3.23.31 \(\int \frac {1+x^5}{\sqrt {1+x^4} (-1+x^5)} \, dx\) [2231]

Optimal. Leaf size=166 \[ -\frac {1}{5} \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{1-2 x+x^2+\sqrt {1+x^4}}\right )+\frac {4}{5} \text {RootSum}\left [4+4 \text {$\#$1}+2 \text {$\#$1}^3+\text {$\#$1}^4\& ,\frac {-2 \log (x)+2 \log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}+\log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}-\log (x) \text {$\#$1}^2+\log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{2+3 \text {$\#$1}^2+2 \text {$\#$1}^3}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 0.69, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1+x^5}{\sqrt {1+x^4} \left (-1+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + x^5)/(Sqrt[1 + x^4]*(-1 + x^5)),x]

[Out]

-1/5*ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/Sqrt[2] - ArcTanh[(1 + x^2)/(Sqrt[2]*Sqrt[1 + x^4])]/(5*Sqrt[2]) + (2*
(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(5*Sqrt[1 + x^4]) - (8*Defer[Int][1/(Sqrt[1
+ x^4]*(1 + x + x^2 + x^3 + x^4)), x])/5 - (6*Defer[Int][x/(Sqrt[1 + x^4]*(1 + x + x^2 + x^3 + x^4)), x])/5 -
(4*Defer[Int][x^2/(Sqrt[1 + x^4]*(1 + x + x^2 + x^3 + x^4)), x])/5 - (2*Defer[Int][x^3/(Sqrt[1 + x^4]*(1 + x
+ x^2 + x^3 + x^4)), x])/5

Rubi steps

\begin {align*} \int \frac {1+x^5}{\sqrt {1+x^4} \left (-1+x^5\right )} \, dx &=\int \left (\frac {1}{\sqrt {1+x^4}}+\frac {2}{\sqrt {1+x^4} \left (-1+x^5\right )}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt {1+x^4} \left (-1+x^5\right )} \, dx+\int \frac {1}{\sqrt {1+x^4}} \, dx\\ &=\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}+2 \int \left (\frac {1}{5 (-1+x) \sqrt {1+x^4}}+\frac {-4-3 x-2 x^2-x^3}{5 \sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )}\right ) \, dx\\ &=\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}+\frac {2}{5} \int \frac {1}{(-1+x) \sqrt {1+x^4}} \, dx+\frac {2}{5} \int \frac {-4-3 x-2 x^2-x^3}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx\\ &=\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {2}{5} \int \frac {1}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx-\frac {2}{5} \int \frac {x}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx+\frac {2}{5} \int \left (-\frac {4}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )}-\frac {3 x}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )}-\frac {2 x^2}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )}-\frac {x^3}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )}\right ) \, dx\\ &=\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {1}{5} \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{5} \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{5} \text {Subst}\left (\int \frac {1}{(1-x) \sqrt {1+x^2}} \, dx,x,x^2\right )-\frac {2}{5} \int \frac {x^3}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {4}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {6}{5} \int \frac {x}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {8}{5} \int \frac {1}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx\\ &=\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{5 \sqrt {1+x^4}}-\frac {1}{5} \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )+\frac {1}{5} \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\frac {-1-x^2}{\sqrt {1+x^4}}\right )-\frac {2}{5} \int \frac {x^3}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {4}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {6}{5} \int \frac {x}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {8}{5} \int \frac {1}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{5 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {1+x^2}{\sqrt {2} \sqrt {1+x^4}}\right )}{5 \sqrt {2}}+\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{5 \sqrt {1+x^4}}-\frac {2}{5} \int \frac {x^3}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {4}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {6}{5} \int \frac {x}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {8}{5} \int \frac {1}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx\\ \end {align*}

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Mathematica [A]
time = 1.87, size = 186, normalized size = 1.12 \begin {gather*} -\frac {1}{5} \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{1-2 x+x^2+\sqrt {1+x^4}}\right )+\frac {4}{5} \text {RootSum}\left [11+14 \text {$\#$1}+12 \text {$\#$1}^2+6 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-4 \log (x)+4 \log \left (1-x+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right )-3 \log (x) \text {$\#$1}+3 \log \left (1-x+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}-\log (x) \text {$\#$1}^2+\log \left (1-x+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{7+12 \text {$\#$1}+9 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^5)/(Sqrt[1 + x^4]*(-1 + x^5)),x]

[Out]

-1/5*(Sqrt[2]*ArcTanh[(Sqrt[2]*x)/(1 - 2*x + x^2 + Sqrt[1 + x^4])]) + (4*RootSum[11 + 14*#1 + 12*#1^2 + 6*#1^3
+ #1^4 & , (-4*Log[x] + 4*Log[1 - x + x^2 + Sqrt[1 + x^4] - x*#1] - 3*Log[x]*#1 + 3*Log[1 - x + x^2 + Sqrt[1
+ x^4] - x*#1]*#1 - Log[x]*#1^2 + Log[1 - x + x^2 + Sqrt[1 + x^4] - x*#1]*#1^2)/(7 + 12*#1 + 9*#1^2 + 2*#1^3)
& ])/5

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 1.70, size = 271, normalized size = 1.63

method	result	size
elliptic	\(-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-\textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (\left (-x^{2}+\sqrt {x^{4}+1}\right )^{2}+\left (\textit {\_R}^{2}-\textit {\_R} -1\right ) \left (-x^{2}+\sqrt {x^{4}+1}\right )+\textit {\_R}^{2}-\textit {\_R} \right )\right )}{5}-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4 \sqrt {\left (x^{2}-1\right )^{2}+2 x^{2}}}\right )}{10}+\frac {\left (-\frac {4 \arctanh \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x \sqrt {\sqrt {5}-1}}\right )}{5 \sqrt {\sqrt {5}-1}}+\frac {4 \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x \sqrt {\sqrt {5}+1}}\right )}{5 \sqrt {\sqrt {5}+1}}+\frac {\ln \left (-1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{10}-\frac {\ln \left (1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{10}\right ) \sqrt {2}}{2}\)	\(206\)
default	\(\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{3}+x^{2}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha }\, \sqrt {x^{4}+1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha }}+\frac {2 \left (-1\right )^{\frac {3}{4}} \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha -1\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i \underline {\hspace {1.25 ex}}\alpha ^{3}, i\right )}{\sqrt {x^{4}+1}}\right )\right )}{5}-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4 \sqrt {x^{4}+1}}\right )}{10}+\frac {2 \left (-1\right )^{\frac {3}{4}} \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i, -\sqrt {-i}\, \left (-1\right )^{\frac {3}{4}}\right )}{5 \sqrt {x^{4}+1}}\)	\(271\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}-2\right ) x +\sqrt {x^{4}+1}+\RootOf \left (\textit {\_Z}^{2}-2\right )}{\left (-1+x \right )^{2}}\right )}{10}+\frac {\RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right ) \ln \left (-\frac {625 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right ) \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{4} x -50 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right ) \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} x^{2}+50 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right ) \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} x -50 \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right )+150 \sqrt {x^{4}+1}\, \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right ) x^{2}-4 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right )+16 \sqrt {x^{4}+1}}{25 \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} x -4 x^{2}-4 x -4}\right )}{5}-\RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right ) \ln \left (\frac {625 \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{5} x +50 x^{2} \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{3}-250 x \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{3}+50 \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{3}+30 \sqrt {x^{4}+1}\, \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-12 \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right ) x^{2}+24 \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right ) x -12 \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )-8 \sqrt {x^{4}+1}}{4 x^{2}+25 \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} x +4}\right )\)	\(561\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5+1)/(x^4+1)^(1/2)/(x^5-1),x,method=_RETURNVERBOSE)

[Out]

1/(1/2*2^(1/2)+1/2*I*2^(1/2))*(1-I*x^2)^(1/2)*(1+I*x^2)^(1/2)/(x^4+1)^(1/2)*EllipticF(x*(1/2*2^(1/2)+1/2*I*2^(
1/2)),I)+1/5*sum(_alpha*(-1/(-_alpha^3-_alpha^2-_alpha)^(1/2)*arctanh(_alpha^2*(_alpha^3+x^2)/(-_alpha^3-_alph
a^2-_alpha)^(1/2)/(x^4+1)^(1/2))+2*(-1)^(3/4)*(-_alpha^3-_alpha^2-_alpha-1)*(1-I*x^2)^(1/2)*(1+I*x^2)^(1/2)/(x
^4+1)^(1/2)*EllipticPi((-1)^(1/4)*x,-I*_alpha^3,I)),_alpha=RootOf(_Z^4+_Z^3+_Z^2+_Z+1))-1/10*2^(1/2)*arctanh(1
/4*(2*x^2+2)*2^(1/2)/(x^4+1)^(1/2))+2/5*(-1)^(3/4)*(1-I*x^2)^(1/2)*(1+I*x^2)^(1/2)/(x^4+1)^(1/2)*EllipticPi((-
1)^(1/4)*x,-I,(-I)^(1/2)/(-1)^(1/4))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+1)/(x^4+1)^(1/2)/(x^5-1),x, algorithm="maxima")

[Out]

integrate((x^5 + 1)/((x^5 - 1)*sqrt(x^4 + 1)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.48, size = 448, normalized size = 2.70 \begin {gather*} \frac {2}{5} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \arctan \left (-\frac {\sqrt {2} \sqrt {x^{4} + 1} {\left (3 \, x^{2} + \sqrt {5} {\left (x^{2} + 2 \, x + 1\right )} + 2 \, x + 3\right )} \sqrt {\sqrt {5} - 1} + \sqrt {2} {\left (3 \, x^{4} - 4 \, x^{3} - 8 \, x^{2} - \sqrt {5} {\left (x^{4} - 2 \, x^{3} - 4 \, x^{2} - 2 \, x + 1\right )} - 4 \, x + 3\right )} \sqrt {13 \, \sqrt {5} + 29} \sqrt {\sqrt {5} - 1}}{4 \, {\left (x^{4} - 2 \, x^{3} - 2 \, x^{2} - 2 \, x + 1\right )}}\right ) - \frac {1}{10} \, \sqrt {2} \sqrt {\sqrt {5} + 1} \log \left (-\frac {2 \, {\left (\sqrt {2} {\left (4 \, x^{4} - 7 \, x^{3} - 14 \, x^{2} - \sqrt {5} {\left (2 \, x^{4} - 3 \, x^{3} - 6 \, x^{2} - 3 \, x + 2\right )} - 7 \, x + 4\right )} \sqrt {\sqrt {5} + 1} + 2 \, \sqrt {x^{4} + 1} {\left (7 \, x^{2} - \sqrt {5} {\left (3 \, x^{2} + 5 \, x + 3\right )} + 11 \, x + 7\right )}\right )}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) + \frac {1}{10} \, \sqrt {2} \sqrt {\sqrt {5} + 1} \log \left (\frac {2 \, {\left (\sqrt {2} {\left (4 \, x^{4} - 7 \, x^{3} - 14 \, x^{2} - \sqrt {5} {\left (2 \, x^{4} - 3 \, x^{3} - 6 \, x^{2} - 3 \, x + 2\right )} - 7 \, x + 4\right )} \sqrt {\sqrt {5} + 1} - 2 \, \sqrt {x^{4} + 1} {\left (7 \, x^{2} - \sqrt {5} {\left (3 \, x^{2} + 5 \, x + 3\right )} + 11 \, x + 7\right )}\right )}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) + \frac {1}{20} \, \sqrt {2} \log \left (-\frac {3 \, x^{4} - 4 \, x^{3} - 2 \, \sqrt {2} \sqrt {x^{4} + 1} {\left (x^{2} - x + 1\right )} + 6 \, x^{2} - 4 \, x + 3}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+1)/(x^4+1)^(1/2)/(x^5-1),x, algorithm="fricas")

[Out]

2/5*sqrt(2)*sqrt(sqrt(5) - 1)*arctan(-1/4*(sqrt(2)*sqrt(x^4 + 1)*(3*x^2 + sqrt(5)*(x^2 + 2*x + 1) + 2*x + 3)*s
qrt(sqrt(5) - 1) + sqrt(2)*(3*x^4 - 4*x^3 - 8*x^2 - sqrt(5)*(x^4 - 2*x^3 - 4*x^2 - 2*x + 1) - 4*x + 3)*sqrt(13
*sqrt(5) + 29)*sqrt(sqrt(5) - 1))/(x^4 - 2*x^3 - 2*x^2 - 2*x + 1)) - 1/10*sqrt(2)*sqrt(sqrt(5) + 1)*log(-2*(sq
rt(2)*(4*x^4 - 7*x^3 - 14*x^2 - sqrt(5)*(2*x^4 - 3*x^3 - 6*x^2 - 3*x + 2) - 7*x + 4)*sqrt(sqrt(5) + 1) + 2*sqr
t(x^4 + 1)*(7*x^2 - sqrt(5)*(3*x^2 + 5*x + 3) + 11*x + 7))/(x^4 + x^3 + x^2 + x + 1)) + 1/10*sqrt(2)*sqrt(sqrt
(5) + 1)*log(2*(sqrt(2)*(4*x^4 - 7*x^3 - 14*x^2 - sqrt(5)*(2*x^4 - 3*x^3 - 6*x^2 - 3*x + 2) - 7*x + 4)*sqrt(sq
rt(5) + 1) - 2*sqrt(x^4 + 1)*(7*x^2 - sqrt(5)*(3*x^2 + 5*x + 3) + 11*x + 7))/(x^4 + x^3 + x^2 + x + 1)) + 1/20
*sqrt(2)*log(-(3*x^4 - 4*x^3 - 2*sqrt(2)*sqrt(x^4 + 1)*(x^2 - x + 1) + 6*x^2 - 4*x + 3)/(x^4 - 4*x^3 + 6*x^2 -
4*x + 1))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}{\left (x - 1\right ) \sqrt {x^{4} + 1} \left (x^{4} + x^{3} + x^{2} + x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**5+1)/(x**4+1)**(1/2)/(x**5-1),x)

[Out]

Integral((x + 1)*(x**4 - x**3 + x**2 - x + 1)/((x - 1)*sqrt(x**4 + 1)*(x**4 + x**3 + x**2 + x + 1)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+1)/(x^4+1)^(1/2)/(x^5-1),x, algorithm="giac")

[Out]

integrate((x^5 + 1)/((x^5 - 1)*sqrt(x^4 + 1)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^5+1}{\sqrt {x^4+1}\,\left (x^5-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5 + 1)/((x^4 + 1)^(1/2)*(x^5 - 1)),x)

[Out]

int((x^5 + 1)/((x^4 + 1)^(1/2)*(x^5 - 1)), x)

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