3.13.21 \(\int \frac {1-x+x^2}{(-1+x^2) \sqrt {1-x+x^2-x^3+x^4}} \, dx\)

Optimal. Leaf size=89 \[ \frac {1}{2} \tanh ^{-1}\left (\frac {x}{x^2-\sqrt {x^4-x^3+x^2-x+1}-2 x+1}\right )+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {5} x}{x^2-\sqrt {x^4-x^3+x^2-x+1}+2 x+1}\right )}{2 \sqrt {5}} \]

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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]

Int[(1 - x + x^2)/((-1 + x^2)*Sqrt[1 - x + x^2 - x^3 + x^4]),x]

[Out]

Defer[Int][1/Sqrt[1 - x + x^2 - x^3 + x^4], x] - Defer[Int][1/((1 - x)*Sqrt[1 - x + x^2 - x^3 + x^4]), x]/2 -
(3*Defer[Int][1/((1 + x)*Sqrt[1 - x + x^2 - x^3 + x^4]), x])/2

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*}

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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]

Integrate[(1 - x + x^2)/((-1 + x^2)*Sqrt[1 - x + x^2 - x^3 + x^4]),x]

[Out]

-(((-1)^(3/5)*Sqrt[(1 - (-1)^(1/5) + (-1)^(2/5) + (-1)^(4/5) - x)/((-1 + (-1)^(1/5))*((-1)^(2/5) + x))]*((-1)^
(2/5) + x)*((-1)^(4/5) + x)*(2*(2*Sqrt[-(((-1)^(1/5)*(-1 + (-1)^(1/5))*((-1)^(1/5) - x))/((1 - (-1)^(1/5) + (-
1)^(2/5))*((-1)^(2/5) + x)))] + (-1)^(2/5)*Sqrt[-(((-1)^(1/5)*(-1 + (-1)^(1/5))*((-1)^(1/5) - x))/((1 - (-1)^(
1/5) + (-1)^(2/5))*((-1)^(2/5) + x)))] - Sqrt[(-1 + (-1)^(1/5) - (-1)^(4/5) + (-1)^(1/5)*(-1 + (-1)^(1/5))*x)/
((1 - (-1)^(1/5) + (-1)^(2/5))*((-1)^(2/5) + x))] + (-1)^(4/5)*Sqrt[(-1 + (-1)^(1/5) - (-1)^(4/5) + (-1)^(1/5)
*(-1 + (-1)^(1/5))*x)/((1 - (-1)^(1/5) + (-1)^(2/5))*((-1)^(2/5) + x))])*EllipticF[ArcSin[Sqrt[-(((-1)^(1/5)*(
-1 + (-1)^(1/5))*((-1)^(1/5) - x))/((1 - (-1)^(1/5) + (-1)^(2/5))*((-1)^(2/5) + x)))]], (1 - (-1)^(1/5) + (-1)
^(2/5))/(-1 + (-1)^(1/5))^2] + (-1)^(1/5)*Sqrt[-(((-1)^(1/5)*(-1 + (-1)^(1/5))*((-1)^(1/5) - x))/((1 - (-1)^(1
/5) + (-1)^(2/5))*((-1)^(2/5) + x)))]*((1 + (-1)^(1/5))^2*EllipticPi[((-1)^(4/5)*(1 + (-1)^(2/5))*(1 - (-1)^(1
/5) + (-1)^(2/5)))/(-1 + (-1)^(1/5))^2, ArcSin[Sqrt[-(((-1)^(1/5)*(-1 + (-1)^(1/5))*((-1)^(1/5) - x))/((1 - (-
1)^(1/5) + (-1)^(2/5))*((-1)^(2/5) + x)))]], (1 - (-1)^(1/5) + (-1)^(2/5))/(-1 + (-1)^(1/5))^2] - 3*(1 + (-1)^
(2/5))*EllipticPi[1 - (-1)^(1/5) + (-1)^(4/5), ArcSin[Sqrt[-(((-1)^(1/5)*(-1 + (-1)^(1/5))*((-1)^(1/5) - x))/(
(1 - (-1)^(1/5) + (-1)^(2/5))*((-1)^(2/5) + x)))]], (1 - (-1)^(1/5) + (-1)^(2/5))/(-1 + (-1)^(1/5))^2])))/((-1
+ (-1)^(2/5))^2*(1 + (-1)^(2/5))*Sqrt[((-1)^(4/5) + x)/((1 - (-1)^(1/5) + (-1)^(2/5))*((-1)^(2/5) + x))]*Sqrt
[1 - x + x^2 - x^3 + x^4]))

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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]

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

[Out]

ArcTanh[x/(1 - 2*x + x^2 - Sqrt[1 - x + x^2 - x^3 + x^4])]/2 + (3*ArcTanh[(Sqrt[5]*x)/(1 + 2*x + x^2 - Sqrt[1
- x + x^2 - x^3 + x^4])])/(2*Sqrt[5])

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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]

integrate((x^2-x+1)/(x^2-1)/(x^4-x^3+x^2-x+1)^(1/2),x, algorithm="fricas")

[Out]

3/40*sqrt(5)*log(-(9*x^4 - 4*x^3 + 4*sqrt(5)*sqrt(x^4 - x^3 + x^2 - x + 1)*(x^2 + 1) + 14*x^2 - 4*x + 9)/(x^4
+ 4*x^3 + 6*x^2 + 4*x + 1)) + 1/4*log((3*x^2 - 4*x - 2*sqrt(x^4 - x^3 + x^2 - x + 1) + 3)/(x^2 - 2*x + 1))

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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]

integrate((x^2-x+1)/(x^2-1)/(x^4-x^3+x^2-x+1)^(1/2),x, algorithm="giac")

[Out]

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

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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]

int((x^2-x+1)/(x^2-1)/(x^4-x^3+x^2-x+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/4*ln(-(3*x^2+2*(x^4-x^3+x^2-x+1)^(1/2)-4*x+3)/(-1+x)^2)-3/20*RootOf(_Z^2-5)*ln((RootOf(_Z^2-5)*x^2+RootOf(_
Z^2-5)-2*(x^4-x^3+x^2-x+1)^(1/2))/(1+x)^2)

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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]

integrate((x^2-x+1)/(x^2-1)/(x^4-x^3+x^2-x+1)^(1/2),x, algorithm="maxima")

[Out]

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

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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]

int((x^2 - x + 1)/((x^2 - 1)*(x^2 - x - x^3 + x^4 + 1)^(1/2)),x)

[Out]

int((x^2 - x + 1)/((x^2 - 1)*(x^2 - x - x^3 + x^4 + 1)^(1/2)), x)

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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]

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

[Out]

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

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