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

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

Optimal. Leaf size=70 \[ \frac {2 \tan ^{-1}\left (\frac {x \sqrt {2 a-c}}{-\sqrt {a x^4+a+b x^3+b x+c x^2}+\sqrt {a} x^2+\sqrt {a}}\right )}{\sqrt {2 a-c}} \]

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Rubi [F] time = 1.19, 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] - I*Defer[Int][1/((I - x)*Sqrt[a + b*x + c*x^2 + b*x^3

+ a*x^4]), x] - I*Defer[Int][1/((I + 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\\ &=-\left (2 \int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\right )+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\\ &=-\left (2 \int \left (\frac {i}{2 (i-x) \sqrt {a+b x+c x^2+b x^3+a x^4}}+\frac {i}{2 (i+x) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx\right )+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\\ &=-\left (i \int \frac {1}{(i-x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\right )-i \int \frac {1}{(i+x) \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\\ \end {align*}

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Mathematica [C] time = 2.17, size = 3600, normalized size = 51.43 \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)]*(((-I)*(EllipticF[ArcSin[Sqrt[((x - R

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

) + EllipticPi[((-I + 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]))/((-I + 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])))/((-I + Root[a + b*#1 + c*#1^2 + b*#1^3 +

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

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

+ 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] - Ro

ot[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 = 0.59, size = 70, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {2 a-c} x}{\sqrt {a}+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {2 a-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]

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

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fricas [B] time = 0.76, size = 291, normalized size = 4.16 \begin {gather*} \left [-\frac {\sqrt {-2 \, a + c} \log \left (-\frac {{\left (8 \, a^{2} - b^{2} - 4 \, a c\right )} x^{4} + 8 \, {\left (2 \, a b - b c\right )} x^{3} - 2 \, {\left (8 \, a^{2} + b^{2} - 12 \, a c + 4 \, c^{2}\right )} x^{2} + 8 \, a^{2} + 4 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (b x^{2} - 2 \, {\left (2 \, a - c\right )} x + b\right )} \sqrt {-2 \, a + c} - b^{2} - 4 \, a c + 8 \, {\left (2 \, a b - b c\right )} x}{x^{4} + 2 \, x^{2} + 1}\right )}{2 \, {\left (2 \, a - c\right )}}, -\frac {\arctan \left (-\frac {\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (b x^{2} - 2 \, {\left (2 \, a - c\right )} x + b\right )} \sqrt {2 \, a - c}}{2 \, {\left ({\left (2 \, a^{2} - a c\right )} x^{4} + {\left (2 \, a b - b c\right )} x^{3} + {\left (2 \, a c - c^{2}\right )} x^{2} + 2 \, a^{2} - a c + {\left (2 \, a b - b c\right )} x\right )}}\right )}{\sqrt {2 \, a - c}}\right ] \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="fricas")

[Out]

[-1/2*sqrt(-2*a + c)*log(-((8*a^2 - b^2 - 4*a*c)*x^4 + 8*(2*a*b - b*c)*x^3 - 2*(8*a^2 + b^2 - 12*a*c + 4*c^2)*

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

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

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

(2*a*b - b*c)*x))/sqrt(2*a - c)]

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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 = 0.17, size = 79345, normalized size = 1133.50


method	result	size

default	\(\text {Expression too large to display}\)	\(79345\)
elliptic	\(\text {Expression too large to display}\)	\(90322\)

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]

result too large to display

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

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