3.10.82 \(\int \frac {\sqrt [4]{-1+2 x^4} (-2+x^8)}{x^6 (-1+x^4)^2} \, dx\) [982]

3.10.82.1 Optimal result

Integrand size = 27, antiderivative size = 74 \[ \int \frac {\sqrt [4]{-1+2 x^4} \left (-2+x^8\right )}{x^6 \left (-1+x^4\right )^2} \, dx=\frac {\sqrt [4]{-1+2 x^4} \left (-8-56 x^4+69 x^8\right )}{20 x^5 \left (-1+x^4\right )}+\frac {15}{8} \arctan \left (\frac {x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {15}{8} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+2 x^4}}\right ) \]

output

1/20*(2*x^4-1)^(1/4)*(69*x^8-56*x^4-8)/x^5/(x^4-1)+15/8*arctan(x/(2*x^4-1) 
^(1/4))-15/8*arctanh(x/(2*x^4-1)^(1/4))
 

3.10.82.2 Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{-1+2 x^4} \left (-2+x^8\right )}{x^6 \left (-1+x^4\right )^2} \, dx=\frac {\sqrt [4]{-1+2 x^4} \left (-8-56 x^4+69 x^8\right )}{20 x^5 \left (-1+x^4\right )}+\frac {15}{8} \arctan \left (\frac {x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {15}{8} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+2 x^4}}\right ) \]

input

Integrate[((-1 + 2*x^4)^(1/4)*(-2 + x^8))/(x^6*(-1 + x^4)^2),x]
 

output

((-1 + 2*x^4)^(1/4)*(-8 - 56*x^4 + 69*x^8))/(20*x^5*(-1 + x^4)) + (15*ArcT 
an[x/(-1 + 2*x^4)^(1/4)])/8 - (15*ArcTanh[x/(-1 + 2*x^4)^(1/4)])/8
 

3.10.82.3 Rubi [F]

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.

\(\displaystyle \int \frac {\sqrt [4]{2 x^4-1} \left (x^8-2\right )}{x^6 \left (x^4-1\right )^2} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\sqrt [4]{2 x^4-1}}{16 (x-1)^2}-\frac {\sqrt [4]{2 x^4-1}}{16 (x+1)^2}-\frac {2 \sqrt [4]{2 x^4-1}}{x^6}+\frac {17 \sqrt [4]{2 x^4-1}}{8 \left (x^2-1\right )}+\frac {2 \sqrt [4]{2 x^4-1}}{x^2+1}-\frac {4 \sqrt [4]{2 x^4-1}}{x^2}+\frac {\sqrt [4]{2 x^4-1}}{4 \left (x^2+1\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\sqrt [4]{2 x^4-1} \left (x^8-2\right )}{x^6 \left (1-x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\sqrt [4]{2 x^4-1}}{16 (x-1)^2}-\frac {\sqrt [4]{2 x^4-1}}{16 (x+1)^2}-\frac {2 \sqrt [4]{2 x^4-1}}{x^6}+\frac {17 \sqrt [4]{2 x^4-1}}{8 \left (x^2-1\right )}+\frac {2 \sqrt [4]{2 x^4-1}}{x^2+1}-\frac {4 \sqrt [4]{2 x^4-1}}{x^2}+\frac {\sqrt [4]{2 x^4-1}}{4 \left (x^2+1\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\sqrt [4]{2 x^4-1} \left (x^8-2\right )}{x^6 \left (1-x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\sqrt [4]{2 x^4-1}}{16 (x-1)^2}-\frac {\sqrt [4]{2 x^4-1}}{16 (x+1)^2}-\frac {2 \sqrt [4]{2 x^4-1}}{x^6}+\frac {17 \sqrt [4]{2 x^4-1}}{8 \left (x^2-1\right )}+\frac {2 \sqrt [4]{2 x^4-1}}{x^2+1}-\frac {4 \sqrt [4]{2 x^4-1}}{x^2}+\frac {\sqrt [4]{2 x^4-1}}{4 \left (x^2+1\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\sqrt [4]{2 x^4-1} \left (x^8-2\right )}{x^6 \left (1-x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\sqrt [4]{2 x^4-1}}{16 (x-1)^2}-\frac {\sqrt [4]{2 x^4-1}}{16 (x+1)^2}-\frac {2 \sqrt [4]{2 x^4-1}}{x^6}+\frac {17 \sqrt [4]{2 x^4-1}}{8 \left (x^2-1\right )}+\frac {2 \sqrt [4]{2 x^4-1}}{x^2+1}-\frac {4 \sqrt [4]{2 x^4-1}}{x^2}+\frac {\sqrt [4]{2 x^4-1}}{4 \left (x^2+1\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\sqrt [4]{2 x^4-1} \left (x^8-2\right )}{x^6 \left (1-x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\sqrt [4]{2 x^4-1}}{16 (x-1)^2}-\frac {\sqrt [4]{2 x^4-1}}{16 (x+1)^2}-\frac {2 \sqrt [4]{2 x^4-1}}{x^6}+\frac {17 \sqrt [4]{2 x^4-1}}{8 \left (x^2-1\right )}+\frac {2 \sqrt [4]{2 x^4-1}}{x^2+1}-\frac {4 \sqrt [4]{2 x^4-1}}{x^2}+\frac {\sqrt [4]{2 x^4-1}}{4 \left (x^2+1\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\sqrt [4]{2 x^4-1} \left (x^8-2\right )}{x^6 \left (1-x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\sqrt [4]{2 x^4-1}}{16 (x-1)^2}-\frac {\sqrt [4]{2 x^4-1}}{16 (x+1)^2}-\frac {2 \sqrt [4]{2 x^4-1}}{x^6}+\frac {17 \sqrt [4]{2 x^4-1}}{8 \left (x^2-1\right )}+\frac {2 \sqrt [4]{2 x^4-1}}{x^2+1}-\frac {4 \sqrt [4]{2 x^4-1}}{x^2}+\frac {\sqrt [4]{2 x^4-1}}{4 \left (x^2+1\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\sqrt [4]{2 x^4-1} \left (x^8-2\right )}{x^6 \left (1-x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\sqrt [4]{2 x^4-1}}{16 (x-1)^2}-\frac {\sqrt [4]{2 x^4-1}}{16 (x+1)^2}-\frac {2 \sqrt [4]{2 x^4-1}}{x^6}+\frac {17 \sqrt [4]{2 x^4-1}}{8 \left (x^2-1\right )}+\frac {2 \sqrt [4]{2 x^4-1}}{x^2+1}-\frac {4 \sqrt [4]{2 x^4-1}}{x^2}+\frac {\sqrt [4]{2 x^4-1}}{4 \left (x^2+1\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\sqrt [4]{2 x^4-1} \left (x^8-2\right )}{x^6 \left (1-x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\sqrt [4]{2 x^4-1}}{16 (x-1)^2}-\frac {\sqrt [4]{2 x^4-1}}{16 (x+1)^2}-\frac {2 \sqrt [4]{2 x^4-1}}{x^6}+\frac {17 \sqrt [4]{2 x^4-1}}{8 \left (x^2-1\right )}+\frac {2 \sqrt [4]{2 x^4-1}}{x^2+1}-\frac {4 \sqrt [4]{2 x^4-1}}{x^2}+\frac {\sqrt [4]{2 x^4-1}}{4 \left (x^2+1\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\sqrt [4]{2 x^4-1} \left (x^8-2\right )}{x^6 \left (1-x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\sqrt [4]{2 x^4-1}}{16 (x-1)^2}-\frac {\sqrt [4]{2 x^4-1}}{16 (x+1)^2}-\frac {2 \sqrt [4]{2 x^4-1}}{x^6}+\frac {17 \sqrt [4]{2 x^4-1}}{8 \left (x^2-1\right )}+\frac {2 \sqrt [4]{2 x^4-1}}{x^2+1}-\frac {4 \sqrt [4]{2 x^4-1}}{x^2}+\frac {\sqrt [4]{2 x^4-1}}{4 \left (x^2+1\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\sqrt [4]{2 x^4-1} \left (x^8-2\right )}{x^6 \left (1-x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\sqrt [4]{2 x^4-1}}{16 (x-1)^2}-\frac {\sqrt [4]{2 x^4-1}}{16 (x+1)^2}-\frac {2 \sqrt [4]{2 x^4-1}}{x^6}+\frac {17 \sqrt [4]{2 x^4-1}}{8 \left (x^2-1\right )}+\frac {2 \sqrt [4]{2 x^4-1}}{x^2+1}-\frac {4 \sqrt [4]{2 x^4-1}}{x^2}+\frac {\sqrt [4]{2 x^4-1}}{4 \left (x^2+1\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\sqrt [4]{2 x^4-1} \left (x^8-2\right )}{x^6 \left (1-x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\sqrt [4]{2 x^4-1}}{16 (x-1)^2}-\frac {\sqrt [4]{2 x^4-1}}{16 (x+1)^2}-\frac {2 \sqrt [4]{2 x^4-1}}{x^6}+\frac {17 \sqrt [4]{2 x^4-1}}{8 \left (x^2-1\right )}+\frac {2 \sqrt [4]{2 x^4-1}}{x^2+1}-\frac {4 \sqrt [4]{2 x^4-1}}{x^2}+\frac {\sqrt [4]{2 x^4-1}}{4 \left (x^2+1\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\sqrt [4]{2 x^4-1} \left (x^8-2\right )}{x^6 \left (1-x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\sqrt [4]{2 x^4-1}}{16 (x-1)^2}-\frac {\sqrt [4]{2 x^4-1}}{16 (x+1)^2}-\frac {2 \sqrt [4]{2 x^4-1}}{x^6}+\frac {17 \sqrt [4]{2 x^4-1}}{8 \left (x^2-1\right )}+\frac {2 \sqrt [4]{2 x^4-1}}{x^2+1}-\frac {4 \sqrt [4]{2 x^4-1}}{x^2}+\frac {\sqrt [4]{2 x^4-1}}{4 \left (x^2+1\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\sqrt [4]{2 x^4-1} \left (x^8-2\right )}{x^6 \left (1-x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\sqrt [4]{2 x^4-1}}{16 (x-1)^2}-\frac {\sqrt [4]{2 x^4-1}}{16 (x+1)^2}-\frac {2 \sqrt [4]{2 x^4-1}}{x^6}+\frac {17 \sqrt [4]{2 x^4-1}}{8 \left (x^2-1\right )}+\frac {2 \sqrt [4]{2 x^4-1}}{x^2+1}-\frac {4 \sqrt [4]{2 x^4-1}}{x^2}+\frac {\sqrt [4]{2 x^4-1}}{4 \left (x^2+1\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\sqrt [4]{2 x^4-1} \left (x^8-2\right )}{x^6 \left (1-x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\sqrt [4]{2 x^4-1}}{16 (x-1)^2}-\frac {\sqrt [4]{2 x^4-1}}{16 (x+1)^2}-\frac {2 \sqrt [4]{2 x^4-1}}{x^6}+\frac {17 \sqrt [4]{2 x^4-1}}{8 \left (x^2-1\right )}+\frac {2 \sqrt [4]{2 x^4-1}}{x^2+1}-\frac {4 \sqrt [4]{2 x^4-1}}{x^2}+\frac {\sqrt [4]{2 x^4-1}}{4 \left (x^2+1\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\sqrt [4]{2 x^4-1} \left (x^8-2\right )}{x^6 \left (1-x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\sqrt [4]{2 x^4-1}}{16 (x-1)^2}-\frac {\sqrt [4]{2 x^4-1}}{16 (x+1)^2}-\frac {2 \sqrt [4]{2 x^4-1}}{x^6}+\frac {17 \sqrt [4]{2 x^4-1}}{8 \left (x^2-1\right )}+\frac {2 \sqrt [4]{2 x^4-1}}{x^2+1}-\frac {4 \sqrt [4]{2 x^4-1}}{x^2}+\frac {\sqrt [4]{2 x^4-1}}{4 \left (x^2+1\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\sqrt [4]{2 x^4-1} \left (x^8-2\right )}{x^6 \left (1-x^4\right )^2}dx\)

input

Int[((-1 + 2*x^4)^(1/4)*(-2 + x^8))/(x^6*(-1 + x^4)^2),x]
 

output

$Aborted
 

3.10.82.3.1 Defintions of rubi rules used

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.10.82.4 Maple [A] (verified)

Time = 4.93 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.65

method	result	size
pseudoelliptic	\(\frac {\left (75 x^{9}-75 x^{5}\right ) \ln \left (\frac {\left (2 x^{4}-1\right )^{\frac {1}{4}}-x}{x}\right )+\left (-75 x^{9}+75 x^{5}\right ) \ln \left (\frac {\left (2 x^{4}-1\right )^{\frac {1}{4}}+x}{x}\right )+\left (-150 x^{9}+150 x^{5}\right ) \arctan \left (\frac {\left (2 x^{4}-1\right )^{\frac {1}{4}}}{x}\right )+\left (276 x^{8}-224 x^{4}-32\right ) \left (2 x^{4}-1\right )^{\frac {1}{4}}}{80 x^{9}-80 x^{5}}\)	\(122\)
trager	\(\frac {\left (2 x^{4}-1\right )^{\frac {1}{4}} \left (69 x^{8}-56 x^{4}-8\right )}{20 x^{5} \left (x^{4}-1\right )}+\frac {15 \ln \left (\frac {2 \left (2 x^{4}-1\right )^{\frac {3}{4}} x -2 \sqrt {2 x^{4}-1}\, x^{2}+2 \left (2 x^{4}-1\right )^{\frac {1}{4}} x^{3}-3 x^{4}+1}{\left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{16}+\frac {15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {2 x^{4}-1}\, x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \left (2 x^{4}-1\right )^{\frac {3}{4}} x +2 \left (2 x^{4}-1\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{16}\)	\(195\)
risch	\(\frac {138 x^{12}-181 x^{8}+40 x^{4}+8}{20 x^{5} \left (x^{4}-1\right ) \left (2 x^{4}-1\right )^{\frac {3}{4}}}+\frac {\left (\frac {15 \ln \left (-\frac {-12 x^{12}+8 \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {1}{4}} x^{9}-4 \sqrt {8 x^{12}-12 x^{8}+6 x^{4}-1}\, x^{6}+16 x^{8}+2 \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {3}{4}} x^{3}-8 \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {1}{4}} x^{5}+2 \sqrt {8 x^{12}-12 x^{8}+6 x^{4}-1}\, x^{2}-7 x^{4}+2 \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {1}{4}} x +1}{\left (2 x^{4}-1\right )^{2} \left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{16}-\frac {15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-12 x^{12}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {1}{4}} x^{9}+4 \sqrt {8 x^{12}-12 x^{8}+6 x^{4}-1}\, x^{6}+16 x^{8}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {3}{4}} x^{3}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {1}{4}} x^{5}-2 \sqrt {8 x^{12}-12 x^{8}+6 x^{4}-1}\, x^{2}-7 x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {1}{4}} x +1}{\left (2 x^{4}-1\right )^{2} \left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{16}\right ) {\left (\left (2 x^{4}-1\right )^{3}\right )}^{\frac {1}{4}}}{\left (2 x^{4}-1\right )^{\frac {3}{4}}}\)	\(471\)

input

int((2*x^4-1)^(1/4)*(x^8-2)/x^6/(x^4-1)^2,x,method=_RETURNVERBOSE)
 

output

((75*x^9-75*x^5)*ln(((2*x^4-1)^(1/4)-x)/x)+(-75*x^9+75*x^5)*ln(((2*x^4-1)^ 
(1/4)+x)/x)+(-150*x^9+150*x^5)*arctan(1/x*(2*x^4-1)^(1/4))+(276*x^8-224*x^ 
4-32)*(2*x^4-1)^(1/4))/(80*x^9-80*x^5)
 

3.10.82.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (62) = 124\).

Time = 2.43 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.04 \[ \int \frac {\sqrt [4]{-1+2 x^4} \left (-2+x^8\right )}{x^6 \left (-1+x^4\right )^2} \, dx=\frac {75 \, {\left (x^{9} - x^{5}\right )} \arctan \left (\frac {2 \, {\left ({\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + {\left (2 \, x^{4} - 1\right )}^{\frac {3}{4}} x\right )}}{x^{4} - 1}\right ) + 75 \, {\left (x^{9} - x^{5}\right )} \log \left (-\frac {3 \, x^{4} - 2 \, {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {2 \, x^{4} - 1} x^{2} - 2 \, {\left (2 \, x^{4} - 1\right )}^{\frac {3}{4}} x - 1}{x^{4} - 1}\right ) + 4 \, {\left (69 \, x^{8} - 56 \, x^{4} - 8\right )} {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{80 \, {\left (x^{9} - x^{5}\right )}} \]

input

integrate((2*x^4-1)^(1/4)*(x^8-2)/x^6/(x^4-1)^2,x, algorithm="fricas")
 

output

1/80*(75*(x^9 - x^5)*arctan(2*((2*x^4 - 1)^(1/4)*x^3 + (2*x^4 - 1)^(3/4)*x 
)/(x^4 - 1)) + 75*(x^9 - x^5)*log(-(3*x^4 - 2*(2*x^4 - 1)^(1/4)*x^3 + 2*sq 
rt(2*x^4 - 1)*x^2 - 2*(2*x^4 - 1)^(3/4)*x - 1)/(x^4 - 1)) + 4*(69*x^8 - 56 
*x^4 - 8)*(2*x^4 - 1)^(1/4))/(x^9 - x^5)
 

3.10.82.6 Sympy [F]

\[ \int \frac {\sqrt [4]{-1+2 x^4} \left (-2+x^8\right )}{x^6 \left (-1+x^4\right )^2} \, dx=\int \frac {\sqrt [4]{2 x^{4} - 1} \left (x^{8} - 2\right )}{x^{6} \left (x - 1\right )^{2} \left (x + 1\right )^{2} \left (x^{2} + 1\right )^{2}}\, dx \]

input

integrate((2*x**4-1)**(1/4)*(x**8-2)/x**6/(x**4-1)**2,x)
 

output

Integral((2*x**4 - 1)**(1/4)*(x**8 - 2)/(x**6*(x - 1)**2*(x + 1)**2*(x**2 
+ 1)**2), x)
 

3.10.82.7 Maxima [F]

\[ \int \frac {\sqrt [4]{-1+2 x^4} \left (-2+x^8\right )}{x^6 \left (-1+x^4\right )^2} \, dx=\int { \frac {{\left (x^{8} - 2\right )} {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{{\left (x^{4} - 1\right )}^{2} x^{6}} \,d x } \]

input

integrate((2*x^4-1)^(1/4)*(x^8-2)/x^6/(x^4-1)^2,x, algorithm="maxima")
 

output

integrate((x^8 - 2)*(2*x^4 - 1)^(1/4)/((x^4 - 1)^2*x^6), x)
 

3.10.82.8 Giac [A] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt [4]{-1+2 x^4} \left (-2+x^8\right )}{x^6 \left (-1+x^4\right )^2} \, dx=\frac {2 \, {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (\frac {1}{x^{4}} - 2\right )}}{5 \, x} + \frac {4 \, {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{x} - \frac {{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{4 \, x {\left (\frac {1}{x^{4}} - 1\right )}} - \frac {15}{8} \, \arctan \left (\frac {{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {15}{16} \, \log \left (\frac {{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{x} + 1\right ) + \frac {15}{16} \, \log \left ({\left | \frac {{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{x} - 1 \right |}\right ) \]

input

integrate((2*x^4-1)^(1/4)*(x^8-2)/x^6/(x^4-1)^2,x, algorithm="giac")
 

output

2/5*(2*x^4 - 1)^(1/4)*(1/x^4 - 2)/x + 4*(2*x^4 - 1)^(1/4)/x - 1/4*(2*x^4 - 
 1)^(1/4)/(x*(1/x^4 - 1)) - 15/8*arctan((2*x^4 - 1)^(1/4)/x) - 15/16*log(( 
2*x^4 - 1)^(1/4)/x + 1) + 15/16*log(abs((2*x^4 - 1)^(1/4)/x - 1))
 

3.10.82.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [4]{-1+2 x^4} \left (-2+x^8\right )}{x^6 \left (-1+x^4\right )^2} \, dx=\int \frac {\left (x^8-2\right )\,{\left (2\,x^4-1\right )}^{1/4}}{x^6\,{\left (x^4-1\right )}^2} \,d x \]

input

int(((x^8 - 2)*(2*x^4 - 1)^(1/4))/(x^6*(x^4 - 1)^2),x)
 

output

int(((x^8 - 2)*(2*x^4 - 1)^(1/4))/(x^6*(x^4 - 1)^2), x)