Optimal. Leaf size=89 \[ \frac {1}{2} \tanh ^{-1}\left (\frac {x}{x^2-\sqrt {x^4-x^3+x^2-x+1}-2 x+1}\right )+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {5} x}{x^2-\sqrt {x^4-x^3+x^2-x+1}+2 x+1}\right )}{2 \sqrt {5}} \]
________________________________________________________________________________________
Rubi [F] time = 0.62, 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+x^2}{\left (-1+x^2\right ) \sqrt {1-x+x^2-x^3+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {1-x+x^2}{\left (-1+x^2\right ) \sqrt {1-x+x^2-x^3+x^4}} \, dx &=\int \left (\frac {1}{\sqrt {1-x+x^2-x^3+x^4}}+\frac {2-x}{\left (-1+x^2\right ) \sqrt {1-x+x^2-x^3+x^4}}\right ) \, dx\\ &=\int \frac {1}{\sqrt {1-x+x^2-x^3+x^4}} \, dx+\int \frac {2-x}{\left (-1+x^2\right ) \sqrt {1-x+x^2-x^3+x^4}} \, dx\\ &=\int \frac {1}{\sqrt {1-x+x^2-x^3+x^4}} \, dx+\int \left (-\frac {1}{2 (1-x) \sqrt {1-x+x^2-x^3+x^4}}-\frac {3}{2 (1+x) \sqrt {1-x+x^2-x^3+x^4}}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {1}{(1-x) \sqrt {1-x+x^2-x^3+x^4}} \, dx\right )-\frac {3}{2} \int \frac {1}{(1+x) \sqrt {1-x+x^2-x^3+x^4}} \, dx+\int \frac {1}{\sqrt {1-x+x^2-x^3+x^4}} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 1.59, size = 743, normalized size = 8.35 \begin {gather*} -\frac {(-1)^{3/5} \sqrt {\frac {-x+(-1)^{4/5}+(-1)^{2/5}-\sqrt [5]{-1}+1}{\left (\sqrt [5]{-1}-1\right ) \left (x+(-1)^{2/5}\right )}} \left (x+(-1)^{2/5}\right ) \left (x+(-1)^{4/5}\right ) \left (2 \left ((-1)^{2/5} \sqrt {-\frac {\sqrt [5]{-1} \left (\sqrt [5]{-1}-1\right ) \left (\sqrt [5]{-1}-x\right )}{\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left (x+(-1)^{2/5}\right )}}+2 \sqrt {-\frac {\sqrt [5]{-1} \left (\sqrt [5]{-1}-1\right ) \left (\sqrt [5]{-1}-x\right )}{\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left (x+(-1)^{2/5}\right )}}+(-1)^{4/5} \sqrt {\frac {\sqrt [5]{-1} \left (\sqrt [5]{-1}-1\right ) x-(-1)^{4/5}+\sqrt [5]{-1}-1}{\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left (x+(-1)^{2/5}\right )}}-\sqrt {\frac {\sqrt [5]{-1} \left (\sqrt [5]{-1}-1\right ) x-(-1)^{4/5}+\sqrt [5]{-1}-1}{\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left (x+(-1)^{2/5}\right )}}\right ) F\left (\sin ^{-1}\left (\sqrt {-\frac {\sqrt [5]{-1} \left (-1+\sqrt [5]{-1}\right ) \left (\sqrt [5]{-1}-x\right )}{\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left (x+(-1)^{2/5}\right )}}\right )|\frac {1-\sqrt [5]{-1}+(-1)^{2/5}}{\left (-1+\sqrt [5]{-1}\right )^2}\right )+\sqrt [5]{-1} \sqrt {-\frac {\sqrt [5]{-1} \left (\sqrt [5]{-1}-1\right ) \left (\sqrt [5]{-1}-x\right )}{\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left (x+(-1)^{2/5}\right )}} \left (\left (1+\sqrt [5]{-1}\right )^2 \Pi \left (\frac {(-1)^{4/5} \left (1+(-1)^{2/5}\right ) \left (1-\sqrt [5]{-1}+(-1)^{2/5}\right )}{\left (-1+\sqrt [5]{-1}\right )^2};\sin ^{-1}\left (\sqrt {-\frac {\sqrt [5]{-1} \left (-1+\sqrt [5]{-1}\right ) \left (\sqrt [5]{-1}-x\right )}{\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left (x+(-1)^{2/5}\right )}}\right )|\frac {1-\sqrt [5]{-1}+(-1)^{2/5}}{\left (-1+\sqrt [5]{-1}\right )^2}\right )-3 \left (1+(-1)^{2/5}\right ) \Pi \left (1-\sqrt [5]{-1}+(-1)^{4/5};\sin ^{-1}\left (\sqrt {-\frac {\sqrt [5]{-1} \left (-1+\sqrt [5]{-1}\right ) \left (\sqrt [5]{-1}-x\right )}{\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left (x+(-1)^{2/5}\right )}}\right )|\frac {1-\sqrt [5]{-1}+(-1)^{2/5}}{\left (-1+\sqrt [5]{-1}\right )^2}\right )\right )\right )}{\left ((-1)^{2/5}-1\right )^2 \left (1+(-1)^{2/5}\right ) \sqrt {\frac {x+(-1)^{4/5}}{\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left (x+(-1)^{2/5}\right )}} \sqrt {x^4-x^3+x^2-x+1}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.44, size = 89, normalized size = 1.00 \begin {gather*} \frac {1}{2} \tanh ^{-1}\left (\frac {x}{1-2 x+x^2-\sqrt {1-x+x^2-x^3+x^4}}\right )+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {5} x}{1+2 x+x^2-\sqrt {1-x+x^2-x^3+x^4}}\right )}{2 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.03, size = 121, normalized size = 1.36 \begin {gather*} \frac {3}{40} \, \sqrt {5} \log \left (-\frac {9 \, x^{4} - 4 \, x^{3} + 4 \, \sqrt {5} \sqrt {x^{4} - x^{3} + x^{2} - x + 1} {\left (x^{2} + 1\right )} + 14 \, x^{2} - 4 \, x + 9}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right ) + \frac {1}{4} \, \log \left (\frac {3 \, x^{2} - 4 \, x - 2 \, \sqrt {x^{4} - x^{3} + x^{2} - x + 1} + 3}{x^{2} - 2 \, x + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - x + 1}{\sqrt {x^{4} - x^{3} + x^{2} - x + 1} {\left (x^{2} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 1.11, size = 94, normalized size = 1.06
method | result | size |
trager | \(-\frac {\ln \left (-\frac {3 x^{2}+2 \sqrt {x^{4}-x^{3}+x^{2}-x +1}-4 x +3}{\left (-1+x \right )^{2}}\right )}{4}-\frac {3 \RootOf \left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-5\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}-5\right )-2 \sqrt {x^{4}-x^{3}+x^{2}-x +1}}{\left (1+x \right )^{2}}\right )}{20}\) | \(94\) |
default | \(\text {Expression too large to display}\) | \(2852\) |
elliptic | \(\text {Expression too large to display}\) | \(101423\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - x + 1}{\sqrt {x^{4} - x^{3} + x^{2} - x + 1} {\left (x^{2} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2-x+1}{\left (x^2-1\right )\,\sqrt {x^4-x^3+x^2-x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - x + 1}{\left (x - 1\right ) \left (x + 1\right ) \sqrt {x^{4} - x^{3} + x^{2} - x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________