3.5.21 \(\int \frac {1+x^2}{(-1+x^2) \sqrt {1-x-x^2+x^3+x^4}} \, dx\)
Optimal. Leaf size=34 \[ 2 \tanh ^{-1}\left (\frac {x}{x^2-\sqrt {x^4+x^3-x^2-x+1}-1}\right ) \]
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Rubi [F] time = 0.47, 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] + Defer[Int][1/((-1 + x)*Sqrt[1 - x - x^2 + x^3 + x^4]), x] - D
efer[Int][1/((1 + 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\\ &=2 \int \frac {1}{\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\\ &=2 \int \left (\frac {1}{2 (-1+x) \sqrt {1-x-x^2+x^3+x^4}}-\frac {1}{2 (1+x) \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 {1}{\sqrt {1-x-x^2+x^3+x^4}} \, dx+\int \frac {1}{(-1+x) \sqrt {1-x-x^2+x^3+x^4}} \, dx-\int \frac {1}{(1+x) \sqrt {1-x-x^2+x^3+x^4}} \, dx\\ \end {align*}
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Mathematica [C] time = 1.63, size = 3217, normalized size = 94.62 \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*(x - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 0])^2*Sqrt[((Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 1, 0] - Root
[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 0])*(x - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 3, 0]))/((x - Root[1 - #1 -
#1^2 + #1^3 + #1^4 & , 2, 0])*(Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 1, 0] - 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])*Sqrt[((x
- 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 & , 2, 0])*(x - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 4, 0])*(Root[1 - #1 - #1^2 + #1^3 + #1^4
& , 2, 0] - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 4, 0]))/((x - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 0])^2*(
Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 1, 0] - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 4, 0])^2)]*(-((EllipticF[Arc
Sin[Sqrt[((x - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 1, 0])*(Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 0] - Root[
1 - #1 - #1^2 + #1^3 + #1^4 & , 4, 0]))/((x - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 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
& , 2, 0] - 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])*(Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 0] - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 4, 0]))
)]*(-1 + Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 1, 0]) + EllipticPi[((-1 + Root[1 - #1 - #1^2 + #1^3 + #1^4 & ,
2, 0])*(Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 1, 0] - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 4, 0]))/((-1 + Root[
1 - #1 - #1^2 + #1^3 + #1^4 & , 1, 0])*(Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 0] - 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])*(Root[1 - #1 - #1^2 + #1^3
+ #1^4 & , 2, 0] - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 4, 0]))/((x - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2,
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 & , 2, 0] - 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])*(Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 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 & ,
2, 0]))/((-1 + Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 1, 0])*(-1 + Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 0])*
(Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 0] - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 4, 0]))) + (EllipticF[ArcSi
n[Sqrt[((x - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 1, 0])*(Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 0] - Root[1
- #1 - #1^2 + #1^3 + #1^4 & , 4, 0]))/((x - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 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 &
, 2, 0] - 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])*(Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 0] - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 4, 0])))]
*(1 + Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 1, 0]) + EllipticPi[((1 + Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 0
])*(Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 1, 0] - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 4, 0]))/((1 + Root[1 - #
1 - #1^2 + #1^3 + #1^4 & , 1, 0])*(Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 0] - 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])*(Root[1 - #1 - #1^2 + #1^3 + #1^
4 & , 2, 0] - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 4, 0]))/((x - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 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 & , 2, 0] - 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])*(Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 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 & , 2, 0
]))/((1 + Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 1, 0])*(1 + Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 0])*(Root[1
- #1 - #1^2 + #1^3 + #1^4 & , 2, 0] - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 4, 0])) + EllipticF[ArcSin[Sqrt[((
x - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 1, 0])*(Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 0] - Root[1 - #1 - #1
^2 + #1^3 + #1^4 & , 4, 0]))/((x - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 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 & , 2, 0] -
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])*(Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 0] - Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 4, 0]))]/(-Root[1 - #
1 - #1^2 + #1^3 + #1^4 & , 2, 0] + Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 4, 0])))/(Sqrt[1 - x - x^2 + x^3 + x^4
]*(-Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 1, 0] + Root[1 - #1 - #1^2 + #1^3 + #1^4 & , 2, 0]))
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IntegrateAlgebraic [A] time = 0.14, size = 34, normalized size = 1.00 \begin {gather*} 2 \tanh ^{-1}\left (\frac {x}{-1+x^2-\sqrt {1-x-x^2+x^3+x^4}}\right ) \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*ArcTanh[x/(-1 + x^2 - Sqrt[1 - x - x^2 + x^3 + x^4])]
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fricas [A] time = 0.49, size = 38, normalized size = 1.12 \begin {gather*} \log \left (-\frac {x^{2} + 2 \, x - 2 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} - 1}{x^{2} - 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]
log(-(x^2 + 2*x - 2*sqrt(x^4 + x^3 - x^2 - x + 1) - 1)/(x^2 - 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 [A] time = 1.16, size = 44, normalized size = 1.29
| | |
| method |
result |
size |
| | |
| trager |
\(-\ln \left (-\frac {x^{2}+2 \sqrt {x^{4}+x^{3}-x^{2}-x +1}+2 x -1}{\left (-1+x \right ) \left (1+x \right )}\right )\) |
\(44\) |
| default |
\(\text {Expression too large to display}\) |
\(4112\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(105548\) |
| | |
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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]
-ln(-(x^2+2*(x^4+x^3-x^2-x+1)^(1/2)+2*x-1)/(-1+x)/(1+x))
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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 = -0.03 \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^3 - x^2 - x + x^4 + 1)^(1/2)),x)
[Out]
int((x^2 + 1)/((x^2 - 1)*(x^3 - x^2 - x + 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} + 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+1)/(x**2-1)/(x**4+x**3-x**2-x+1)**(1/2),x)
[Out]
Integral((x**2 + 1)/((x - 1)*(x + 1)*sqrt(x**4 + x**3 - x**2 - x + 1)), x)
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