∫ 1+x2 _________________________________________________(−1+x2 )√ ___________

3.24.70 \(\int \frac {1+x^2}{(-1+x^2) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\)

Optimal. Leaf size=189 \[ -\frac {\sqrt {-2 a-2 b-c} \tan ^{-1}\left (\frac {x \sqrt {-2 a-2 b-c}}{-\sqrt {a x^4+a+b x^3+b x+c x^2}+\sqrt {a} x^2-2 \sqrt {a} x+\sqrt {a}}\right )}{2 a+2 b+c}-\frac {\sqrt {-2 a+2 b-c} \tan ^{-1}\left (\frac {x \sqrt {-2 a+2 b-c}}{-\sqrt {a x^4+a+b x^3+b x+c x^2}+\sqrt {a} x^2+2 \sqrt {a} x+\sqrt {a}}\right )}{2 a-2 b+c} \]

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Rubi [F] time = 0.77, 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 {a+b x+c x^2+b x^3+a x^4}} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

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

Rubi steps

\begin {align*} \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx &=\int \left (\frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}}+\frac {2}{\left (-1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx\\ &=2 \int \frac {1}{\left (-1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\\ &=2 \int \left (\frac {1}{2 (-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}}-\frac {1}{2 (1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\\ &=\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\int \frac {1}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx-\int \frac {1}{(1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\\ \end {align*}

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Mathematica [C] time = 2.12, size = 3575, normalized size = 18.92 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*(x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])^2*Sqrt[((Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & ,

1] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])*(x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 3]))/

((x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] - Root

[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 3]))]*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] - Root[a + b*#

1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4])*Sqrt[((x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1])*(Root[a + b*#

1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])*(x - Root[a + b*#1 + c*

#1^2 + b*#1^3 + a*#1^4 & , 4])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 + c*#1^2 + b*#

1^3 + a*#1^4 & , 4]))/((x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])^2*(Root[a + b*#1 + c*#1^2 + b*#1^

3 + a*#1^4 & , 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4])^2)]*(-((EllipticF[ArcSin[Sqrt[((x - Root[

a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 +

c*#1^2 + b*#1^3 + a*#1^4 & , 4]))/((x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])*(Root[a + b*#1 + c*#

1^2 + b*#1^3 + a*#1^4 & , 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))]], -(((Root[a + b*#1 + c*#1^2

+ b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 3])*(Root[a + b*#1 + c*#1^2 + b*#1^3

+ a*#1^4 & , 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))/((-Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^

4 & , 1] + Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 3])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] -

Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4])))]*(-1 + Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1]) +

EllipticPi[((-1 + Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 &

, 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))/((-1 + Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & ,

1])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4])), ArcS

in[Sqrt[((x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2

] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))/((x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])*(

Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))]], -(((Roo

t[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 3])*(Root[a + b*#1

+ c*#1^2 + b*#1^3 + a*#1^4 & , 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))/((-Root[a + b*#1 + c*#1

^2 + b*#1^3 + a*#1^4 & , 1] + Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 3])*(Root[a + b*#1 + c*#1^2 + b*#1^

3 + a*#1^4 & , 2] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4])))]*(-Root[a + b*#1 + c*#1^2 + b*#1^3 + a*

#1^4 & , 1] + Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2]))/((-1 + Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^

4 & , 1])*(-1 + Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & ,

2] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))) + (EllipticF[ArcSin[Sqrt[((x - Root[a + b*#1 + c*#1^2

+ b*#1^3 + a*#1^4 & , 1])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 + c*#1^2 + b*#1^3

+ a*#1^4 & , 4]))/((x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*

#1^4 & , 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))]], -(((Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^

4 & , 2] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 3])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] -

Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))/((-Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] + Root[a

+ b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 3])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 +

c*#1^2 + b*#1^3 + a*#1^4 & , 4])))]*(1 + Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1]) + EllipticPi[((1 + R

oot[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] - Root[a + b*

#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))/((1 + Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1])*(Root[a + b*#1 +

c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4])), ArcSin[Sqrt[((x - Root[a

+ b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 +

c*#1^2 + b*#1^3 + a*#1^4 & , 4]))/((x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])*(Root[a + b*#1 + c*#1

^2 + b*#1^3 + a*#1^4 & , 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))]], -(((Root[a + b*#1 + c*#1^2

+ b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 3])*(Root[a + b*#1 + c*#1^2 + b*#1^3 +

a*#1^4 & , 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))/((-Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4

& , 1] + Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 3])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] -

Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4])))]*(-Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] + Root[a

+ b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2]))/((1 + Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1])*(1 + Root[a

+ b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 +

c*#1^2 + b*#1^3 + a*#1^4 & , 4])) + EllipticF[ArcSin[Sqrt[((x - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1

])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))/((x -

Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] - Root[a + b

*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4]))]], ((Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 +

c*#1^2 + b*#1^3 + a*#1^4 & , 3])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] - Root[a + b*#1 + c*#1^2 +

b*#1^3 + a*#1^4 & , 4]))/((Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] - Root[a + b*#1 + c*#1^2 + b*#1^3 +

a*#1^4 & , 3])*(Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] - Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 &

, 4]))]/(-Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 2] + Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 4])))

/(Sqrt[x*(b + c*x + b*x^2) + a*(1 + x^4)]*(-Root[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , 1] + Root[a + b*#1 +

c*#1^2 + b*#1^3 + a*#1^4 & , 2]))

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IntegrateAlgebraic [A] time = 1.30, size = 189, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {-2 a-2 b-c} \tan ^{-1}\left (\frac {\sqrt {-2 a-2 b-c} x}{\sqrt {a}-2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{2 a+2 b+c}-\frac {\sqrt {-2 a+2 b-c} \tan ^{-1}\left (\frac {\sqrt {-2 a+2 b-c} x}{\sqrt {a}+2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{2 a-2 b+c} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((Sqrt[-2*a - 2*b - c]*ArcTan[(Sqrt[-2*a - 2*b - c]*x)/(Sqrt[a] - 2*Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x +

c*x^2 + b*x^3 + a*x^4])])/(2*a + 2*b + c)) - (Sqrt[-2*a + 2*b - c]*ArcTan[(Sqrt[-2*a + 2*b - c]*x)/(Sqrt[a] +

2*Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(2*a - 2*b + c)

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fricas [B] time = 1.01, size = 1659, normalized size = 8.78

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*((2*a + 2*b + c)*sqrt(2*a - 2*b + c)*log(((24*a^2 - 16*a*b + b^2 + 4*a*c)*x^4 + 4*(8*a^2 + 4*a*b - 3*b^2

- 2*(2*a - b)*c)*x^3 + 2*(24*a^2 + 3*b^2 - 4*(a + 2*b)*c + 4*c^2)*x^2 + 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a

)*((4*a - b)*x^2 + 2*(2*a + b - c)*x + 4*a - b)*sqrt(2*a - 2*b + c) + 24*a^2 - 16*a*b + b^2 + 4*a*c + 4*(8*a^2

+ 4*a*b - 3*b^2 - 2*(2*a - b)*c)*x)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1)) + sqrt(2*a + 2*b + c)*(2*a - 2*b + c)*lo

g(((24*a^2 + 16*a*b + b^2 + 4*a*c)*x^4 - 4*(8*a^2 - 4*a*b - 3*b^2 - 2*(2*a + b)*c)*x^3 + 2*(24*a^2 + 3*b^2 - 4

*(a - 2*b)*c + 4*c^2)*x^2 - 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4*a + b)*x^2 - 2*(2*a - b - c)*x + 4*a +

b)*sqrt(2*a + 2*b + c) + 24*a^2 + 16*a*b + b^2 + 4*a*c - 4*(8*a^2 - 4*a*b - 3*b^2 - 2*(2*a + b)*c)*x)/(x^4 -

4*x^3 + 6*x^2 - 4*x + 1)))/(4*a^2 - 4*b^2 + 4*a*c + c^2), 1/4*(2*(2*a - 2*b + c)*sqrt(-2*a - 2*b - c)*arctan(1

/2*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4*a + b)*x^2 - 2*(2*a - b - c)*x + 4*a + b)*sqrt(-2*a - 2*b - c)/((

2*a^2 + 2*a*b + a*c)*x^4 + (2*a*b + 2*b^2 + b*c)*x^3 + (2*(a + b)*c + c^2)*x^2 + 2*a^2 + 2*a*b + a*c + (2*a*b

+ 2*b^2 + b*c)*x)) + (2*a + 2*b + c)*sqrt(2*a - 2*b + c)*log(((24*a^2 - 16*a*b + b^2 + 4*a*c)*x^4 + 4*(8*a^2 +

4*a*b - 3*b^2 - 2*(2*a - b)*c)*x^3 + 2*(24*a^2 + 3*b^2 - 4*(a + 2*b)*c + 4*c^2)*x^2 + 4*sqrt(a*x^4 + b*x^3 +

c*x^2 + b*x + a)*((4*a - b)*x^2 + 2*(2*a + b - c)*x + 4*a - b)*sqrt(2*a - 2*b + c) + 24*a^2 - 16*a*b + b^2 + 4

*a*c + 4*(8*a^2 + 4*a*b - 3*b^2 - 2*(2*a - b)*c)*x)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1)))/(4*a^2 - 4*b^2 + 4*a*c +

c^2), 1/4*(2*(2*a + 2*b + c)*sqrt(-2*a + 2*b - c)*arctan(-1/2*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4*a - b

)*x^2 + 2*(2*a + b - c)*x + 4*a - b)*sqrt(-2*a + 2*b - c)/((2*a^2 - 2*a*b + a*c)*x^4 + (2*a*b - 2*b^2 + b*c)*x

^3 + (2*(a - b)*c + c^2)*x^2 + 2*a^2 - 2*a*b + a*c + (2*a*b - 2*b^2 + b*c)*x)) + sqrt(2*a + 2*b + c)*(2*a - 2*

b + c)*log(((24*a^2 + 16*a*b + b^2 + 4*a*c)*x^4 - 4*(8*a^2 - 4*a*b - 3*b^2 - 2*(2*a + b)*c)*x^3 + 2*(24*a^2 +

3*b^2 - 4*(a - 2*b)*c + 4*c^2)*x^2 - 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4*a + b)*x^2 - 2*(2*a - b - c)*

x + 4*a + b)*sqrt(2*a + 2*b + c) + 24*a^2 + 16*a*b + b^2 + 4*a*c - 4*(8*a^2 - 4*a*b - 3*b^2 - 2*(2*a + b)*c)*x

)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)))/(4*a^2 - 4*b^2 + 4*a*c + c^2), 1/2*((2*a + 2*b + c)*sqrt(-2*a + 2*b - c)*a

rctan(-1/2*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4*a - b)*x^2 + 2*(2*a + b - c)*x + 4*a - b)*sqrt(-2*a + 2*b

- c)/((2*a^2 - 2*a*b + a*c)*x^4 + (2*a*b - 2*b^2 + b*c)*x^3 + (2*(a - b)*c + c^2)*x^2 + 2*a^2 - 2*a*b + a*c +

(2*a*b - 2*b^2 + b*c)*x)) + (2*a - 2*b + c)*sqrt(-2*a - 2*b - c)*arctan(1/2*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x

+ a)*((4*a + b)*x^2 - 2*(2*a - b - c)*x + 4*a + b)*sqrt(-2*a - 2*b - c)/((2*a^2 + 2*a*b + a*c)*x^4 + (2*a*b +

2*b^2 + b*c)*x^3 + (2*(a + b)*c + c^2)*x^2 + 2*a^2 + 2*a*b + a*c + (2*a*b + 2*b^2 + b*c)*x)))/(4*a^2 - 4*b^2 +

4*a*c + c^2)]

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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 1}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\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)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

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

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maple [C] time = 0.23, size = 4541, normalized size = 24.03


method	result	size

default	\(\text {Expression too large to display}\)	\(4541\)
elliptic	\(\text {Expression too large to display}\)	\(90178\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+

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

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

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

Z*b+a,index=3))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=3)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-R

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

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

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

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

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

ipticF(((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2))*(x-RootOf(_Z

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

Z*b+a,index=1))/(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2),((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,in

dex=2)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=3))*(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+RootOf(_Z^4*

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

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

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

a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z

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

otOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2)*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2))^2*((RootOf(_

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

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

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

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

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

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

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

(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)+1)*(EllipticF(((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootO

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

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

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

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

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

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

4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1)+1)*EllipticPi(((RootOf(_Z^4*a+

_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b

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

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

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

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

Of(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=3))*(-RootOf(_Z^4*a+_Z^3*b+_Z

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

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

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

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

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

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

^2*c+_Z*b+a,index=2))^2*((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index

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

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

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

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

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

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

)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)-1)*(EllipticF(((RootOf(_Z^4*a+_

Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+

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

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

3*b+_Z^2*c+_Z*b+a,index=3))*(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,i

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

Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2))+(RootOf(_Z^4*a+_Z^3*b

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

=1)-1)*EllipticPi(((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2))*(

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

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

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

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

3*b+_Z^2*c+_Z*b+a,index=2)),((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,i

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

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

+_Z*b+a,index=4)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2)))

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 1}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\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)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate((x^2 + 1)/(sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*(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+1}{\left (x^2-1\right )\,\sqrt {a\,x^4+b\,x^3+c\,x^2+b\,x+a}} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x^2 + 1)/((x^2 - 1)*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(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 {a x^{4} + a + b x^{3} + b x + c x^{2}}}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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