3.18.71 \(\int \frac {1+x^{10}}{\sqrt {1+x^4} (-1+x^{10})} \, dx\)

Optimal. Leaf size=119 \[ -\frac {1}{5} \sqrt {2 \left (\sqrt {5}-1\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {x^4+1}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{5 \sqrt {2}}-\frac {1}{5} \sqrt {2 \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt {x^4+1}}\right ) \]

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Rubi [F]  time = 1.88, 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^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + x^10)/(Sqrt[1 + x^4]*(-1 + x^10)),x]

[Out]

-1/5*ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/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]) - (4*Defer[Int][1/(Sqrt[1 + x^4]*(1 - x + x^2 - x^3 + x^4)), x])/5 + (3*Defer[Int]
[x/(Sqrt[1 + x^4]*(1 - x + x^2 - x^3 + x^4)), x])/5 - (2*Defer[Int][x^2/(Sqrt[1 + x^4]*(1 - x + x^2 - x^3 + x^
4)), x])/5 + Defer[Int][x^3/(Sqrt[1 + x^4]*(1 - x + x^2 - x^3 + x^4)), x]/5 - (4*Defer[Int][1/(Sqrt[1 + x^4]*(
1 + x + x^2 + x^3 + x^4)), x])/5 - (3*Defer[Int][x/(Sqrt[1 + x^4]*(1 + x + x^2 + x^3 + x^4)), x])/5 - (2*Defer
[Int][x^2/(Sqrt[1 + x^4]*(1 + x + x^2 + x^3 + x^4)), x])/5 - 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^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx &=\int \left (\frac {1}{\sqrt {1+x^4}}+\frac {2}{\sqrt {1+x^4} \left (-1+x^{10}\right )}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt {1+x^4} \left (-1+x^{10}\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 \left (-1+x^2\right ) \sqrt {1+x^4}}+\frac {-4+3 x-2 x^2+x^3}{10 \sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )}+\frac {-4-3 x-2 x^2-x^3}{10 \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 {-4+3 x-2 x^2+x^3}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx+\frac {1}{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 {2}{5} \int \frac {1}{\left (-1+x^2\right ) \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}}-\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} \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 {1}{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 {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} \int \frac {x^3}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {1}{5} \int \frac {x^3}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx+\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{-1+2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )-\frac {2}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {2}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx+\frac {3}{5} \int \frac {x}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {3}{5} \int \frac {x}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {4}{5} \int \frac {1}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {4}{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 {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} \int \frac {x^3}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {1}{5} \int \frac {x^3}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {2}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {2}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx+\frac {3}{5} \int \frac {x}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {3}{5} \int \frac {x}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {4}{5} \int \frac {1}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {4}{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 [C]  time = 0.66, size = 371, normalized size = 3.12 \begin {gather*} \frac {\sqrt [4]{-1} \left (\left (-4-\sqrt [5]{-1}+(-1)^{2/5}-(-1)^{3/5}+(-1)^{4/5}\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+2 \left (\Pi \left (i;\left .\sin ^{-1}\left ((-1)^{3/4} x\right )\right |-1\right )+\left (2-\sqrt [5]{-1}+(-1)^{2/5}-(-1)^{3/5}+(-1)^{4/5}\right ) \Pi \left (\sqrt [10]{-1};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-(-1)^{4/5} \Pi \left (-(-1)^{3/10};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+(-1)^{3/5} \Pi \left (-(-1)^{3/10};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-(-1)^{2/5} \Pi \left (-(-1)^{3/10};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\sqrt [5]{-1} \Pi \left (-(-1)^{3/10};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left (-(-1)^{7/10};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-(-1)^{4/5} \Pi \left ((-1)^{9/10};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+(-1)^{3/5} \Pi \left ((-1)^{9/10};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-(-1)^{2/5} \Pi \left ((-1)^{9/10};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\sqrt [5]{-1} \Pi \left ((-1)^{9/10};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right )\right )}{4+\sqrt [5]{-1}-(-1)^{2/5}+(-1)^{3/5}-(-1)^{4/5}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(1 + x^10)/(Sqrt[1 + x^4]*(-1 + x^10)),x]

[Out]

((-1)^(1/4)*((-4 - (-1)^(1/5) + (-1)^(2/5) - (-1)^(3/5) + (-1)^(4/5))*EllipticF[I*ArcSinh[(-1)^(1/4)*x], -1] +
2*(EllipticPi[I, ArcSin[(-1)^(3/4)*x], -1] + (2 - (-1)^(1/5) + (-1)^(2/5) - (-1)^(3/5) + (-1)^(4/5))*Elliptic
Pi[(-1)^(1/10), I*ArcSinh[(-1)^(1/4)*x], -1] + (-1)^(1/5)*EllipticPi[-(-1)^(3/10), I*ArcSinh[(-1)^(1/4)*x], -1
] - (-1)^(2/5)*EllipticPi[-(-1)^(3/10), I*ArcSinh[(-1)^(1/4)*x], -1] + (-1)^(3/5)*EllipticPi[-(-1)^(3/10), I*A
rcSinh[(-1)^(1/4)*x], -1] - (-1)^(4/5)*EllipticPi[-(-1)^(3/10), I*ArcSinh[(-1)^(1/4)*x], -1] + EllipticPi[-(-1
)^(7/10), I*ArcSinh[(-1)^(1/4)*x], -1] + (-1)^(1/5)*EllipticPi[(-1)^(9/10), I*ArcSinh[(-1)^(1/4)*x], -1] - (-1
)^(2/5)*EllipticPi[(-1)^(9/10), I*ArcSinh[(-1)^(1/4)*x], -1] + (-1)^(3/5)*EllipticPi[(-1)^(9/10), I*ArcSinh[(-
1)^(1/4)*x], -1] - (-1)^(4/5)*EllipticPi[(-1)^(9/10), I*ArcSinh[(-1)^(1/4)*x], -1])))/(4 + (-1)^(1/5) - (-1)^(
2/5) + (-1)^(3/5) - (-1)^(4/5))

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IntegrateAlgebraic [A]  time = 2.31, size = 119, normalized size = 1.00 \begin {gather*} -\frac {1}{5} \sqrt {2 \left (-1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {1+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{5 \sqrt {2}}-\frac {1}{5} \sqrt {2 \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {1+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x^10)/(Sqrt[1 + x^4]*(-1 + x^10)),x]

[Out]

-1/5*(Sqrt[2*(-1 + Sqrt[5])]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/Sqrt[1 + x^4]]) - ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x
^4]]/(5*Sqrt[2]) - (Sqrt[2*(1 + Sqrt[5])]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x)/Sqrt[1 + x^4]])/5

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fricas [B]  time = 0.58, size = 431, normalized size = 3.62 \begin {gather*} -\frac {1}{5} \, \sqrt {2 \, \sqrt {5} - 2} \arctan \left (\frac {2 \, {\left (x^{5} + 2 \, x^{3} + \sqrt {5} {\left (x^{5} + x\right )} + x\right )} \sqrt {x^{4} + 1} \sqrt {2 \, \sqrt {5} - 2} - {\left (x^{8} - 5 \, x^{6} + 3 \, x^{4} - 5 \, x^{2} - \sqrt {5} {\left (x^{8} - x^{6} + 3 \, x^{4} - x^{2} + 1\right )} + 1\right )} \sqrt {2 \, \sqrt {5} - 2} \sqrt {\sqrt {5} + 2}}{4 \, {\left (x^{8} - x^{6} + x^{4} - x^{2} + 1\right )}}\right ) + \frac {1}{20} \, \sqrt {2} \log \left (\frac {x^{4} - 2 \, \sqrt {2} \sqrt {x^{4} + 1} x + 2 \, x^{2} + 1}{x^{4} - 2 \, x^{2} + 1}\right ) - \frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 2} \log \left (-\frac {4 \, {\left (3 \, x^{5} - x^{3} - \sqrt {5} {\left (x^{5} - x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{4} + 1} + {\left (3 \, x^{8} - 5 \, x^{6} + 9 \, x^{4} - 5 \, x^{2} - \sqrt {5} {\left (x^{8} - 3 \, x^{6} + 3 \, x^{4} - 3 \, x^{2} + 1\right )} + 3\right )} \sqrt {2 \, \sqrt {5} + 2}}{x^{8} + x^{6} + x^{4} + x^{2} + 1}\right ) + \frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 2} \log \left (-\frac {4 \, {\left (3 \, x^{5} - x^{3} - \sqrt {5} {\left (x^{5} - x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{4} + 1} - {\left (3 \, x^{8} - 5 \, x^{6} + 9 \, x^{4} - 5 \, x^{2} - \sqrt {5} {\left (x^{8} - 3 \, x^{6} + 3 \, x^{4} - 3 \, x^{2} + 1\right )} + 3\right )} \sqrt {2 \, \sqrt {5} + 2}}{x^{8} + x^{6} + x^{4} + x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^10+1)/(x^4+1)^(1/2)/(x^10-1),x, algorithm="fricas")

[Out]

-1/5*sqrt(2*sqrt(5) - 2)*arctan(1/4*(2*(x^5 + 2*x^3 + sqrt(5)*(x^5 + x) + x)*sqrt(x^4 + 1)*sqrt(2*sqrt(5) - 2)
- (x^8 - 5*x^6 + 3*x^4 - 5*x^2 - sqrt(5)*(x^8 - x^6 + 3*x^4 - x^2 + 1) + 1)*sqrt(2*sqrt(5) - 2)*sqrt(sqrt(5)
+ 2))/(x^8 - x^6 + x^4 - x^2 + 1)) + 1/20*sqrt(2)*log((x^4 - 2*sqrt(2)*sqrt(x^4 + 1)*x + 2*x^2 + 1)/(x^4 - 2*x
^2 + 1)) - 1/20*sqrt(2*sqrt(5) + 2)*log(-(4*(3*x^5 - x^3 - sqrt(5)*(x^5 - x^3 + x) + 3*x)*sqrt(x^4 + 1) + (3*x
^8 - 5*x^6 + 9*x^4 - 5*x^2 - sqrt(5)*(x^8 - 3*x^6 + 3*x^4 - 3*x^2 + 1) + 3)*sqrt(2*sqrt(5) + 2))/(x^8 + x^6 +
x^4 + x^2 + 1)) + 1/20*sqrt(2*sqrt(5) + 2)*log(-(4*(3*x^5 - x^3 - sqrt(5)*(x^5 - x^3 + x) + 3*x)*sqrt(x^4 + 1)
- (3*x^8 - 5*x^6 + 9*x^4 - 5*x^2 - sqrt(5)*(x^8 - 3*x^6 + 3*x^4 - 3*x^2 + 1) + 3)*sqrt(2*sqrt(5) + 2))/(x^8 +
x^6 + x^4 + x^2 + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{10} + 1}{{\left (x^{10} - 1\right )} \sqrt {x^{4} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^10+1)/(x^4+1)^(1/2)/(x^10-1),x, algorithm="giac")

[Out]

integrate((x^10 + 1)/((x^10 - 1)*sqrt(x^4 + 1)), x)

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maple [A]  time = 2.72, size = 109, normalized size = 0.92

method	result	size
elliptic	\(\frac {\left (\frac {\ln \left (-1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{10}-\frac {\ln \left (1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{10}-\frac {4 \arctanh \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x \sqrt {\sqrt {5}-1}}\right )}{5 \sqrt {\sqrt {5}-1}}+\frac {4 \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x \sqrt {\sqrt {5}+1}}\right )}{5 \sqrt {\sqrt {5}+1}}\right ) \sqrt {2}}{2}\)	\(109\)
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 {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}}+\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 )}{10}+\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 )}{10}\)	\(377\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}-2\right ) x +\sqrt {x^{4}+1}}{\left (-1+x \right ) \left (1+x \right )}\right )}{10}+\frac {\RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2}-1\right ) \ln \left (\frac {25 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2}-1\right ) \RootOf \left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2} x^{2}-\RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2}-1\right ) x^{4}+2 \sqrt {x^{4}+1}\, x -\RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2}-1\right )}{\left (25 \RootOf \left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2} x -x^{2}-x -1\right ) \left (25 \RootOf \left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2} x +x^{2}-x +1\right )}\right )}{5}-\RootOf \left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right ) \ln \left (\frac {125 \RootOf \left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{3} x^{2}+5 \RootOf \left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right ) x^{4}-5 \RootOf \left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right ) x^{2}+2 \sqrt {x^{4}+1}\, x +5 \RootOf \left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )}{\left (25 \RootOf \left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2} x +x^{2}+1\right ) \left (25 \RootOf \left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2} x -x^{2}-1\right )}\right )\)	\(383\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^10+1)/(x^4+1)^(1/2)/(x^10-1),x,method=_RETURNVERBOSE)

[Out]

1/2*(1/10*ln(-1+1/2*2^(1/2)/x*(x^4+1)^(1/2))-1/10*ln(1+1/2*2^(1/2)/x*(x^4+1)^(1/2))-4/5/(5^(1/2)-1)^(1/2)*arct
anh(2^(1/2)/x*(x^4+1)^(1/2)/(5^(1/2)-1)^(1/2))+4/5/(5^(1/2)+1)^(1/2)*arctan(2^(1/2)/x*(x^4+1)^(1/2)/(5^(1/2)+1
)^(1/2)))*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{10} + 1}{{\left (x^{10} - 1\right )} \sqrt {x^{4} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^10+1)/(x^4+1)^(1/2)/(x^10-1),x, algorithm="maxima")

[Out]

integrate((x^10 + 1)/((x^10 - 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^{10}+1}{\sqrt {x^4+1}\,\left (x^{10}-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^10 + 1)/((x^4 + 1)^(1/2)*(x^10 - 1)),x)

[Out]

int((x^10 + 1)/((x^4 + 1)^(1/2)*(x^10 - 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} + 1\right ) \left (x^{8} - x^{6} + x^{4} - x^{2} + 1\right )}{\left (x - 1\right ) \left (x + 1\right ) \sqrt {x^{4} + 1} \left (x^{4} - x^{3} + x^{2} - x + 1\right ) \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**10+1)/(x**4+1)**(1/2)/(x**10-1),x)

[Out]

Integral((x**2 + 1)*(x**8 - x**6 + x**4 - x**2 + 1)/((x - 1)*(x + 1)*sqrt(x**4 + 1)*(x**4 - x**3 + x**2 - x +
1)*(x**4 + x**3 + x**2 + x + 1)), x)

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