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

Optimal. Leaf size=171 \[ \frac{\sqrt{\sqrt{p^2+4}-p} \tanh ^{-1}\left (\frac{\sqrt{\sqrt{p^2+4}-p} x \left (\sqrt{p^2+4}+p-2 x^2\right )}{2 \sqrt{2} \sqrt{p x^2-x^4+1}}\right )}{2 \sqrt{2}}-\frac{\sqrt{\sqrt{p^2+4}+p} \tan ^{-1}\left (\frac{\sqrt{\sqrt{p^2+4}+p} x \left (-\sqrt{p^2+4}+p-2 x^2\right )}{2 \sqrt{2} \sqrt{p x^2-x^4+1}}\right )}{2 \sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.141518, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{\sqrt{\sqrt{p^2+4}-p} \tanh ^{-1}\left (\frac{\sqrt{\sqrt{p^2+4}-p} x \left (\sqrt{p^2+4}+p-2 x^2\right )}{2 \sqrt{2} \sqrt{p x^2-x^4+1}}\right )}{2 \sqrt{2}}-\frac{\sqrt{\sqrt{p^2+4}+p} \tan ^{-1}\left (\frac{\sqrt{\sqrt{p^2+4}+p} x \left (-\sqrt{p^2+4}+p-2 x^2\right )}{2 \sqrt{2} \sqrt{p x^2-x^4+1}}\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.9404, size = 144, normalized size = 0.84 \[ \frac{\sqrt{2} \sqrt{- p + \sqrt{p^{2} + 4}} \operatorname{atanh}{\left (\frac{\sqrt{2} x \sqrt{- p + \sqrt{p^{2} + 4}} \left (p - 2 x^{2} + \sqrt{p^{2} + 4}\right )}{4 \sqrt{p x^{2} - x^{4} + 1}} \right )}}{4} - \frac{\sqrt{2} \sqrt{p + \sqrt{p^{2} + 4}} \operatorname{atan}{\left (\frac{\sqrt{2} x \sqrt{p + \sqrt{p^{2} + 4}} \left (p - 2 x^{2} - \sqrt{p^{2} + 4}\right )}{4 \sqrt{p x^{2} - x^{4} + 1}} \right )}}{4} \]

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

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Mathematica [C]  time = 0.426933, size = 322, normalized size = 1.88 \[ \frac{\sqrt{\frac{4 x^2}{\sqrt{p^2+4}-p}+2} \sqrt{1-\frac{2 x^2}{\sqrt{p^2+4}+p}} \left (2 i F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{1}{\sqrt{p^2+4}-p}} x\right )|\frac{p-\sqrt{p^2+4}}{p+\sqrt{p^2+4}}\right )-(p+2 i) \Pi \left (\frac{1}{2} i \left (p-\sqrt{p^2+4}\right );i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{1}{\sqrt{p^2+4}-p}} x\right )|\frac{p-\sqrt{p^2+4}}{p+\sqrt{p^2+4}}\right )+(p-2 i) \Pi \left (\frac{1}{2} i \left (\sqrt{p^2+4}-p\right );i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{1}{\sqrt{p^2+4}-p}} x\right )|\frac{p-\sqrt{p^2+4}}{p+\sqrt{p^2+4}}\right )\right )}{4 \sqrt{\frac{1}{\sqrt{p^2+4}-p}} \sqrt{p x^2-x^4+1}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2 + (4*x^2)/(-p + Sqrt[4 + p^2])]*Sqrt[1 - (2*x^2)/(p + Sqrt[4 + p^2])]*((
2*I)*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[(-p + Sqrt[4 + p^2])^(-1)]*x], (p - Sqrt[4
+ p^2])/(p + Sqrt[4 + p^2])] - (2*I + p)*EllipticPi[(I/2)*(p - Sqrt[4 + p^2]),
I*ArcSinh[Sqrt[2]*Sqrt[(-p + Sqrt[4 + p^2])^(-1)]*x], (p - Sqrt[4 + p^2])/(p + S
qrt[4 + p^2])] + (-2*I + p)*EllipticPi[(I/2)*(-p + Sqrt[4 + p^2]), I*ArcSinh[Sqr
t[2]*Sqrt[(-p + Sqrt[4 + p^2])^(-1)]*x], (p - Sqrt[4 + p^2])/(p + Sqrt[4 + p^2])
]))/(4*Sqrt[(-p + Sqrt[4 + p^2])^(-1)]*Sqrt[1 + p*x^2 - x^4])

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Maple [B]  time = 0.101, size = 456, normalized size = 2.7 \[ -{\frac{\sqrt{2}}{32}\sqrt{p+\sqrt{{p}^{2}+4}}\sqrt{{p}^{2}+4}\ln \left ({\frac{\sqrt{2}}{x}\sqrt{p+\sqrt{{p}^{2}+4}}\sqrt{-{x}^{4}+p{x}^{2}+1}}-{\frac{-{x}^{4}+p{x}^{2}+1}{{x}^{2}}}-\sqrt{{p}^{2}+4} \right ) }+{\frac{\sqrt{2}}{4}\arctan \left ({\frac{1}{2} \left ( 2\,\sqrt{p+\sqrt{{p}^{2}+4}}-2\,{\frac{\sqrt{-{x}^{4}+p{x}^{2}+1}\sqrt{2}}{x}} \right ){\frac{1}{\sqrt{-p+\sqrt{{p}^{2}+4}}}}} \right ){\frac{1}{\sqrt{-p+\sqrt{{p}^{2}+4}}}}}+{\frac{\sqrt{2}p}{32}\sqrt{p+\sqrt{{p}^{2}+4}}\ln \left ({\frac{\sqrt{2}}{x}\sqrt{p+\sqrt{{p}^{2}+4}}\sqrt{-{x}^{4}+p{x}^{2}+1}}-{\frac{-{x}^{4}+p{x}^{2}+1}{{x}^{2}}}-\sqrt{{p}^{2}+4} \right ) }+{\frac{\sqrt{2}}{32}\sqrt{p+\sqrt{{p}^{2}+4}}\sqrt{{p}^{2}+4}\ln \left ({\frac{-{x}^{4}+p{x}^{2}+1}{{x}^{2}}}+{\frac{\sqrt{2}}{x}\sqrt{p+\sqrt{{p}^{2}+4}}\sqrt{-{x}^{4}+p{x}^{2}+1}}+\sqrt{{p}^{2}+4} \right ) }-{\frac{\sqrt{2}}{4}\arctan \left ({\frac{1}{2} \left ( 2\,{\frac{\sqrt{-{x}^{4}+p{x}^{2}+1}\sqrt{2}}{x}}+2\,\sqrt{p+\sqrt{{p}^{2}+4}} \right ){\frac{1}{\sqrt{-p+\sqrt{{p}^{2}+4}}}}} \right ){\frac{1}{\sqrt{-p+\sqrt{{p}^{2}+4}}}}}-{\frac{\sqrt{2}p}{32}\sqrt{p+\sqrt{{p}^{2}+4}}\ln \left ({\frac{-{x}^{4}+p{x}^{2}+1}{{x}^{2}}}+{\frac{\sqrt{2}}{x}\sqrt{p+\sqrt{{p}^{2}+4}}\sqrt{-{x}^{4}+p{x}^{2}+1}}+\sqrt{{p}^{2}+4} \right ) } \]

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]

-1/32*2^(1/2)*(p+(p^2+4)^(1/2))^(1/2)*(p^2+4)^(1/2)*ln((p+(p^2+4)^(1/2))^(1/2)*(
-x^4+p*x^2+1)^(1/2)*2^(1/2)/x-(-x^4+p*x^2+1)/x^2-(p^2+4)^(1/2))+1/4*2^(1/2)/(-p+
(p^2+4)^(1/2))^(1/2)*arctan(1/2*(2*(p+(p^2+4)^(1/2))^(1/2)-2*(-x^4+p*x^2+1)^(1/2
)*2^(1/2)/x)/(-p+(p^2+4)^(1/2))^(1/2))+1/32*2^(1/2)*(p+(p^2+4)^(1/2))^(1/2)*p*ln
((p+(p^2+4)^(1/2))^(1/2)*(-x^4+p*x^2+1)^(1/2)*2^(1/2)/x-(-x^4+p*x^2+1)/x^2-(p^2+
4)^(1/2))+1/32*2^(1/2)*(p+(p^2+4)^(1/2))^(1/2)*(p^2+4)^(1/2)*ln((-x^4+p*x^2+1)/x
^2+(p+(p^2+4)^(1/2))^(1/2)*(-x^4+p*x^2+1)^(1/2)*2^(1/2)/x+(p^2+4)^(1/2))-1/4*2^(
1/2)/(-p+(p^2+4)^(1/2))^(1/2)*arctan(1/2*(2*(-x^4+p*x^2+1)^(1/2)*2^(1/2)/x+2*(p+
(p^2+4)^(1/2))^(1/2))/(-p+(p^2+4)^(1/2))^(1/2))-1/32*2^(1/2)*(p+(p^2+4)^(1/2))^(
1/2)*p*ln((-x^4+p*x^2+1)/x^2+(p+(p^2+4)^(1/2))^(1/2)*(-x^4+p*x^2+1)^(1/2)*2^(1/2
)/x+(p^2+4)^(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 = 0.339132, size = 2419, normalized size = 14.15 \[ \text{result too large to display} \]

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*((p^2 + 4)^(1/4)*(p - sqrt(p^2 + 4))*log(-((p^4 + 5*p^2 + 4)*x^4 - p^4 - (p
^5 + 5*p^3 + 4*p)*x^2 - 5*p^2 + 2*(sqrt(-x^4 + p*x^2 + 1)*sqrt(p^2 + 4)*(p^2 + 1
)*x - sqrt(-x^4 + p*x^2 + 1)*(p^3 + 3*p)*x)*(p^2 + 4)^(1/4)*sqrt((p^2 - sqrt(p^2
+ 4)*p + 4)/(p^2 - sqrt(p^2 + 4)*p + 2)) - ((p^3 + 3*p)*x^4 - p^3 - (p^4 + 3*p^
2)*x^2 - 3*p)*sqrt(p^2 + 4) + ((p^3 + 3*p)*sqrt(p^2 + 4)*x^2 - (p^4 + 5*p^2 + 4)
*x^2)*sqrt(p^2 + 4) - 4)/((p^4 + 5*p^2 + 4)*x^4 + p^4 + 5*p^2 - ((p^3 + 3*p)*x^4
+ p^3 + 3*p)*sqrt(p^2 + 4) + 4)) - (p^2 + 4)^(1/4)*(p - sqrt(p^2 + 4))*log(-((p
^4 + 5*p^2 + 4)*x^4 - p^4 - (p^5 + 5*p^3 + 4*p)*x^2 - 5*p^2 - 2*(sqrt(-x^4 + p*x
^2 + 1)*sqrt(p^2 + 4)*(p^2 + 1)*x - sqrt(-x^4 + p*x^2 + 1)*(p^3 + 3*p)*x)*(p^2 +
4)^(1/4)*sqrt((p^2 - sqrt(p^2 + 4)*p + 4)/(p^2 - sqrt(p^2 + 4)*p + 2)) - ((p^3
+ 3*p)*x^4 - p^3 - (p^4 + 3*p^2)*x^2 - 3*p)*sqrt(p^2 + 4) + ((p^3 + 3*p)*sqrt(p^
2 + 4)*x^2 - (p^4 + 5*p^2 + 4)*x^2)*sqrt(p^2 + 4) - 4)/((p^4 + 5*p^2 + 4)*x^4 +
p^4 + 5*p^2 - ((p^3 + 3*p)*x^4 + p^3 + 3*p)*sqrt(p^2 + 4) + 4)) - 8*(p^2 + 4)^(1
/4)*arctan(((sqrt(-x^4 + p*x^2 + 1)*p - sqrt(-x^4 + p*x^2 + 1)*sqrt(p^2 + 4))*sq
rt((p^2 - sqrt(p^2 + 4)*p + 4)/(p^2 - sqrt(p^2 + 4)*p + 2)) + (p*x^3 - sqrt(p^2
+ 4)*x^3 + 2*x)*(p^2 + 4)^(1/4))/((p*x^4 - (x^4 + 1)*sqrt(p^2 + 4) + p)*sqrt(-((
p^4 + 5*p^2 + 4)*x^4 - p^4 - (p^5 + 5*p^3 + 4*p)*x^2 - 5*p^2 + 2*(sqrt(-x^4 + p*
x^2 + 1)*sqrt(p^2 + 4)*(p^2 + 1)*x - sqrt(-x^4 + p*x^2 + 1)*(p^3 + 3*p)*x)*(p^2
+ 4)^(1/4)*sqrt((p^2 - sqrt(p^2 + 4)*p + 4)/(p^2 - sqrt(p^2 + 4)*p + 2)) - ((p^3
+ 3*p)*x^4 - p^3 - (p^4 + 3*p^2)*x^2 - 3*p)*sqrt(p^2 + 4) + ((p^3 + 3*p)*sqrt(p
^2 + 4)*x^2 - (p^4 + 5*p^2 + 4)*x^2)*sqrt(p^2 + 4) - 4)/((p^4 + 5*p^2 + 4)*x^4 +
p^4 + 5*p^2 - ((p^3 + 3*p)*x^4 + p^3 + 3*p)*sqrt(p^2 + 4) + 4))*sqrt((p^2 - sqr
t(p^2 + 4)*p + 4)/(p^2 - sqrt(p^2 + 4)*p + 2)) - (sqrt(-x^4 + p*x^2 + 1)*p*x^2 -
sqrt(-x^4 + p*x^2 + 1)*sqrt(p^2 + 4)*x^2)*sqrt((p^2 - sqrt(p^2 + 4)*p + 4)/(p^2
- sqrt(p^2 + 4)*p + 2)) - (2*x^3 - p*x + sqrt(p^2 + 4)*x)*(p^2 + 4)^(1/4))) + 8
*(p^2 + 4)^(1/4)*arctan(((sqrt(-x^4 + p*x^2 + 1)*p - sqrt(-x^4 + p*x^2 + 1)*sqrt
(p^2 + 4))*sqrt((p^2 - sqrt(p^2 + 4)*p + 4)/(p^2 - sqrt(p^2 + 4)*p + 2)) - (p*x^
3 - sqrt(p^2 + 4)*x^3 + 2*x)*(p^2 + 4)^(1/4))/((p*x^4 - (x^4 + 1)*sqrt(p^2 + 4)
+ p)*sqrt(-((p^4 + 5*p^2 + 4)*x^4 - p^4 - (p^5 + 5*p^3 + 4*p)*x^2 - 5*p^2 - 2*(s
qrt(-x^4 + p*x^2 + 1)*sqrt(p^2 + 4)*(p^2 + 1)*x - sqrt(-x^4 + p*x^2 + 1)*(p^3 +
3*p)*x)*(p^2 + 4)^(1/4)*sqrt((p^2 - sqrt(p^2 + 4)*p + 4)/(p^2 - sqrt(p^2 + 4)*p
+ 2)) - ((p^3 + 3*p)*x^4 - p^3 - (p^4 + 3*p^2)*x^2 - 3*p)*sqrt(p^2 + 4) + ((p^3
+ 3*p)*sqrt(p^2 + 4)*x^2 - (p^4 + 5*p^2 + 4)*x^2)*sqrt(p^2 + 4) - 4)/((p^4 + 5*p
^2 + 4)*x^4 + p^4 + 5*p^2 - ((p^3 + 3*p)*x^4 + p^3 + 3*p)*sqrt(p^2 + 4) + 4))*sq
rt((p^2 - sqrt(p^2 + 4)*p + 4)/(p^2 - sqrt(p^2 + 4)*p + 2)) - (sqrt(-x^4 + p*x^2
+ 1)*p*x^2 - sqrt(-x^4 + p*x^2 + 1)*sqrt(p^2 + 4)*x^2)*sqrt((p^2 - sqrt(p^2 + 4
)*p + 4)/(p^2 - sqrt(p^2 + 4)*p + 2)) + (2*x^3 - p*x + sqrt(p^2 + 4)*x)*(p^2 + 4
)^(1/4))))/((p - sqrt(p^2 + 4))*sqrt((p^2 - sqrt(p^2 + 4)*p + 4)/(p^2 - sqrt(p^2
+ 4)*p + 2)))

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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)