3.6.91 \(\int \frac {-1+x^2}{(1+x^2) \sqrt {1-x-x^2-x^3+x^4}} \, dx\)
Optimal. Leaf size=46 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {3} x}{x^2-\sqrt {x^4-x^3-x^2-x+1}+1}\right )}{\sqrt {3}} \]
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Rubi [F] time = 0.48, 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^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^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] - I*Defer[Int][1/((I - x)*Sqrt[1 - x - x^2 - x^3 + x^4]), x] -
I*Defer[Int][1/((I + x)*Sqrt[1 - x - x^2 - x^3 + x^4]), x]
Rubi steps
\begin {align*} \int \frac {-1+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}{\left (1+x^2\right ) \sqrt {1-x-x^2-x^3+x^4}}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\left (1+x^2\right ) \sqrt {1-x-x^2-x^3+x^4}} \, dx\right )+\int \frac {1}{\sqrt {1-x-x^2-x^3+x^4}} \, dx\\ &=-\left (2 \int \left (\frac {i}{2 (i-x) \sqrt {1-x-x^2-x^3+x^4}}+\frac {i}{2 (i+x) \sqrt {1-x-x^2-x^3+x^4}}\right ) \, dx\right )+\int \frac {1}{\sqrt {1-x-x^2-x^3+x^4}} \, dx\\ &=-\left (i \int \frac {1}{(i-x) \sqrt {1-x-x^2-x^3+x^4}} \, dx\right )-i \int \frac {1}{(i+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 = 3.22, size = 3173, normalized size = 68.98 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[(-1 + x^2)/((1 + x^2)*Sqrt[1 - x - x^2 - x^3 + x^4]),x]
[Out]
(2*(1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*x)^2*Sqrt[(-x + Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 3, 0])/((1
+ Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*x)*(-Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0] + Root[1 - #1 - #1^2
- #1^3 + #1^4 & , 3, 0]))]*Sqrt[-(((x - Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0])*(1 + Sqrt[13] + Sqrt[2*(-
1 + Sqrt[13])] - 4*Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0]))/((1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*x
)*(Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0] - Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0])))]*Sqrt[(-x + Root
[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0])/((1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*x)*(-Root[1 - #1 - #1^2 -
#1^3 + #1^4 & , 1, 0] + Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0]))]*(((-4*I)*((EllipticPi[(((1 - 4*I) + Sqr
t[13] + Sqrt[2*(-1 + Sqrt[13])])*(Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0] - Root[1 - #1 - #1^2 - #1^3 + #1^
4 & , 4, 0]))/((-I + Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0])*(1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*R
oot[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0])), ArcSin[Sqrt[-(((x - Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0])*(
1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0]))/((1 + Sqrt[13] + Sqrt[
2*(-1 + Sqrt[13])] - 4*x)*(Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0] - Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4
, 0])))]], -(((1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 3, 0])*(Root[1
- #1 - #1^2 - #1^3 + #1^4 & , 1, 0] - Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0]))/((-Root[1 - #1 - #1^2 - #1^
3 + #1^4 & , 1, 0] + Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 3, 0])*(1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*R
oot[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0])))]*(1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*Root[1 - #1 - #1^2 -
#1^3 + #1^4 & , 1, 0]))/4 + EllipticF[ArcSin[Sqrt[-(((x - Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0])*(1 + Sq
rt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0]))/((1 + Sqrt[13] + Sqrt[2*(-1
+ Sqrt[13])] - 4*x)*(Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0] - Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0]))
)]], -(((1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 3, 0])*(Root[1 - #1 -
#1^2 - #1^3 + #1^4 & , 1, 0] - Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0]))/((-Root[1 - #1 - #1^2 - #1^3 + #1
^4 & , 1, 0] + Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 3, 0])*(1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*Root[1
- #1 - #1^2 - #1^3 + #1^4 & , 4, 0])))]*(-I + Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0]))*(Root[1 - #1 - #1^2
- #1^3 + #1^4 & , 1, 0] - Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0]))/(((1 - 4*I) + Sqrt[13] + Sqrt[2*(-1 +
Sqrt[13])])*(-I + Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0])) + ((4*I)*((EllipticPi[(((1 + 4*I) + Sqrt[13] +
Sqrt[2*(-1 + Sqrt[13])])*(Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0] - Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4,
0]))/((I + Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0])*(1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*Root[1 - #
1 - #1^2 - #1^3 + #1^4 & , 4, 0])), ArcSin[Sqrt[-(((x - Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0])*(1 + Sqrt[
13] + Sqrt[2*(-1 + Sqrt[13])] - 4*Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0]))/((1 + Sqrt[13] + Sqrt[2*(-1 + S
qrt[13])] - 4*x)*(Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0] - Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0])))]]
, -(((1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 3, 0])*(Root[1 - #1 - #1
^2 - #1^3 + #1^4 & , 1, 0] - Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0]))/((-Root[1 - #1 - #1^2 - #1^3 + #1^4
& , 1, 0] + Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 3, 0])*(1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*Root[1 - #
1 - #1^2 - #1^3 + #1^4 & , 4, 0])))]*(1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*Root[1 - #1 - #1^2 - #1^3 + #
1^4 & , 1, 0]))/4 + EllipticF[ArcSin[Sqrt[-(((x - Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0])*(1 + Sqrt[13] +
Sqrt[2*(-1 + Sqrt[13])] - 4*Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0]))/((1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13
])] - 4*x)*(Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0] - Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0])))]], -(((
1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 3, 0])*(Root[1 - #1 - #1^2 - #
1^3 + #1^4 & , 1, 0] - Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0]))/((-Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1,
0] + Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 3, 0])*(1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*Root[1 - #1 - #1
^2 - #1^3 + #1^4 & , 4, 0])))]*(I + Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0]))*(Root[1 - #1 - #1^2 - #1^3 +
#1^4 & , 1, 0] - Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0]))/(((1 + 4*I) + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])]
)*(I + Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0])) + EllipticF[ArcSin[Sqrt[-(((x - Root[1 - #1 - #1^2 - #1^3
+ #1^4 & , 1, 0])*(1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0]))/((1
+ Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*x)*(Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0] - Root[1 - #1 - #1^2
- #1^3 + #1^4 & , 4, 0])))]], ((1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[13])] - 4*Root[1 - #1 - #1^2 - #1^3 + #1^4 &
, 3, 0])*(Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0] - Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0]))/((Root[1 -
#1 - #1^2 - #1^3 + #1^4 & , 1, 0] - Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 3, 0])*(1 + Sqrt[13] + Sqrt[2*(-1 +
Sqrt[13])] - 4*Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0]))]*(-Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 1, 0] + Ro
ot[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0])))/(Sqrt[1 - x - x^2 - x^3 + x^4]*(1 + Sqrt[13] + Sqrt[2*(-1 + Sqrt[1
3])] - 4*Root[1 - #1 - #1^2 - #1^3 + #1^4 & , 4, 0]))
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IntegrateAlgebraic [A] time = 0.17, size = 46, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {3} x}{1+x^2-\sqrt {1-x-x^2-x^3+x^4}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*Sqrt[1 - x - x^2 - x^3 + x^4]),x]
[Out]
(2*ArcTan[(Sqrt[3]*x)/(1 + x^2 - Sqrt[1 - x - x^2 - x^3 + x^4])])/Sqrt[3]
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fricas [A] time = 0.50, size = 39, normalized size = 0.85 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (x^{2} + 6 \, x + 1\right )}}{6 \, \sqrt {x^{4} - x^{3} - x^{2} - x + 1}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2-1)/(x^2+1)/(x^4-x^3-x^2-x+1)^(1/2),x, algorithm="fricas")
[Out]
-1/3*sqrt(3)*arctan(1/6*sqrt(3)*(x^2 + 6*x + 1)/sqrt(x^4 - x^3 - x^2 - x + 1))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - 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^2-1)/(x^2+1)/(x^4-x^3-x^2-x+1)^(1/2),x, algorithm="giac")
[Out]
integrate((x^2 - 1)/(sqrt(x^4 - x^3 - x^2 - x + 1)*(x^2 + 1)), x)
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maple [C] time = 1.17, size = 67, normalized size = 1.46
| | |
| method |
result |
size |
| | |
| trager |
\(\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) x^{2}+6 \RootOf \left (\textit {\_Z}^{2}+3\right ) x +6 \sqrt {x^{4}-x^{3}-x^{2}-x +1}+\RootOf \left (\textit {\_Z}^{2}+3\right )}{x^{2}+1}\right )}{3}\) |
\(67\) |
| default |
\(\text {Expression too large to display}\) |
\(100440\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(114996\) |
| | |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^2-1)/(x^2+1)/(x^4-x^3-x^2-x+1)^(1/2),x,method=_RETURNVERBOSE)
[Out]
1/3*RootOf(_Z^2+3)*ln(-(RootOf(_Z^2+3)*x^2+6*RootOf(_Z^2+3)*x+6*(x^4-x^3-x^2-x+1)^(1/2)+RootOf(_Z^2+3))/(x^2+1
))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - 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^2-1)/(x^2+1)/(x^4-x^3-x^2-x+1)^(1/2),x, algorithm="maxima")
[Out]
integrate((x^2 - 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 = -0.02 \begin {gather*} \int \frac {x^2-1}{\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 - 1)/((x^2 + 1)*(x^4 - x^2 - x^3 - x + 1)^(1/2)),x)
[Out]
int((x^2 - 1)/((x^2 + 1)*(x^4 - x^2 - x^3 - x + 1)^(1/2)), 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 + 1\right )}{\left (x^{2} + 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**2-1)/(x**2+1)/(x**4-x**3-x**2-x+1)**(1/2),x)
[Out]
Integral((x - 1)*(x + 1)/((x**2 + 1)*sqrt(x**4 - x**3 - x**2 - x + 1)), x)
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