Integrand size = 31, antiderivative size = 65 \[ \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{4-7 x^4+4 x^8} \, dx=\frac {1}{8} \arctan \left (\frac {-1+\frac {x^2}{2}+x^4}{x \sqrt {1-x^4}}\right )-\frac {1}{8} \text {arctanh}\left (\frac {-1-\frac {x^2}{2}+x^4}{x \sqrt {1-x^4}}\right ) \] Output:
1/8*arctan((-1+1/2*x^2+x^4)/x/(-x^4+1)^(1/2))-1/8*arctanh((-1-1/2*x^2+x^4) /x/(-x^4+1)^(1/2))
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{4-7 x^4+4 x^8} \, dx=\left (\frac {1}{8}-\frac {i}{8}\right ) \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) x}{\sqrt {1-x^4}}\right )-\left (\frac {1}{8}+\frac {i}{8}\right ) \arctan \left (\frac {(1+i) \sqrt {1-x^4}}{x}\right ) \] Input:
Integrate[(Sqrt[1 - x^4]*(1 + x^4))/(4 - 7*x^4 + 4*x^8),x]
Output:
(1/8 - I/8)*ArcTan[((1/2 + I/2)*x)/Sqrt[1 - x^4]] - (1/8 + I/8)*ArcTan[((1 + I)*Sqrt[1 - x^4])/x]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.64 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.38, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {7279, 2009}
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.
\(\displaystyle \int \frac {\sqrt {1-x^4} \left (x^4+1\right )}{4 x^8-7 x^4+4} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {\left (1-i \sqrt {15}\right ) \sqrt {1-x^4}}{8 x^4-i \sqrt {15}-7}+\frac {\left (1+i \sqrt {15}\right ) \sqrt {1-x^4}}{8 x^4+i \sqrt {15}-7}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{8} \left (1+i \sqrt {15}\right ) \operatorname {EllipticF}(\arcsin (x),-1)-\frac {1}{8} \left (1-i \sqrt {15}\right ) \operatorname {EllipticF}(\arcsin (x),-1)+\frac {1}{8} \operatorname {EllipticPi}\left (-\frac {2}{\sqrt {\frac {1}{2} \left (7-i \sqrt {15}\right )}},\arcsin (x),-1\right )+\frac {1}{8} \operatorname {EllipticPi}\left (\frac {2}{\sqrt {\frac {1}{2} \left (7-i \sqrt {15}\right )}},\arcsin (x),-1\right )+\frac {1}{8} \operatorname {EllipticPi}\left (-\frac {2}{\sqrt {\frac {1}{2} \left (7+i \sqrt {15}\right )}},\arcsin (x),-1\right )+\frac {1}{8} \operatorname {EllipticPi}\left (\frac {2}{\sqrt {\frac {1}{2} \left (7+i \sqrt {15}\right )}},\arcsin (x),-1\right )\) |
Input:
Int[(Sqrt[1 - x^4]*(1 + x^4))/(4 - 7*x^4 + 4*x^8),x]
Output:
-1/8*((1 - I*Sqrt[15])*EllipticF[ArcSin[x], -1]) - ((1 + I*Sqrt[15])*Ellip ticF[ArcSin[x], -1])/8 + EllipticPi[-2/Sqrt[(7 - I*Sqrt[15])/2], ArcSin[x] , -1]/8 + EllipticPi[2/Sqrt[(7 - I*Sqrt[15])/2], ArcSin[x], -1]/8 + Ellipt icPi[-2/Sqrt[(7 + I*Sqrt[15])/2], ArcSin[x], -1]/8 + EllipticPi[2/Sqrt[(7 + I*Sqrt[15])/2], ArcSin[x], -1]/8
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains complex when optimal does not.
Time = 1.60 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.68
method | result | size |
pseudoelliptic | \(\left (-\frac {1}{8}+\frac {i}{8}\right ) \arctan \left (\frac {\left (1-i\right ) \sqrt {-x^{4}+1}}{x}\right )+\left (-\frac {1}{8}-\frac {i}{8}\right ) \arctan \left (\frac {\left (1+i\right ) \sqrt {-x^{4}+1}}{x}\right )\) | \(44\) |
default | \(-\frac {\ln \left (\frac {\frac {-x^{4}+1}{2 x^{2}}-\frac {\sqrt {-x^{4}+1}}{2 x}+\frac {1}{4}}{\frac {-x^{4}+1}{2 x^{2}}+\frac {\sqrt {-x^{4}+1}}{2 x}+\frac {1}{4}}\right )}{16}-\frac {\arctan \left (\frac {2 \sqrt {-x^{4}+1}}{x}+1\right )}{8}-\frac {\arctan \left (\frac {2 \sqrt {-x^{4}+1}}{x}-1\right )}{8}\) | \(102\) |
elliptic | \(-\frac {\ln \left (\frac {\frac {-x^{4}+1}{2 x^{2}}-\frac {\sqrt {-x^{4}+1}}{2 x}+\frac {1}{4}}{\frac {-x^{4}+1}{2 x^{2}}+\frac {\sqrt {-x^{4}+1}}{2 x}+\frac {1}{4}}\right )}{16}-\frac {\arctan \left (\frac {2 \sqrt {-x^{4}+1}}{x}+1\right )}{8}-\frac {\arctan \left (\frac {2 \sqrt {-x^{4}+1}}{x}-1\right )}{8}\) | \(102\) |
trager | \(\operatorname {RootOf}\left (128 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (128 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) x^{4}+64 \operatorname {RootOf}\left (128 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2} x^{2}-4 x^{2} \operatorname {RootOf}\left (128 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )-\sqrt {-x^{4}+1}\, x -8 \operatorname {RootOf}\left (128 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )}{-2 x^{4}+16 x^{2} \operatorname {RootOf}\left (128 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )-x^{2}+2}\right )+\frac {\ln \left (\frac {-16 \operatorname {RootOf}\left (128 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) x^{4}+128 \operatorname {RootOf}\left (128 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2} x^{2}+2 x^{4}-24 x^{2} \operatorname {RootOf}\left (128 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )-2 \sqrt {-x^{4}+1}\, x +x^{2}+16 \operatorname {RootOf}\left (128 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )-2}{2 x^{4}+16 x^{2} \operatorname {RootOf}\left (128 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )-x^{2}-2}\right )}{8}-\ln \left (\frac {-16 \operatorname {RootOf}\left (128 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) x^{4}+128 \operatorname {RootOf}\left (128 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2} x^{2}+2 x^{4}-24 x^{2} \operatorname {RootOf}\left (128 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )-2 \sqrt {-x^{4}+1}\, x +x^{2}+16 \operatorname {RootOf}\left (128 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )-2}{2 x^{4}+16 x^{2} \operatorname {RootOf}\left (128 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )-x^{2}-2}\right ) \operatorname {RootOf}\left (128 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )\) | \(371\) |
Input:
int((-x^4+1)^(1/2)*(x^4+1)/(4*x^8-7*x^4+4),x,method=_RETURNVERBOSE)
Output:
(-1/8+1/8*I)*arctan((1-I)*(-x^4+1)^(1/2)/x)-(1/8+1/8*I)*arctan((1+I)*(-x^4 +1)^(1/2)/x)
Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (57) = 114\).
Time = 0.18 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.00 \[ \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{4-7 x^4+4 x^8} \, dx=-\frac {1}{8} \, \arctan \left (-\frac {4 \, x^{8} - 7 \, x^{4} - 4 \, {\left (2 \, x^{5} + x^{3} - 2 \, x\right )} \sqrt {-x^{4} + 1} + 4}{4 \, x^{8} + 8 \, x^{6} - 7 \, x^{4} - 8 \, x^{2} + 4}\right ) + \frac {1}{16} \, \log \left (\frac {4 \, x^{8} - 8 \, x^{6} - 7 \, x^{4} + 8 \, x^{2} - 4 \, {\left (2 \, x^{5} - x^{3} - 2 \, x\right )} \sqrt {-x^{4} + 1} + 4}{4 \, x^{8} - 7 \, x^{4} + 4}\right ) \] Input:
integrate((-x^4+1)^(1/2)*(x^4+1)/(4*x^8-7*x^4+4),x, algorithm="fricas")
Output:
-1/8*arctan(-(4*x^8 - 7*x^4 - 4*(2*x^5 + x^3 - 2*x)*sqrt(-x^4 + 1) + 4)/(4 *x^8 + 8*x^6 - 7*x^4 - 8*x^2 + 4)) + 1/16*log((4*x^8 - 8*x^6 - 7*x^4 + 8*x ^2 - 4*(2*x^5 - x^3 - 2*x)*sqrt(-x^4 + 1) + 4)/(4*x^8 - 7*x^4 + 4))
\[ \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{4-7 x^4+4 x^8} \, dx=\int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{4} + 1\right )}{4 x^{8} - 7 x^{4} + 4}\, dx \] Input:
integrate((-x**4+1)**(1/2)*(x**4+1)/(4*x**8-7*x**4+4),x)
Output:
Integral(sqrt(-(x - 1)*(x + 1)*(x**2 + 1))*(x**4 + 1)/(4*x**8 - 7*x**4 + 4 ), x)
\[ \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{4-7 x^4+4 x^8} \, dx=\int { \frac {{\left (x^{4} + 1\right )} \sqrt {-x^{4} + 1}}{4 \, x^{8} - 7 \, x^{4} + 4} \,d x } \] Input:
integrate((-x^4+1)^(1/2)*(x^4+1)/(4*x^8-7*x^4+4),x, algorithm="maxima")
Output:
integrate((x^4 + 1)*sqrt(-x^4 + 1)/(4*x^8 - 7*x^4 + 4), x)
\[ \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{4-7 x^4+4 x^8} \, dx=\int { \frac {{\left (x^{4} + 1\right )} \sqrt {-x^{4} + 1}}{4 \, x^{8} - 7 \, x^{4} + 4} \,d x } \] Input:
integrate((-x^4+1)^(1/2)*(x^4+1)/(4*x^8-7*x^4+4),x, algorithm="giac")
Output:
integrate((x^4 + 1)*sqrt(-x^4 + 1)/(4*x^8 - 7*x^4 + 4), x)
Timed out. \[ \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{4-7 x^4+4 x^8} \, dx=\int \frac {\sqrt {1-x^4}\,\left (x^4+1\right )}{4\,x^8-7\,x^4+4} \,d x \] Input:
int(((1 - x^4)^(1/2)*(x^4 + 1))/(4*x^8 - 7*x^4 + 4),x)
Output:
int(((1 - x^4)^(1/2)*(x^4 + 1))/(4*x^8 - 7*x^4 + 4), x)
\[ \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{4-7 x^4+4 x^8} \, dx=\int \frac {\sqrt {-x^{4}+1}}{4 x^{8}-7 x^{4}+4}d x +\int \frac {\sqrt {-x^{4}+1}\, x^{4}}{4 x^{8}-7 x^{4}+4}d x \] Input:
int((-x^4+1)^(1/2)*(x^4+1)/(4*x^8-7*x^4+4),x)
Output:
int(sqrt( - x**4 + 1)/(4*x**8 - 7*x**4 + 4),x) + int((sqrt( - x**4 + 1)*x* *4)/(4*x**8 - 7*x**4 + 4),x)