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

3.1.68.1 Optimal result

Integrand size = 24, antiderivative size = 171 \[ \int \frac {\sqrt {1+p x^2-x^4}}{1+x^4} \, dx=-\frac {\sqrt {p+\sqrt {4+p^2}} \arctan \left (\frac {\sqrt {p+\sqrt {4+p^2}} x \left (p-\sqrt {4+p^2}-2 x^2\right )}{2 \sqrt {2} \sqrt {1+p x^2-x^4}}\right )}{2 \sqrt {2}}+\frac {\sqrt {-p+\sqrt {4+p^2}} \text {arctanh}\left (\frac {\sqrt {-p+\sqrt {4+p^2}} x \left (p+\sqrt {4+p^2}-2 x^2\right )}{2 \sqrt {2} \sqrt {1+p x^2-x^4}}\right )}{2 \sqrt {2}} \]

output

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)
 

3.1.68.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.38 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt {1+p x^2-x^4}}{1+x^4} \, dx=\frac {1}{4} i \left (\sqrt {-2 i-p} \arctan \left (\frac {\sqrt {-2 i-p} x}{\sqrt {1+p x^2-x^4}}\right )-\sqrt {2 i-p} \arctan \left (\frac {\sqrt {2 i-p} x}{\sqrt {1+p x^2-x^4}}\right )\right ) \]

input

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

output

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

3.1.68.3 Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {2518}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2518

\(\displaystyle \frac {\sqrt {\sqrt {p^2+4}-p} \text {arctanh}\left (\frac {\sqrt {\sqrt {p^2+4}-p} x \left (\sqrt {p^2+4}+p-2 x^2\right )}{2 \sqrt {2} \sqrt {p x^2-x^4+1}}\right )}{2 \sqrt {2}}-\frac {\sqrt {\sqrt {p^2+4}+p} \arctan \left (\frac {\sqrt {\sqrt {p^2+4}+p} x \left (-\sqrt {p^2+4}+p-2 x^2\right )}{2 \sqrt {2} \sqrt {p x^2-x^4+1}}\right )}{2 \sqrt {2}}\)

input

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

output

-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])
 

3.1.68.3.1 Defintions of rubi rules used

rule 2518

Int[Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]/((d_) + (e_.)*(x_)^4), x_Symbo 
l] :> 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]
 

3.1.68.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.92 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(-\frac {i \sqrt {2 i+p}\, \sqrt {2 i-p}\, \ln \left (2\right )+i \sqrt {2 i+p}\, \sqrt {2 i-p}\, \ln \left (\frac {\sqrt {2 i+p}\, \sqrt {-x^{4}+p \,x^{2}+1}+2 x \left (i+\frac {p}{2}\right )}{x^{2}+i}\right )+i p \arctan \left (\frac {\sqrt {-x^{4}+p \,x^{2}+1}}{x \sqrt {2 i-p}}\right )+2 \arctan \left (\frac {\sqrt {-x^{4}+p \,x^{2}+1}}{x \sqrt {2 i-p}}\right )}{4 \sqrt {2 i-p}}\)	\(149\)
default	\(\frac {\left (-\frac {\sqrt {p +\sqrt {p^{2}+4}}\, \sqrt {p^{2}+4}\, \ln \left (\frac {\sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}\, \sqrt {p +\sqrt {p^{2}+4}}}{x}-\frac {-x^{4}+p \,x^{2}+1}{x^{2}}-\sqrt {p^{2}+4}\right )}{16}+\frac {\sqrt {p^{2}+4}\, \left (p +\sqrt {p^{2}+4}\right ) \arctan \left (\frac {2 \sqrt {p +\sqrt {p^{2}+4}}-\frac {2 \sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}}{x}}{2 \sqrt {-p +\sqrt {p^{2}+4}}}\right )}{8 \sqrt {-p +\sqrt {p^{2}+4}}}+\frac {\sqrt {p +\sqrt {p^{2}+4}}\, p \ln \left (\frac {\sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}\, \sqrt {p +\sqrt {p^{2}+4}}}{x}-\frac {-x^{4}+p \,x^{2}+1}{x^{2}}-\sqrt {p^{2}+4}\right )}{16}-\frac {p \left (p +\sqrt {p^{2}+4}\right ) \arctan \left (\frac {2 \sqrt {p +\sqrt {p^{2}+4}}-\frac {2 \sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}}{x}}{2 \sqrt {-p +\sqrt {p^{2}+4}}}\right )}{8 \sqrt {-p +\sqrt {p^{2}+4}}}+\frac {\sqrt {p +\sqrt {p^{2}+4}}\, \sqrt {p^{2}+4}\, \ln \left (\frac {-x^{4}+p \,x^{2}+1}{x^{2}}+\frac {\sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}\, \sqrt {p +\sqrt {p^{2}+4}}}{x}+\sqrt {p^{2}+4}\right )}{16}-\frac {\sqrt {p^{2}+4}\, \left (p +\sqrt {p^{2}+4}\right ) \arctan \left (\frac {\frac {2 \sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}}{x}+2 \sqrt {p +\sqrt {p^{2}+4}}}{2 \sqrt {-p +\sqrt {p^{2}+4}}}\right )}{8 \sqrt {-p +\sqrt {p^{2}+4}}}-\frac {\sqrt {p +\sqrt {p^{2}+4}}\, p \ln \left (\frac {-x^{4}+p \,x^{2}+1}{x^{2}}+\frac {\sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}\, \sqrt {p +\sqrt {p^{2}+4}}}{x}+\sqrt {p^{2}+4}\right )}{16}+\frac {p \left (p +\sqrt {p^{2}+4}\right ) \arctan \left (\frac {\frac {2 \sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}}{x}+2 \sqrt {p +\sqrt {p^{2}+4}}}{2 \sqrt {-p +\sqrt {p^{2}+4}}}\right )}{8 \sqrt {-p +\sqrt {p^{2}+4}}}\right ) \sqrt {2}}{2}\)	\(629\)
elliptic	\(\frac {\left (-\frac {\sqrt {p +\sqrt {p^{2}+4}}\, \sqrt {p^{2}+4}\, \ln \left (\frac {\sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}\, \sqrt {p +\sqrt {p^{2}+4}}}{x}-\frac {-x^{4}+p \,x^{2}+1}{x^{2}}-\sqrt {p^{2}+4}\right )}{16}+\frac {\sqrt {p^{2}+4}\, \left (p +\sqrt {p^{2}+4}\right ) \arctan \left (\frac {2 \sqrt {p +\sqrt {p^{2}+4}}-\frac {2 \sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}}{x}}{2 \sqrt {-p +\sqrt {p^{2}+4}}}\right )}{8 \sqrt {-p +\sqrt {p^{2}+4}}}+\frac {\sqrt {p +\sqrt {p^{2}+4}}\, p \ln \left (\frac {\sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}\, \sqrt {p +\sqrt {p^{2}+4}}}{x}-\frac {-x^{4}+p \,x^{2}+1}{x^{2}}-\sqrt {p^{2}+4}\right )}{16}-\frac {p \left (p +\sqrt {p^{2}+4}\right ) \arctan \left (\frac {2 \sqrt {p +\sqrt {p^{2}+4}}-\frac {2 \sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}}{x}}{2 \sqrt {-p +\sqrt {p^{2}+4}}}\right )}{8 \sqrt {-p +\sqrt {p^{2}+4}}}+\frac {\sqrt {p +\sqrt {p^{2}+4}}\, \sqrt {p^{2}+4}\, \ln \left (\frac {-x^{4}+p \,x^{2}+1}{x^{2}}+\frac {\sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}\, \sqrt {p +\sqrt {p^{2}+4}}}{x}+\sqrt {p^{2}+4}\right )}{16}-\frac {\sqrt {p^{2}+4}\, \left (p +\sqrt {p^{2}+4}\right ) \arctan \left (\frac {\frac {2 \sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}}{x}+2 \sqrt {p +\sqrt {p^{2}+4}}}{2 \sqrt {-p +\sqrt {p^{2}+4}}}\right )}{8 \sqrt {-p +\sqrt {p^{2}+4}}}-\frac {\sqrt {p +\sqrt {p^{2}+4}}\, p \ln \left (\frac {-x^{4}+p \,x^{2}+1}{x^{2}}+\frac {\sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}\, \sqrt {p +\sqrt {p^{2}+4}}}{x}+\sqrt {p^{2}+4}\right )}{16}+\frac {p \left (p +\sqrt {p^{2}+4}\right ) \arctan \left (\frac {\frac {2 \sqrt {-x^{4}+p \,x^{2}+1}\, \sqrt {2}}{x}+2 \sqrt {p +\sqrt {p^{2}+4}}}{2 \sqrt {-p +\sqrt {p^{2}+4}}}\right )}{8 \sqrt {-p +\sqrt {p^{2}+4}}}\right ) \sqrt {2}}{2}\)	\(629\)

input

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

output

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

3.1.68.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt {1+p x^2-x^4}}{1+x^4} \, dx=\frac {1}{8} \, \sqrt {-p + 2 i} \log \left (-\frac {\sqrt {-x^{4} + p x^{2} + 1} {\left (x^{2} + i\right )} - {\left (i \, x^{3} - x\right )} \sqrt {-p + 2 i}}{x^{4} + 1}\right ) - \frac {1}{8} \, \sqrt {-p + 2 i} \log \left (-\frac {\sqrt {-x^{4} + p x^{2} + 1} {\left (x^{2} + i\right )} - {\left (-i \, x^{3} + x\right )} \sqrt {-p + 2 i}}{x^{4} + 1}\right ) - \frac {1}{8} \, \sqrt {-p - 2 i} \log \left (-\frac {\sqrt {-x^{4} + p x^{2} + 1} {\left (x^{2} - i\right )} - {\left (i \, x^{3} + x\right )} \sqrt {-p - 2 i}}{x^{4} + 1}\right ) + \frac {1}{8} \, \sqrt {-p - 2 i} \log \left (-\frac {\sqrt {-x^{4} + p x^{2} + 1} {\left (x^{2} - i\right )} - {\left (-i \, x^{3} - x\right )} \sqrt {-p - 2 i}}{x^{4} + 1}\right ) \]

input

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

output

1/8*sqrt(-p + 2*I)*log(-(sqrt(-x^4 + p*x^2 + 1)*(x^2 + I) - (I*x^3 - x)*sq 
rt(-p + 2*I))/(x^4 + 1)) - 1/8*sqrt(-p + 2*I)*log(-(sqrt(-x^4 + p*x^2 + 1) 
*(x^2 + I) - (-I*x^3 + x)*sqrt(-p + 2*I))/(x^4 + 1)) - 1/8*sqrt(-p - 2*I)* 
log(-(sqrt(-x^4 + p*x^2 + 1)*(x^2 - I) - (I*x^3 + x)*sqrt(-p - 2*I))/(x^4 
+ 1)) + 1/8*sqrt(-p - 2*I)*log(-(sqrt(-x^4 + p*x^2 + 1)*(x^2 - I) - (-I*x^ 
3 - x)*sqrt(-p - 2*I))/(x^4 + 1))
 

3.1.68.6 Sympy [F]

\[ \int \frac {\sqrt {1+p x^2-x^4}}{1+x^4} \, dx=\int \frac {\sqrt {p x^{2} - x^{4} + 1}}{x^{4} + 1}\, dx \]

input

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

output

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

3.1.68.7 Maxima [F]

\[ \int \frac {\sqrt {1+p x^2-x^4}}{1+x^4} \, dx=\int { \frac {\sqrt {-x^{4} + p x^{2} + 1}}{x^{4} + 1} \,d x } \]

input

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

output

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

3.1.68.8 Giac [F]

\[ \int \frac {\sqrt {1+p x^2-x^4}}{1+x^4} \, dx=\int { \frac {\sqrt {-x^{4} + p x^{2} + 1}}{x^{4} + 1} \,d x } \]

input

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

output

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

3.1.68.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {1+p x^2-x^4}}{1+x^4} \, dx=\int \frac {\sqrt {-x^4+p\,x^2+1}}{x^4+1} \,d x \]

input

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

output

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

3.1.68.10 Reduce [F]

\[ \int \frac {\sqrt {1+p x^2-x^4}}{1+x^4} \, dx=\int \frac {\sqrt {-x^{4}+p \,x^{2}+1}}{x^{4}+1}d x \]

input

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

output

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