{40 x^{5}}\) | \(77\) |
trager | \(\text {Expression too large to display}\) | \(4166\) |
risch | \(\text {Expression too large to display}\) | \(6505\) |
Input:
int((x^4+2)^(1/4)*(x^8-4)/x^6/(x^8-2*x^4-4),x,method=_RETURNVERBOSE)
Output:
1/40*(5*sum(ln((-_R*x+(x^4+2)^(1/4))/x)/_R^3/(2*_R^4-1),_R=RootOf(_Z^8-_Z^ 4-1))*x^5+16*(x^4+2)^(1/4)*x^4-8*(x^4+2)^(1/4))/x^5
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 4.56 (sec) , antiderivative size = 1446, normalized size of antiderivative = 18.08 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\text {Too large to display} \] Input:
integrate((x^4+2)^(1/4)*(x^8-4)/x^6/(x^8-2*x^4-4),x, algorithm="fricas")
Output:
-1/80*(5*sqrt(1/5)*x^5*sqrt(-sqrt(sqrt(5) - 2))*log((sqrt(1/5)*(2*(5*x^6 + sqrt(5)*(x^6 + 4*x^2))*sqrt(x^4 + 2) - (15*x^8 + 30*x^4 + sqrt(5)*(7*x^8 + 14*x^4 + 4) + 20)*sqrt(sqrt(5) - 2))*sqrt(-sqrt(sqrt(5) - 2)) + 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))/(x^8 - 2*x^4 - 4)) - 5*sqrt(1/5)*x^5* sqrt(-sqrt(sqrt(5) - 2))*log(-(sqrt(1/5)*(2*(5*x^6 + sqrt(5)*(x^6 + 4*x^2) )*sqrt(x^4 + 2) - (15*x^8 + 30*x^4 + sqrt(5)*(7*x^8 + 14*x^4 + 4) + 20)*sq rt(sqrt(5) - 2))*sqrt(-sqrt(sqrt(5) - 2)) - 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))/(x^8 - 2*x^4 - 4)) + 5*sqrt(1/5)*x^5*sqrt(-sqrt(-sqrt(5) - 2))*log((sqrt(1/5)*(2*(5*x^6 - sqrt(5)*(x^6 + 4*x^2))*sqrt(x^4 + 2) - (15* x^8 + 30*x^4 - sqrt(5)*(7*x^8 + 14*x^4 + 4) + 20)*sqrt(-sqrt(5) - 2))*sqrt (-sqrt(-sqrt(5) - 2)) + 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))/(x^8 - 2*x^4 - 4)) - 5*sqrt(1/5)*x^5*sqrt(-sqrt(-sqrt(5) - 2))*log(-(sqrt(1/5) *(2*(5*x^6 - sqrt(5)*(x^6 + 4*x^2))*sqrt(x^4 + 2) - (15*x^8 + 30*x^4 - sqr t(5)*(7*x^8 + 14*x^4 + 4) + 20)*sqrt(-sqrt(5) - 2))*sqrt(-sqrt(-sqrt(5) - 2)) - 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))/(x^8 - 2*x^4 - 4)) + 5 *sqrt(1/5)*x^5*(sqrt(5) - 2)^(1/4)*log((4*(x^5 + sqrt(5)*x - x)*(x^4 + ...
Timed out. \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\text {Timed out} \] Input:
integrate((x**4+2)**(1/4)*(x**8-4)/x**6/(x**8-2*x**4-4),x)
Output:
Timed out
Not integrable
Time = 0.17 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.38 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\int { \frac {{\left (x^{8} - 4\right )} {\left (x^{4} + 2\right )}^{\frac {1}{4}}}{{\left (x^{8} - 2 \, x^{4} - 4\right )} x^{6}} \,d x } \] Input:
integrate((x^4+2)^(1/4)*(x^8-4)/x^6/(x^8-2*x^4-4),x, algorithm="maxima")
Output:
integrate((x^8 - 4)*(x^4 + 2)^(1/4)/((x^8 - 2*x^4 - 4)*x^6), x)
Not integrable
Time = 0.41 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.38 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\int { \frac {{\left (x^{8} - 4\right )} {\left (x^{4} + 2\right )}^{\frac {1}{4}}}{{\left (x^{8} - 2 \, x^{4} - 4\right )} x^{6}} \,d x } \] Input:
integrate((x^4+2)^(1/4)*(x^8-4)/x^6/(x^8-2*x^4-4),x, algorithm="giac")
Output:
integrate((x^8 - 4)*(x^4 + 2)^(1/4)/((x^8 - 2*x^4 - 4)*x^6), x)
Not integrable
Time = 8.56 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=-\int \frac {{\left (x^4+2\right )}^{1/4}\,\left (x^8-4\right )}{x^6\,\left (-x^8+2\,x^4+4\right )} \,d x \] Input:
int(-((x^4 + 2)^(1/4)*(x^8 - 4))/(x^6*(2*x^4 - x^8 + 4)),x)
Output:
-int(((x^4 + 2)^(1/4)*(x^8 - 4))/(x^6*(2*x^4 - x^8 + 4)), x)
Not integrable
Time = 0.18 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.48 \[ \int \frac {\sqrt [4]{2+x^4} \left (-4+x^8\right )}{x^6 \left (-4-2 x^4+x^8\right )} \, dx=\frac {-3 \left (x^{4}+2\right )^{\frac {1}{4}} x^{4}-\left (x^{4}+2\right )^{\frac {1}{4}}+40 \left (\int \frac {\left (x^{4}+2\right )^{\frac {1}{4}}}{x^{14}-8 x^{6}-8 x^{2}}d x \right ) x^{5}-5 \left (\int \frac {\left (x^{4}+2\right )^{\frac {1}{4}} x^{6}}{x^{12}-8 x^{4}-8}d x \right ) x^{5}+20 \left (\int \frac {\left (x^{4}+2\right )^{\frac {1}{4}} x^{2}}{x^{12}-8 x^{4}-8}d x \right ) x^{5}}{5 x^{5}} \] Input:
int((x^4+2)^(1/4)*(x^8-4)/x^6/(x^8-2*x^4-4),x)
Output:
( - 3*(x**4 + 2)**(1/4)*x**4 - (x**4 + 2)**(1/4) + 40*int((x**4 + 2)**(1/4 )/(x**14 - 8*x**6 - 8*x**2),x)*x**5 - 5*int(((x**4 + 2)**(1/4)*x**6)/(x**1 2 - 8*x**4 - 8),x)*x**5 + 20*int(((x**4 + 2)**(1/4)*x**2)/(x**12 - 8*x**4 - 8),x)*x**5)/(5*x**5)