3.23.30 \(\int \frac {-1+x^5}{\sqrt {1+x^4} (1+x^5)} \, dx\)

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

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Rubi [F]  time = 0.90, 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\\ &=-\left (2 \int \frac {1}{\sqrt {1+x^4} \left (1+x^5\right )} \, dx\right )+\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} \operatorname {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} \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )-\frac {1}{5} \operatorname {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 [C]  time = 8.04, size = 841, normalized size = 5.07 \begin {gather*} \frac {\left (3 \left (1-\sqrt [5]{-1}+(-1)^{2/5}-(-1)^{3/5}+(-1)^{4/5}\right ) x^8-12 \left (1-\sqrt [5]{-1}+(-1)^{2/5}-(-1)^{3/5}+(-1)^{4/5}\right ) x^6+\left (16-17 \sqrt [5]{-1}+18 (-1)^{2/5}-19 (-1)^{3/5}+20 (-1)^{4/5}\right ) x^4-12 \left (1-\sqrt [5]{-1}+(-1)^{2/5}-(-1)^{3/5}+(-1)^{4/5}\right ) x^2+3 \left (1-\sqrt [5]{-1}+(-1)^{2/5}-(-1)^{3/5}+(-1)^{4/5}\right )\right ) \left (\frac {\sqrt {2} \tanh ^{-1}\left (\frac {x^2+1}{\sqrt {2} \sqrt {x^4+1}}\right )}{-4-\sqrt [5]{-1}+(-1)^{2/5}-(-1)^{3/5}+(-1)^{4/5}}+2 \left (\frac {(-1)^{4/5} \tanh ^{-1}\left (\frac {1-\sqrt [5]{-1} x^2}{\sqrt {1+(-1)^{2/5}} \sqrt {x^4+1}}\right )}{\sqrt {1+(-1)^{2/5}} \left (1+4 \sqrt [5]{-1}+(-1)^{2/5}-(-1)^{3/5}+(-1)^{4/5}\right )}-\frac {(-1)^{2/5} \tanh ^{-1}\left (\frac {(-1)^{2/5} x^2+1}{\sqrt {1+(-1)^{4/5}} \sqrt {x^4+1}}\right )}{\left (1+\sqrt [5]{-1}\right )^4 \left (1+(-1)^{4/5}\right )^{5/2}}+\frac {\tanh ^{-1}\left (\frac {1-(-1)^{3/5} x^2}{\sqrt {1-\sqrt [5]{-1}} \sqrt {x^4+1}}\right )}{\left (1-\sqrt [5]{-1}\right )^{3/2} \left (1+\sqrt [5]{-1}\right )^4 \left (1-\sqrt [5]{-1}+(-1)^{2/5}\right )}+\frac {(-1)^{2/5} \tanh ^{-1}\left (\frac {(-1)^{4/5} x^2+1}{\sqrt {1-(-1)^{3/5}} \sqrt {x^4+1}}\right )}{\sqrt {1-(-1)^{3/5}} \left (-4-\sqrt [5]{-1}+(-1)^{2/5}-(-1)^{3/5}+(-1)^{4/5}\right )}\right )\right )}{2 \left (-1+\sqrt [5]{-1}\right )^2 \left (1+\sqrt [5]{-1}\right )^3 \left (1+(-1)^{2/5}\right ) \left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) x^4}-\sqrt [4]{-1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\frac {2 \sqrt [4]{-1} \Pi \left (i;\left .\sin ^{-1}\left ((-1)^{3/4} x\right )\right |-1\right )}{4+\sqrt [5]{-1}-(-1)^{2/5}+(-1)^{3/5}-(-1)^{4/5}}-\frac {2 (-1)^{17/20} \Pi \left (\sqrt [10]{-1};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )}{\left (-1+\sqrt [5]{-1}\right )^2 \left (1+\sqrt [5]{-1}\right )^4}+\frac {2 (-1)^{9/20} \Pi \left (-(-1)^{3/10};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )}{1+4 \sqrt [5]{-1}+(-1)^{2/5}-(-1)^{3/5}+(-1)^{4/5}}+\frac {2 \sqrt [4]{-1} \Pi \left (-(-1)^{7/10};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )}{4+\sqrt [5]{-1}-(-1)^{2/5}+(-1)^{3/5}-(-1)^{4/5}}+\frac {2 (-1)^{9/20} \Pi \left ((-1)^{9/10};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )}{1+4 \sqrt [5]{-1}+(-1)^{2/5}-(-1)^{3/5}+(-1)^{4/5}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((3*(1 - (-1)^(1/5) + (-1)^(2/5) - (-1)^(3/5) + (-1)^(4/5)) - 12*(1 - (-1)^(1/5) + (-1)^(2/5) - (-1)^(3/5) + (
-1)^(4/5))*x^2 + (16 - 17*(-1)^(1/5) + 18*(-1)^(2/5) - 19*(-1)^(3/5) + 20*(-1)^(4/5))*x^4 - 12*(1 - (-1)^(1/5)
+ (-1)^(2/5) - (-1)^(3/5) + (-1)^(4/5))*x^6 + 3*(1 - (-1)^(1/5) + (-1)^(2/5) - (-1)^(3/5) + (-1)^(4/5))*x^8)*
((Sqrt[2]*ArcTanh[(1 + x^2)/(Sqrt[2]*Sqrt[1 + x^4])])/(-4 - (-1)^(1/5) + (-1)^(2/5) - (-1)^(3/5) + (-1)^(4/5))
+ 2*(((-1)^(4/5)*ArcTanh[(1 - (-1)^(1/5)*x^2)/(Sqrt[1 + (-1)^(2/5)]*Sqrt[1 + x^4])])/(Sqrt[1 + (-1)^(2/5)]*(1
+ 4*(-1)^(1/5) + (-1)^(2/5) - (-1)^(3/5) + (-1)^(4/5))) - ((-1)^(2/5)*ArcTanh[(1 + (-1)^(2/5)*x^2)/(Sqrt[1 +
(-1)^(4/5)]*Sqrt[1 + x^4])])/((1 + (-1)^(1/5))^4*(1 + (-1)^(4/5))^(5/2)) + ArcTanh[(1 - (-1)^(3/5)*x^2)/(Sqrt[
1 - (-1)^(1/5)]*Sqrt[1 + x^4])]/((1 - (-1)^(1/5))^(3/2)*(1 + (-1)^(1/5))^4*(1 - (-1)^(1/5) + (-1)^(2/5))) + ((
-1)^(2/5)*ArcTanh[(1 + (-1)^(4/5)*x^2)/(Sqrt[1 - (-1)^(3/5)]*Sqrt[1 + x^4])])/(Sqrt[1 - (-1)^(3/5)]*(-4 - (-1)
^(1/5) + (-1)^(2/5) - (-1)^(3/5) + (-1)^(4/5))))))/(2*(-1 + (-1)^(1/5))^2*(1 + (-1)^(1/5))^3*(1 + (-1)^(2/5))*
(1 - (-1)^(1/5) + (-1)^(2/5))*x^4) - (-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*x], -1] + (2*(-1)^(1/4)*Ellipti
cPi[I, ArcSin[(-1)^(3/4)*x], -1])/(4 + (-1)^(1/5) - (-1)^(2/5) + (-1)^(3/5) - (-1)^(4/5)) - (2*(-1)^(17/20)*El
lipticPi[(-1)^(1/10), I*ArcSinh[(-1)^(1/4)*x], -1])/((-1 + (-1)^(1/5))^2*(1 + (-1)^(1/5))^4) + (2*(-1)^(9/20)*
EllipticPi[-(-1)^(3/10), I*ArcSinh[(-1)^(1/4)*x], -1])/(1 + 4*(-1)^(1/5) + (-1)^(2/5) - (-1)^(3/5) + (-1)^(4/5
)) + (2*(-1)^(1/4)*EllipticPi[-(-1)^(7/10), I*ArcSinh[(-1)^(1/4)*x], -1])/(4 + (-1)^(1/5) - (-1)^(2/5) + (-1)^
(3/5) - (-1)^(4/5)) + (2*(-1)^(9/20)*EllipticPi[(-1)^(9/10), I*ArcSinh[(-1)^(1/4)*x], -1])/(1 + 4*(-1)^(1/5) +
(-1)^(2/5) - (-1)^(3/5) + (-1)^(4/5))

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IntegrateAlgebraic [A]  time = 0.00, size = 166, normalized size = 1.00 \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 [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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-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[4 - 4*#1 - 2*#1^3 + #1^4 & ,
(-2*Log[x] + 2*Log[1 + x^2 + Sqrt[1 + x^4] - x*#1] + Log[x]*#1 - Log[1 + x^2 + Sqrt[1 + x^4] - x*#1]*#1 - Log[
x]*#1^2 + Log[1 + x^2 + Sqrt[1 + x^4] - x*#1]*#1^2)/(-2 - 3*#1^2 + 2*#1^3) & ])/5

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fricas [B]  time = 0.79, size = 454, normalized size = 2.73 \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)*
sqrt(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(1
3*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*(s
qrt(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*sq
rt(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(sqr
t(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(s
qrt(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)) + 1/2
0*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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5} - 1}{{\left (x^{5} + 1\right )} \sqrt {x^{4} + 1}}\,{d x} \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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maple [B]  time = 2.70, size = 206, normalized size = 1.24

method	result	size
elliptic	\(\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4 \sqrt {\left (x^{2}-1\right )^{2}+2 x^{2}}}\right )}{10}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-\textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (\left (-x^{2}+\sqrt {x^{4}+1}\right )^{2}+\left (\textit {\_R}^{2}-\textit {\_R} -1\right ) \left (-x^{2}+\sqrt {x^{4}+1}\right )+\textit {\_R}^{2}-\textit {\_R} \right )\right )}{5}+\frac {\left (\frac {\ln \left (-1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{10}-\frac {\ln \left (1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{10}+\frac {4 \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x \sqrt {\sqrt {5}+1}}\right )}{5 \sqrt {\sqrt {5}+1}}-\frac {4 \arctanh \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x \sqrt {\sqrt {5}-1}}\right )}{5 \sqrt {\sqrt {5}-1}}\right ) \sqrt {2}}{2}\)	\(206\)
default	\(\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}+\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4 \sqrt {x^{4}+1}}\right )}{10}+\frac {2 \left (-1\right )^{\frac {3}{4}} \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i, -\sqrt {-i}\, \left (-1\right )^{\frac {3}{4}}\right )}{5 \sqrt {x^{4}+1}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+x^{2}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha }\, \sqrt {x^{4}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha }}+\frac {2 \left (-1\right )^{\frac {3}{4}} \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +1\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \underline {\hspace {1.25 ex}}\alpha ^{3}, i\right )}{\sqrt {x^{4}+1}}\right )\right )}{5}\)	\(267\)
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 +\RootOf \left (\textit {\_Z}^{2}-2\right )+\sqrt {x^{4}+1}}{\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 )+4 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right ) x^{2}+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 )+16 \sqrt {x^{4}+1}}{4 x^{2}+25 \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} x -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}}{25 \RootOf \left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} x -4 x^{2}-4}\right )\)	\(560\)

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]

1/10*2^(1/2)*arctanh(1/4*(2*x^2+2)*2^(1/2)/((x^2-1)^2+2*x^2)^(1/2))+1/5*sum(_R*ln((-x^2+(x^4+1)^(1/2))^2+(_R^2
-_R-1)*(-x^2+(x^4+1)^(1/2))+_R^2-_R),_R=RootOf(_Z^4-_Z^2-1))+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)*arctan(2^(1/2)/x*(x^4+1)^(1/2)/(5^(1/2)+1)^(1/2))-4/5
/(5^(1/2)-1)^(1/2)*arctanh(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^{5} - 1}{{\left (x^{5} + 1\right )} \sqrt {x^{4} + 1}}\,{d x} \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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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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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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