3.1.68 \(\int \frac {\sqrt {1+p x^2-x^4}}{1+x^4} \, dx\) [68]

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

[Out]

1/4*arctanh(1/4*x*(p-2*x^2+(p^2+4)^(1/2))*(-p+(p^2+4)^(1/2))^(1/2)*2^(1/2)/(-x^4+p*x^2+1)^(1/2))*(-p+(p^2+4)^(
1/2))^(1/2)*2^(1/2)-1/4*arctan(1/4*x*(p-2*x^2-(p^2+4)^(1/2))*(p+(p^2+4)^(1/2))^(1/2)*2^(1/2)/(-x^4+p*x^2+1)^(1
/2))*(p+(p^2+4)^(1/2))^(1/2)*2^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {2097} \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(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])])/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])

Rule 2097

Int[Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]/((d_) + (e_.)*(x_)^4), x_Symbol] :> With[{q = Sqrt[b^2 - 4*a*c]},
Simp[(-a)*(Sqrt[b + q]/(2*Sqrt[2]*Rt[(-a)*c, 2]*d))*ArcTan[Sqrt[b + q]*x*((b - q + 2*c*x^2)/(2*Sqrt[2]*Rt[(-a
)*c, 2]*Sqrt[a + b*x^2 + c*x^4]))], x] + Simp[a*(Sqrt[-b + q]/(2*Sqrt[2]*Rt[(-a)*c, 2]*d))*ArcTanh[Sqrt[-b + q
]*x*((b + q + 2*c*x^2)/(2*Sqrt[2]*Rt[(-a)*c, 2]*Sqrt[a + b*x^2 + c*x^4]))], x]] /; FreeQ[{a, b, c, d, e}, x] &
& EqQ[c*d + a*e, 0] && NegQ[a*c]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.23, size = 92, normalized size = 0.54 \begin {gather*} \frac {1}{4} i \left (\sqrt {-2 i-p} \tan ^{-1}\left (\frac {\sqrt {-2 i-p} x}{\sqrt {1+p x^2-x^4}}\right )-\sqrt {2 i-p} \tan ^{-1}\left (\frac {\sqrt {2 i-p} x}{\sqrt {1+p x^2-x^4}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

cought exception: maximum recursion depth exceeded while calling a Python object

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(624\) vs. \(2(131)=262\).
time = 0.11, size = 625, normalized size = 3.65

method	result	size
default	\(\frac {\left (-\frac {\sqrt {p +\sqrt {p^{2}+4}}\, \sqrt {p^{2}+4}\, \ln \left (\frac {-x^{4}+p \,x^{2}+1}{x^{2}}-\frac {\sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}\, \sqrt {p +\sqrt {p^{2}+4}}}{x}+\sqrt {p^{2}+4}\right )}{16}-\frac {\sqrt {p^{2}+4}\, \left (p +\sqrt {p^{2}+4}\right ) \arctan \left (\frac {\frac {2 \sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}}{x}-2 \sqrt {p +\sqrt {p^{2}+4}}}{2 \sqrt {-p +\sqrt {p^{2}+4}}}\right )}{8 \sqrt {-p +\sqrt {p^{2}+4}}}+\frac {\sqrt {p +\sqrt {p^{2}+4}}\, p \ln \left (\frac {-x^{4}+p \,x^{2}+1}{x^{2}}-\frac {\sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}\, \sqrt {p +\sqrt {p^{2}+4}}}{x}+\sqrt {p^{2}+4}\right )}{16}+\frac {p \left (p +\sqrt {p^{2}+4}\right ) \arctan \left (\frac {\frac {2 \sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}}{x}-2 \sqrt {p +\sqrt {p^{2}+4}}}{2 \sqrt {-p +\sqrt {p^{2}+4}}}\right )}{8 \sqrt {-p +\sqrt {p^{2}+4}}}+\frac {\sqrt {p +\sqrt {p^{2}+4}}\, \sqrt {p^{2}+4}\, \ln \left (\frac {-x^{4}+p \,x^{2}+1}{x^{2}}+\frac {\sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}\, \sqrt {p +\sqrt {p^{2}+4}}}{x}+\sqrt {p^{2}+4}\right )}{16}-\frac {\sqrt {p^{2}+4}\, \left (p +\sqrt {p^{2}+4}\right ) \arctan \left (\frac {\frac {2 \sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}}{x}+2 \sqrt {p +\sqrt {p^{2}+4}}}{2 \sqrt {-p +\sqrt {p^{2}+4}}}\right )}{8 \sqrt {-p +\sqrt {p^{2}+4}}}-\frac {\sqrt {p +\sqrt {p^{2}+4}}\, p \ln \left (\frac {-x^{4}+p \,x^{2}+1}{x^{2}}+\frac {\sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}\, \sqrt {p +\sqrt {p^{2}+4}}}{x}+\sqrt {p^{2}+4}\right )}{16}+\frac {p \left (p +\sqrt {p^{2}+4}\right ) \arctan \left (\frac {\frac {2 \sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}}{x}+2 \sqrt {p +\sqrt {p^{2}+4}}}{2 \sqrt {-p +\sqrt {p^{2}+4}}}\right )}{8 \sqrt {-p +\sqrt {p^{2}+4}}}\right ) \sqrt {2}}{2}\)	\(625\)
elliptic	\(\frac {\left (-\frac {\sqrt {p +\sqrt {p^{2}+4}}\, \sqrt {p^{2}+4}\, \ln \left (\frac {-x^{4}+p \,x^{2}+1}{x^{2}}-\frac {\sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}\, \sqrt {p +\sqrt {p^{2}+4}}}{x}+\sqrt {p^{2}+4}\right )}{16}-\frac {\sqrt {p^{2}+4}\, \left (p +\sqrt {p^{2}+4}\right ) \arctan \left (\frac {\frac {2 \sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}}{x}-2 \sqrt {p +\sqrt {p^{2}+4}}}{2 \sqrt {-p +\sqrt {p^{2}+4}}}\right )}{8 \sqrt {-p +\sqrt {p^{2}+4}}}+\frac {\sqrt {p +\sqrt {p^{2}+4}}\, p \ln \left (\frac {-x^{4}+p \,x^{2}+1}{x^{2}}-\frac {\sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}\, \sqrt {p +\sqrt {p^{2}+4}}}{x}+\sqrt {p^{2}+4}\right )}{16}+\frac {p \left (p +\sqrt {p^{2}+4}\right ) \arctan \left (\frac {\frac {2 \sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}}{x}-2 \sqrt {p +\sqrt {p^{2}+4}}}{2 \sqrt {-p +\sqrt {p^{2}+4}}}\right )}{8 \sqrt {-p +\sqrt {p^{2}+4}}}+\frac {\sqrt {p +\sqrt {p^{2}+4}}\, \sqrt {p^{2}+4}\, \ln \left (\frac {-x^{4}+p \,x^{2}+1}{x^{2}}+\frac {\sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}\, \sqrt {p +\sqrt {p^{2}+4}}}{x}+\sqrt {p^{2}+4}\right )}{16}-\frac {\sqrt {p^{2}+4}\, \left (p +\sqrt {p^{2}+4}\right ) \arctan \left (\frac {\frac {2 \sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}}{x}+2 \sqrt {p +\sqrt {p^{2}+4}}}{2 \sqrt {-p +\sqrt {p^{2}+4}}}\right )}{8 \sqrt {-p +\sqrt {p^{2}+4}}}-\frac {\sqrt {p +\sqrt {p^{2}+4}}\, p \ln \left (\frac {-x^{4}+p \,x^{2}+1}{x^{2}}+\frac {\sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}\, \sqrt {p +\sqrt {p^{2}+4}}}{x}+\sqrt {p^{2}+4}\right )}{16}+\frac {p \left (p +\sqrt {p^{2}+4}\right ) \arctan \left (\frac {\frac {2 \sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}}{x}+2 \sqrt {p +\sqrt {p^{2}+4}}}{2 \sqrt {-p +\sqrt {p^{2}+4}}}\right )}{8 \sqrt {-p +\sqrt {p^{2}+4}}}\right ) \sqrt {2}}{2}\)	\(625\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^4+p*x^2+1)^(1/2)/(x^4+1),x,method=_RETURNVERBOSE)

[Out]

1/2*(-1/16*(p+(p^2+4)^(1/2))^(1/2)*(p^2+4)^(1/2)*ln((-x^4+p*x^2+1)/x^2-(-x^4+p*x^2+1)^(1/2)*2^(1/2)/x*(p+(p^2+
4)^(1/2))^(1/2)+(p^2+4)^(1/2))-1/8*(p^2+4)^(1/2)*(p+(p^2+4)^(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))+1/16*(p+(p^2+4)^(1/2))^(1/2)*p
*ln((-x^4+p*x^2+1)/x^2-(-x^4+p*x^2+1)^(1/2)*2^(1/2)/x*(p+(p^2+4)^(1/2))^(1/2)+(p^2+4)^(1/2))+1/8*p*(p+(p^2+4)^
(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))+1/16*(p+(p^2+4)^(1/2))^(1/2)*(p^2+4)^(1/2)*ln((-x^4+p*x^2+1)/x^2+(-x^4+p*x^2+1)^(1/2)*2^(1
/2)/x*(p+(p^2+4)^(1/2))^(1/2)+(p^2+4)^(1/2))-1/8*(p^2+4)^(1/2)*(p+(p^2+4)^(1/2))/(-p+(p^2+4)^(1/2))^(1/2)*arct
an(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))-1/16*(p+(p^2+4)^
(1/2))^(1/2)*p*ln((-x^4+p*x^2+1)/x^2+(-x^4+p*x^2+1)^(1/2)*2^(1/2)/x*(p+(p^2+4)^(1/2))^(1/2)+(p^2+4)^(1/2))+1/8
*p*(p+(p^2+4)^(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)))*2^(1/2)

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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^4+p*x^2+1)^(1/2)/(x^4+1),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2667 vs. \(2 (135) = 270\).
time = 1.75, size = 2667, normalized size = 15.60

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(8*sqrt(2)*sqrt(p^2 + sqrt(p^2 + 4)*p + 4)*(p^2 + 4)^(3/4)*arctan(1/4*(2*(p^3 + 4*p)*x^12 - 2*(p^4 - 2*p
^2 - 24)*x^10 - 20*(p^3 + 4*p)*x^8 + 2*(3*p^4 + 4*p^2 - 32)*x^6 + 10*(p^3 + 4*p)*x^4 + 4*(p^2 + 4)*x^2 - 2*((p
^2 + 4)*x^12 - (p^3 + 4*p)*x^10 - (p^3 + 4*p)*x^6 - (p^2 + 4)*x^4 + (p*x^12 - (p^2 - 6)*x^10 - 10*p*x^8 + (3*p
^2 - 8)*x^6 + 5*p*x^4 + 2*x^2)*sqrt(p^2 + 4))*sqrt(p^2 + 4) + 2*((p^2 + 4)*x^12 - (p^3 + 4*p)*x^10 - (p^3 + 4*
p)*x^6 - (p^2 + 4)*x^4)*sqrt(p^2 + 4) + sqrt(p^2 + sqrt(p^2 + 4)*p + 4)*(2*(sqrt(2)*(x^9 - p*x^7 - x^5)*sqrt(-
x^4 + p*x^2 + 1)*sqrt(p^2 + 4) + sqrt(2)*(x^11 - 2*p*x^9 + (p^2 - 2)*x^7 + 2*p*x^5 + x^3)*sqrt(-x^4 + p*x^2 +
1))*(p^2 + 4)^(3/4) - (sqrt(2)*(p*x^9 + 8*x^7 - 6*p*x^5 + 2*p^2*x^3 + p*x)*sqrt(-x^4 + p*x^2 + 1)*sqrt(p^2 + 4
) + sqrt(2)*((p^2 + 4)*x^9 + 4*(p^2 + 4)*x^5 - 2*(p^3 + 4*p)*x^3 - (p^2 + 4)*x)*sqrt(-x^4 + p*x^2 + 1))*(p^2 +
4)^(1/4)) - (2*((p^3 + 4*p)*x^8 + 4*(p^2 + 4)*x^6 - (p^3 + 4*p)*x^4)*sqrt(-x^4 + p*x^2 + 1)*sqrt(p^2 + 4) + 2
*((p^4 + 6*p^2 + 8)*x^8 + 4*(p^3 + 4*p)*x^6 - (p^4 - 4*p^2 - 32)*x^4 - 4*(p^3 + 4*p)*x^2 - 2*p^2 - 8)*sqrt(-x^
4 + p*x^2 + 1) - 2*((p*x^10 - (p^2 - 4)*x^8 - 6*p*x^6 + (p^2 - 4)*x^4 + p*x^2)*sqrt(-x^4 + p*x^2 + 1)*sqrt(p^2
+ 4) + ((p^2 + 4)*x^10 - (p^3 + 4*p)*x^8 - 2*(p^2 + 4)*x^6 + (p^3 + 4*p)*x^4 + (p^2 + 4)*x^2)*sqrt(-x^4 + p*x
^2 + 1))*sqrt(p^2 + 4) - sqrt(p^2 + sqrt(p^2 + 4)*p + 4)*((sqrt(2)*(x^11 - p*x^9 - p*x^5 - x^3)*sqrt(p^2 + 4)
+ sqrt(2)*(2*x^13 - 5*p*x^11 + (3*p^2 - 8)*x^9 + 10*p*x^7 - (p^2 - 6)*x^5 - p*x^3))*(p^2 + 4)^(3/4) - (sqrt(2)
*(p*x^11 - (p^2 - 6)*x^9 - 10*p*x^7 + (3*p^2 - 8)*x^5 + 5*p*x^3 + 2*x)*sqrt(p^2 + 4) + sqrt(2)*((p^2 + 4)*x^11
- (p^3 + 4*p)*x^9 - (p^3 + 4*p)*x^5 - (p^2 + 4)*x^3))*(p^2 + 4)^(1/4)))*sqrt(-((p^2 + 4)*x^4 - (p^2 + 4)^(3/2
)*x^2 - sqrt(2)*sqrt(-x^4 + p*x^2 + 1)*sqrt(p^2 + sqrt(p^2 + 4)*p + 4)*(p^2 + 4)^(3/4)*x - (p^3 + 4*p)*x^2 - p
^2 - 4)/((p^2 + 4)*x^4 + p^2 + 4)))/((p^2 + 4)*x^12 - 3*(p^3 + 4*p)*x^10 + (2*p^4 + p^2 - 28)*x^8 + 10*(p^3 +
4*p)*x^6 - (2*p^4 + p^2 - 28)*x^4 - 3*(p^3 + 4*p)*x^2 - p^2 - 4)) + 8*sqrt(2)*sqrt(p^2 + sqrt(p^2 + 4)*p + 4)*
(p^2 + 4)^(3/4)*arctan(-1/4*(2*(p^3 + 4*p)*x^12 - 2*(p^4 - 2*p^2 - 24)*x^10 - 20*(p^3 + 4*p)*x^8 + 2*(3*p^4 +
4*p^2 - 32)*x^6 + 10*(p^3 + 4*p)*x^4 + 4*(p^2 + 4)*x^2 - 2*((p^2 + 4)*x^12 - (p^3 + 4*p)*x^10 - (p^3 + 4*p)*x^
6 - (p^2 + 4)*x^4 + (p*x^12 - (p^2 - 6)*x^10 - 10*p*x^8 + (3*p^2 - 8)*x^6 + 5*p*x^4 + 2*x^2)*sqrt(p^2 + 4))*sq
rt(p^2 + 4) + 2*((p^2 + 4)*x^12 - (p^3 + 4*p)*x^10 - (p^3 + 4*p)*x^6 - (p^2 + 4)*x^4)*sqrt(p^2 + 4) - sqrt(p^2
+ sqrt(p^2 + 4)*p + 4)*(2*(sqrt(2)*(x^9 - p*x^7 - x^5)*sqrt(-x^4 + p*x^2 + 1)*sqrt(p^2 + 4) + sqrt(2)*(x^11 -
2*p*x^9 + (p^2 - 2)*x^7 + 2*p*x^5 + x^3)*sqrt(-x^4 + p*x^2 + 1))*(p^2 + 4)^(3/4) - (sqrt(2)*(p*x^9 + 8*x^7 -
6*p*x^5 + 2*p^2*x^3 + p*x)*sqrt(-x^4 + p*x^2 + 1)*sqrt(p^2 + 4) + sqrt(2)*((p^2 + 4)*x^9 + 4*(p^2 + 4)*x^5 - 2
*(p^3 + 4*p)*x^3 - (p^2 + 4)*x)*sqrt(-x^4 + p*x^2 + 1))*(p^2 + 4)^(1/4)) - (2*((p^3 + 4*p)*x^8 + 4*(p^2 + 4)*x
^6 - (p^3 + 4*p)*x^4)*sqrt(-x^4 + p*x^2 + 1)*sqrt(p^2 + 4) + 2*((p^4 + 6*p^2 + 8)*x^8 + 4*(p^3 + 4*p)*x^6 - (p
^4 - 4*p^2 - 32)*x^4 - 4*(p^3 + 4*p)*x^2 - 2*p^2 - 8)*sqrt(-x^4 + p*x^2 + 1) - 2*((p*x^10 - (p^2 - 4)*x^8 - 6*
p*x^6 + (p^2 - 4)*x^4 + p*x^2)*sqrt(-x^4 + p*x^2 + 1)*sqrt(p^2 + 4) + ((p^2 + 4)*x^10 - (p^3 + 4*p)*x^8 - 2*(p
^2 + 4)*x^6 + (p^3 + 4*p)*x^4 + (p^2 + 4)*x^2)*sqrt(-x^4 + p*x^2 + 1))*sqrt(p^2 + 4) + sqrt(p^2 + sqrt(p^2 + 4
)*p + 4)*((sqrt(2)*(x^11 - p*x^9 - p*x^5 - x^3)*sqrt(p^2 + 4) + sqrt(2)*(2*x^13 - 5*p*x^11 + (3*p^2 - 8)*x^9 +
10*p*x^7 - (p^2 - 6)*x^5 - p*x^3))*(p^2 + 4)^(3/4) - (sqrt(2)*(p*x^11 - (p^2 - 6)*x^9 - 10*p*x^7 + (3*p^2 - 8
)*x^5 + 5*p*x^3 + 2*x)*sqrt(p^2 + 4) + sqrt(2)*((p^2 + 4)*x^11 - (p^3 + 4*p)*x^9 - (p^3 + 4*p)*x^5 - (p^2 + 4)
*x^3))*(p^2 + 4)^(1/4)))*sqrt(-((p^2 + 4)*x^4 - (p^2 + 4)^(3/2)*x^2 + sqrt(2)*sqrt(-x^4 + p*x^2 + 1)*sqrt(p^2
+ sqrt(p^2 + 4)*p + 4)*(p^2 + 4)^(3/4)*x - (p^3 + 4*p)*x^2 - p^2 - 4)/((p^2 + 4)*x^4 + p^2 + 4)))/((p^2 + 4)*x
^12 - 3*(p^3 + 4*p)*x^10 + (2*p^4 + p^2 - 28)*x^8 + 10*(p^3 + 4*p)*x^6 - (2*p^4 + p^2 - 28)*x^4 - 3*(p^3 + 4*p
)*x^2 - p^2 - 4)) - (sqrt(2)*sqrt(p^2 + 4)*p - sqrt(2)*(p^2 + 4))*sqrt(p^2 + sqrt(p^2 + 4)*p + 4)*(p^2 + 4)^(1
/4)*log(-((p^2 + 4)*x^4 - (p^2 + 4)^(3/2)*x^2 + sqrt(2)*sqrt(-x^4 + p*x^2 + 1)*sqrt(p^2 + sqrt(p^2 + 4)*p + 4)
*(p^2 + 4)^(3/4)*x - (p^3 + 4*p)*x^2 - p^2 - 4)/((p^2 + 4)*x^4 + p^2 + 4)) + (sqrt(2)*sqrt(p^2 + 4)*p - sqrt(2
)*(p^2 + 4))*sqrt(p^2 + sqrt(p^2 + 4)*p + 4)*(p^2 + 4)^(1/4)*log(-((p^2 + 4)*x^4 - (p^2 + 4)^(3/2)*x^2 - sqrt(
2)*sqrt(-x^4 + p*x^2 + 1)*sqrt(p^2 + sqrt(p^2 + 4)*p + 4)*(p^2 + 4)^(3/4)*x - (p^3 + 4*p)*x^2 - p^2 - 4)/((p^2
+ 4)*x^4 + p^2 + 4)))/(p^2 + 4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {p x^{2} - x^{4} + 1}}{x^{4} + 1}\, dx \end {gather*}

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 [F] N/A
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^4+p*x^2+1)^(1/2)/(x^4+1),x)

[Out]

Could not integrate

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {-x^4+p\,x^2+1}}{x^4+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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