3.4.27 \(\int \frac {1-x^2}{(1+2 a x+x^2) \sqrt {1+2 a x+2 b x^2+2 a x^3+x^4}} \, dx\) [327]

Optimal. Leaf size=74 \[ \frac {\tan ^{-1}\left (\frac {a+2 \left (1+a^2-b\right ) x+a x^2}{\sqrt {2} \sqrt {1-b} \sqrt {1+2 a x+2 b x^2+2 a x^3+x^4}}\right )}{\sqrt {2} \sqrt {1-b}} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2109} \begin {gather*} \frac {\text {ArcTan}\left (\frac {2 x \left (a^2-b+1\right )+a x^2+a}{\sqrt {2} \sqrt {1-b} \sqrt {2 a x^3+2 a x+2 b x^2+x^4+1}}\right )}{\sqrt {2} \sqrt {1-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2109

Int[((f_) + (g_.)*(x_)^2)/(((d_) + (e_.)*(x_) + (d_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (b_.)*(x
_)^3 + (a_.)*(x_)^4]), x_Symbol] :> Simp[a*(f/(d*Rt[a^2*(2*a - c), 2]))*ArcTan[(a*b + (4*a^2 + b^2 - 2*a*c)*x
+ a*b*x^2)/(2*Rt[a^2*(2*a - c), 2]*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])], x] /; FreeQ[{a, b, c, d, e, f, g},
x] && EqQ[b*d - a*e, 0] && EqQ[f + g, 0] && PosQ[a^2*(2*a - c)]

Rubi steps

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

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Mathematica [A]
time = 0.63, size = 65, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {-1+b} x}{1+2 a x+x^2-\sqrt {1+2 b x^2+x^4+2 a \left (x+x^3\right )}}\right )}{\sqrt {-1+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.17, size = 247419, normalized size = 3343.50

method	result	size
default	\(\text {Expression too large to display}\)	\(247419\)
elliptic	\(\text {Expression too large to display}\)	\(258804\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2+1)/(2*a*x+x^2+1)/(2*a*x^3+x^4+2*b*x^2+2*a*x+1)^(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*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.78, size = 252, normalized size = 3.41 \begin {gather*} \left [\frac {\sqrt {2} \log \left (\frac {4 \, a^{3} x^{3} + {\left (a^{2} + 2 \, b - 2\right )} x^{4} + 4 \, a^{3} x + 2 \, {\left (2 \, a^{4} + 5 \, a^{2} - 2 \, {\left (2 \, a^{2} + 3\right )} b + 4 \, b^{2} + 2\right )} x^{2} + a^{2} - \frac {2 \, \sqrt {2} \sqrt {2 \, a x^{3} + x^{4} + 2 \, b x^{2} + 2 \, a x + 1} {\left ({\left (a b - a\right )} x^{2} + a b - 2 \, {\left (a^{2} - {\left (a^{2} + 2\right )} b + b^{2} + 1\right )} x - a\right )}}{\sqrt {b - 1}} + 2 \, b - 2}{4 \, a x^{3} + x^{4} + 2 \, {\left (2 \, a^{2} + 1\right )} x^{2} + 4 \, a x + 1}\right )}{4 \, \sqrt {b - 1}}, \frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{b - 1}} \arctan \left (\frac {\sqrt {2} \sqrt {2 \, a x^{3} + x^{4} + 2 \, b x^{2} + 2 \, a x + 1} {\left (b - 1\right )} \sqrt {-\frac {1}{b - 1}}}{a x^{2} + 2 \, {\left (a^{2} - b + 1\right )} x + a}\right )\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*sqrt(2)*log((4*a^3*x^3 + (a^2 + 2*b - 2)*x^4 + 4*a^3*x + 2*(2*a^4 + 5*a^2 - 2*(2*a^2 + 3)*b + 4*b^2 + 2)*
x^2 + a^2 - 2*sqrt(2)*sqrt(2*a*x^3 + x^4 + 2*b*x^2 + 2*a*x + 1)*((a*b - a)*x^2 + a*b - 2*(a^2 - (a^2 + 2)*b +
b^2 + 1)*x - a)/sqrt(b - 1) + 2*b - 2)/(4*a*x^3 + x^4 + 2*(2*a^2 + 1)*x^2 + 4*a*x + 1))/sqrt(b - 1), 1/2*sqrt(
2)*sqrt(-1/(b - 1))*arctan(sqrt(2)*sqrt(2*a*x^3 + x^4 + 2*b*x^2 + 2*a*x + 1)*(b - 1)*sqrt(-1/(b - 1))/(a*x^2 +
2*(a^2 - b + 1)*x + a))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x^{2}}{2 a x \sqrt {2 a x^{3} + 2 a x + 2 b x^{2} + x^{4} + 1} + x^{2} \sqrt {2 a x^{3} + 2 a x + 2 b x^{2} + x^{4} + 1} + \sqrt {2 a x^{3} + 2 a x + 2 b x^{2} + x^{4} + 1}}\, dx - \int \left (- \frac {1}{2 a x \sqrt {2 a x^{3} + 2 a x + 2 b x^{2} + x^{4} + 1} + x^{2} \sqrt {2 a x^{3} + 2 a x + 2 b x^{2} + x^{4} + 1} + \sqrt {2 a x^{3} + 2 a x + 2 b x^{2} + x^{4} + 1}}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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