3.65 \(\int \frac{\sqrt{1+p x^2+x^4}}{1-x^4} \, dx\)

Optimal. Leaf size=75 \[ \frac{1}{4} \sqrt{2-p} \tan ^{-1}\left (\frac{\sqrt{2-p} x}{\sqrt{p x^2+x^4+1}}\right )+\frac{1}{4} \sqrt{p+2} \tanh ^{-1}\left (\frac{\sqrt{p+2} x}{\sqrt{p x^2+x^4+1}}\right ) \]

[Out]

(Sqrt[2 - p]*ArcTan[(Sqrt[2 - p]*x)/Sqrt[1 + p*x^2 + x^4]])/4 + (Sqrt[2 + p]*Arc
Tanh[(Sqrt[2 + p]*x)/Sqrt[1 + p*x^2 + x^4]])/4

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Rubi [A]  time = 0.161053, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{1}{4} \sqrt{2-p} \tan ^{-1}\left (\frac{\sqrt{2-p} x}{\sqrt{p x^2+x^4+1}}\right )+\frac{1}{4} \sqrt{p+2} \tanh ^{-1}\left (\frac{\sqrt{p+2} x}{\sqrt{p x^2+x^4+1}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + p*x^2 + x^4]/(1 - x^4),x]

[Out]

(Sqrt[2 - p]*ArcTan[(Sqrt[2 - p]*x)/Sqrt[1 + p*x^2 + x^4]])/4 + (Sqrt[2 + p]*Arc
Tanh[(Sqrt[2 + p]*x)/Sqrt[1 + p*x^2 + x^4]])/4

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Rubi in Sympy [A]  time = 9.10875, size = 63, normalized size = 0.84 \[ \frac{\sqrt{- p + 2} \operatorname{atan}{\left (\frac{x \sqrt{- p + 2}}{\sqrt{p x^{2} + x^{4} + 1}} \right )}}{4} + \frac{\sqrt{p + 2} \operatorname{atanh}{\left (\frac{x \sqrt{p + 2}}{\sqrt{p x^{2} + x^{4} + 1}} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**4+p*x**2+1)**(1/2)/(-x**4+1),x)

[Out]

sqrt(-p + 2)*atan(x*sqrt(-p + 2)/sqrt(p*x**2 + x**4 + 1))/4 + sqrt(p + 2)*atanh(
x*sqrt(p + 2)/sqrt(p*x**2 + x**4 + 1))/4

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Mathematica [A]  time = 0.329071, size = 0, normalized size = 0. \[ \int \frac{\sqrt{1+p x^2+x^4}}{1-x^4} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[1 + p*x^2 + x^4]/(1 - x^4),x]

[Out]

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

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Maple [C]  time = 0.098, size = 1421, normalized size = 19. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4+p*x^2+1)^(1/2)/(-x^4+1),x)

[Out]

-1/2*(1+p)/(-2*p+2*(p^2-4)^(1/2))^(1/2)*(1-(-1/2*p+1/2*(p^2-4)^(1/2))*x^2)^(1/2)
*(1-(-1/2*p-1/2*(p^2-4)^(1/2))*x^2)^(1/2)/(x^4+p*x^2+1)^(1/2)*EllipticF(1/2*x*(-
2*p+2*(p^2-4)^(1/2))^(1/2),(-1-p*(-1/2*p-1/2*(p^2-4)^(1/2)))^(1/2))+2/(-2*p+2*(p
^2-4)^(1/2))^(1/2)*(1-(-1/2*p+1/2*(p^2-4)^(1/2))*x^2)^(1/2)*(1-(-1/2*p-1/2*(p^2-
4)^(1/2))*x^2)^(1/2)/(x^4+p*x^2+1)^(1/2)/(p+(p^2-4)^(1/2))*(EllipticF(1/2*x*(-2*
p+2*(p^2-4)^(1/2))^(1/2),(-1-p*(-1/2*p-1/2*(p^2-4)^(1/2)))^(1/2))-EllipticE(1/2*
x*(-2*p+2*(p^2-4)^(1/2))^(1/2),(-1-p*(-1/2*p-1/2*(p^2-4)^(1/2)))^(1/2)))+1/(-1/2
*p+1/2*(p^2-4)^(1/2))^(1/2)*(1-(-1/2*p+1/2*(p^2-4)^(1/2))*x^2)^(1/2)*(1-(-1/2*p-
1/2*(p^2-4)^(1/2))*x^2)^(1/2)/(x^4+p*x^2+1)^(1/2)*EllipticPi((-1/2*p+1/2*(p^2-4)
^(1/2))^(1/2)*x,1/(-1/2*p+1/2*(p^2-4)^(1/2)),(-1/2*p-1/2*(p^2-4)^(1/2))^(1/2)/(-
1/2*p+1/2*(p^2-4)^(1/2))^(1/2))+1/2*p/(-1/2*p+1/2*(p^2-4)^(1/2))^(1/2)*(1-(-1/2*
p+1/2*(p^2-4)^(1/2))*x^2)^(1/2)*(1-(-1/2*p-1/2*(p^2-4)^(1/2))*x^2)^(1/2)/(x^4+p*
x^2+1)^(1/2)*EllipticPi((-1/2*p+1/2*(p^2-4)^(1/2))^(1/2)*x,1/(-1/2*p+1/2*(p^2-4)
^(1/2)),(-1/2*p-1/2*(p^2-4)^(1/2))^(1/2)/(-1/2*p+1/2*(p^2-4)^(1/2))^(1/2))+1/2*(
-1-p)/(-2*p+2*(p^2-4)^(1/2))^(1/2)*(1-(-1/2*p+1/2*(p^2-4)^(1/2))*x^2)^(1/2)*(1-(
-1/2*p-1/2*(p^2-4)^(1/2))*x^2)^(1/2)/(x^4+p*x^2+1)^(1/2)*EllipticF(1/2*x*(-2*p+2
*(p^2-4)^(1/2))^(1/2),(-1-p*(-1/2*p-1/2*(p^2-4)^(1/2)))^(1/2))+1/(-2*p+2*(p^2-4)
^(1/2))^(1/2)*(1+1/2*p*x^2-1/2*x^2*(p^2-4)^(1/2))^(1/2)*(1+1/2*p*x^2+1/2*x^2*(p^
2-4)^(1/2))^(1/2)/(x^4+p*x^2+1)^(1/2)*EllipticF(1/2*x*(-2*p+2*(p^2-4)^(1/2))^(1/
2),(-1-p*(-1/2*p-1/2*(p^2-4)^(1/2)))^(1/2))*p-1/(-2*p+2*(p^2-4)^(1/2))^(1/2)*(1+
1/2*p*x^2-1/2*x^2*(p^2-4)^(1/2))^(1/2)*(1+1/2*p*x^2+1/2*x^2*(p^2-4)^(1/2))^(1/2)
/(x^4+p*x^2+1)^(1/2)*EllipticF(1/2*x*(-2*p+2*(p^2-4)^(1/2))^(1/2),(-1-p*(-1/2*p-
1/2*(p^2-4)^(1/2)))^(1/2))-2/(-2*p+2*(p^2-4)^(1/2))^(1/2)*(1+1/2*p*x^2-1/2*x^2*(
p^2-4)^(1/2))^(1/2)*(1+1/2*p*x^2+1/2*x^2*(p^2-4)^(1/2))^(1/2)/(x^4+p*x^2+1)^(1/2
)/(p+(p^2-4)^(1/2))*EllipticF(1/2*x*(-2*p+2*(p^2-4)^(1/2))^(1/2),(-1-p*(-1/2*p-1
/2*(p^2-4)^(1/2)))^(1/2))+2/(-2*p+2*(p^2-4)^(1/2))^(1/2)*(1+1/2*p*x^2-1/2*x^2*(p
^2-4)^(1/2))^(1/2)*(1+1/2*p*x^2+1/2*x^2*(p^2-4)^(1/2))^(1/2)/(x^4+p*x^2+1)^(1/2)
/(p+(p^2-4)^(1/2))*EllipticE(1/2*x*(-2*p+2*(p^2-4)^(1/2))^(1/2),(-1-p*(-1/2*p-1/
2*(p^2-4)^(1/2)))^(1/2))+1/(-1/2*p+1/2*(p^2-4)^(1/2))^(1/2)*(1+1/2*p*x^2-1/2*x^2
*(p^2-4)^(1/2))^(1/2)*(1+1/2*p*x^2+1/2*x^2*(p^2-4)^(1/2))^(1/2)/(x^4+p*x^2+1)^(1
/2)*EllipticPi((-1/2*p+1/2*(p^2-4)^(1/2))^(1/2)*x,-1/(-1/2*p+1/2*(p^2-4)^(1/2)),
(-1/2*p-1/2*(p^2-4)^(1/2))^(1/2)/(-1/2*p+1/2*(p^2-4)^(1/2))^(1/2))-1/2*p/(-1/2*p
+1/2*(p^2-4)^(1/2))^(1/2)*(1+1/2*p*x^2-1/2*x^2*(p^2-4)^(1/2))^(1/2)*(1+1/2*p*x^2
+1/2*x^2*(p^2-4)^(1/2))^(1/2)/(x^4+p*x^2+1)^(1/2)*EllipticPi((-1/2*p+1/2*(p^2-4)
^(1/2))^(1/2)*x,-1/(-1/2*p+1/2*(p^2-4)^(1/2)),(-1/2*p-1/2*(p^2-4)^(1/2))^(1/2)/(
-1/2*p+1/2*(p^2-4)^(1/2))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{\sqrt{x^{4} + p x^{2} + 1}}{x^{4} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(x^4 + p*x^2 + 1)/(x^4 - 1),x, algorithm="maxima")

[Out]

-integrate(sqrt(x^4 + p*x^2 + 1)/(x^4 - 1), x)

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Fricas [A]  time = 0.27045, size = 1, normalized size = 0.01 \[ \left [\frac{1}{8} \, \sqrt{p - 2} \log \left (\frac{x^{4} + 2 \,{\left (p - 1\right )} x^{2} - 2 \, \sqrt{x^{4} + p x^{2} + 1} \sqrt{p - 2} x + 1}{x^{4} + 2 \, x^{2} + 1}\right ) + \frac{1}{8} \, \sqrt{p + 2} \log \left (\frac{x^{4} + 2 \,{\left (p + 1\right )} x^{2} + 2 \, \sqrt{x^{4} + p x^{2} + 1} \sqrt{p + 2} x + 1}{x^{4} - 2 \, x^{2} + 1}\right ), \frac{1}{4} \, \sqrt{-p + 2} \arctan \left (\frac{\sqrt{-p + 2} x}{\sqrt{x^{4} + p x^{2} + 1}}\right ) + \frac{1}{8} \, \sqrt{p + 2} \log \left (\frac{x^{4} + 2 \,{\left (p + 1\right )} x^{2} + 2 \, \sqrt{x^{4} + p x^{2} + 1} \sqrt{p + 2} x + 1}{x^{4} - 2 \, x^{2} + 1}\right ), \frac{1}{4} \, \sqrt{-p - 2} \arctan \left (\frac{\sqrt{x^{4} + p x^{2} + 1}}{\sqrt{-p - 2} x}\right ) + \frac{1}{8} \, \sqrt{p - 2} \log \left (\frac{x^{4} + 2 \,{\left (p - 1\right )} x^{2} - 2 \, \sqrt{x^{4} + p x^{2} + 1} \sqrt{p - 2} x + 1}{x^{4} + 2 \, x^{2} + 1}\right ), \frac{1}{4} \, \sqrt{-p + 2} \arctan \left (\frac{\sqrt{-p + 2} x}{\sqrt{x^{4} + p x^{2} + 1}}\right ) + \frac{1}{4} \, \sqrt{-p - 2} \arctan \left (\frac{\sqrt{x^{4} + p x^{2} + 1}}{\sqrt{-p - 2} x}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(x^4 + p*x^2 + 1)/(x^4 - 1),x, algorithm="fricas")

[Out]

[1/8*sqrt(p - 2)*log((x^4 + 2*(p - 1)*x^2 - 2*sqrt(x^4 + p*x^2 + 1)*sqrt(p - 2)*
x + 1)/(x^4 + 2*x^2 + 1)) + 1/8*sqrt(p + 2)*log((x^4 + 2*(p + 1)*x^2 + 2*sqrt(x^
4 + p*x^2 + 1)*sqrt(p + 2)*x + 1)/(x^4 - 2*x^2 + 1)), 1/4*sqrt(-p + 2)*arctan(sq
rt(-p + 2)*x/sqrt(x^4 + p*x^2 + 1)) + 1/8*sqrt(p + 2)*log((x^4 + 2*(p + 1)*x^2 +
2*sqrt(x^4 + p*x^2 + 1)*sqrt(p + 2)*x + 1)/(x^4 - 2*x^2 + 1)), 1/4*sqrt(-p - 2)
*arctan(sqrt(x^4 + p*x^2 + 1)/(sqrt(-p - 2)*x)) + 1/8*sqrt(p - 2)*log((x^4 + 2*(
p - 1)*x^2 - 2*sqrt(x^4 + p*x^2 + 1)*sqrt(p - 2)*x + 1)/(x^4 + 2*x^2 + 1)), 1/4*
sqrt(-p + 2)*arctan(sqrt(-p + 2)*x/sqrt(x^4 + p*x^2 + 1)) + 1/4*sqrt(-p - 2)*arc
tan(sqrt(x^4 + p*x^2 + 1)/(sqrt(-p - 2)*x))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{\sqrt{p x^{2} + x^{4} + 1}}{x^{4} - 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**4+p*x**2+1)**(1/2)/(-x**4+1),x)

[Out]

-Integral(sqrt(p*x**2 + x**4 + 1)/(x**4 - 1), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{\sqrt{x^{4} + p x^{2} + 1}}{x^{4} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(x^4 + p*x^2 + 1)/(x^4 - 1),x, algorithm="giac")

[Out]

integrate(-sqrt(x^4 + p*x^2 + 1)/(x^4 - 1), x)