∫ 1+x2 ___________________________________________(−1+x2 )√ _________

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 \[ 2 \tanh ^{-1}\left (\frac {x}{x^2-\sqrt {x^4+x^3-x^2-x+1}-1}\right ) \]

________________________________________________________________________________________

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*}

________________________________________________________________________________________

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]))

________________________________________________________________________________________

IntegrateAlgebraic [A] time = 0.17, size = 34, normalized size = 1.00 \begin {gather*} 2 \tanh ^{-1}\left (\frac {x}{x^2-\sqrt {x^4+x^3-x^2-x+1}-1}\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])]

________________________________________________________________________________________

fricas [A] time = 0.43, 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))

________________________________________________________________________________________

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)

________________________________________________________________________________________

maple [C] time = 1.81, size = 4112, normalized size = 120.94 \begin {gather*} \text {output too large to display} \end {gather*}

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)

[Out]

2*(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4))*((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index

=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,i

ndex=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)))^(1/2)*(x-RootOf(_Z^4+_Z^

3-_Z^2-_Z+1,index=2))^2*((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))*(x-RootOf(_

Z^4+_Z^3-_Z^2-_Z+1,index=3))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(x-Root

Of(_Z^4+_Z^3-_Z^2-_Z+1,index=2)))^(1/2)*((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index

=1))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,i

ndex=1))/(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)))^(1/2)/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-

_Z^2-_Z+1,index=2))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/((x-RootOf(_Z^4+

_Z^3-_Z^2-_Z+1,index=1))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3))*(x-Ro

otOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)))^(1/2)*EllipticF(((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^

2-_Z+1,index=2))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3

-_Z^2-_Z+1,index=1))/(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)))^(1/2),((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-Root

Of(_Z^4+_Z^3-_Z^2-_Z+1,index=3))*(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4))/(Ro

otOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-Ro

otOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)))^(1/2))+2*(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,i

ndex=4))*((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2))*(x-RootOf(_Z^4+_Z^3-_Z^2-_

Z+1,index=1))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(x-RootOf(_Z^4+_Z^3-_Z

^2-_Z+1,index=2)))^(1/2)*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2))^2*((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootO

f(_Z^4+_Z^3-_Z^2-_Z+1,index=1))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3)-R

ootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)))^(1/2)*((RootOf(_Z^4+_Z^3-_Z^2-_Z+

1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4))/(RootOf(_Z^4+_Z^3-_Z^2

-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)))^(1/2)/(RootOf(_Z^

4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^

4+_Z^3-_Z^2-_Z+1,index=1))/((x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2))*(x

-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)))^(1/2)/(RootOf(_Z^4+_Z^3-_Z^2-_Z

+1,index=2)-1)*(EllipticF(((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2))*(x-RootOf

(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(x-Ro

otOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)))^(1/2),((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,ind

ex=3))*(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,i

ndex=1)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,i

ndex=4)))^(1/2))+(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(RootOf(_Z^4+_Z^3-_

Z^2-_Z+1,index=1)-1)*EllipticPi(((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2))*(x-

RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))

/(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)))^(1/2),(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-1)*(RootOf(_Z^4+_Z^3-_Z^2

-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)-1)/(RootOf(_Z^4+_Z^3-

_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)),((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3

-_Z^2-_Z+1,index=3))*(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4))/(RootOf(_Z^4+_Z

^3-_Z^2-_Z+1,index=1)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z

^3-_Z^2-_Z+1,index=4)))^(1/2)))-2*(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4))*((

RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1

))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,ind

ex=2)))^(1/2)*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2))^2*((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3

-_Z^2-_Z+1,index=1))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3)-RootOf(_Z^4+

_Z^3-_Z^2-_Z+1,index=1))/(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)))^(1/2)*((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-

RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index

=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)))^(1/2)/(RootOf(_Z^4+_Z^3-_Z^2

-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2

-_Z+1,index=1))/((x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2))*(x-RootOf(_Z^

4+_Z^3-_Z^2-_Z+1,index=3))*(x-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)))^(1/2)/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)

+1)*(EllipticF(((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2))*(x-RootOf(_Z^4+_Z^3-

_Z^2-_Z+1,index=1))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(x-RootOf(_Z^4+_

Z^3-_Z^2-_Z+1,index=2)))^(1/2),((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3))*(Roo

tOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)-Roo

tOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)))^(

1/2))+(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,in

dex=1)+1)*EllipticPi(((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2))*(x-RootOf(_Z^4

+_Z^3-_Z^2-_Z+1,index=1))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(x-RootOf(

_Z^4+_Z^3-_Z^2-_Z+1,index=2)))^(1/2),(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)+1)*(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index

=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)+1)/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,i

ndex=4)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)),((RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,

index=3))*(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=1)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=4))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+

1,index=1)-RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=3))/(RootOf(_Z^4+_Z^3-_Z^2-_Z+1,index=2)-RootOf(_Z^4+_Z^3-_Z^2-_Z+

1,index=4)))^(1/2)))

________________________________________________________________________________________

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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]