Integrand size = 53, antiderivative size = 76 \[ \int \frac {\left (-1+x^4\right ) \left (1-x^2+x^4\right ) \sqrt {4-x^2+4 x^4}}{\left (1+x^4\right ) \left (4+7 x^4+4 x^8\right )} \, dx=\arctan \left (\frac {x}{\sqrt {4-x^2+4 x^4}}\right )-\frac {3}{4} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{\sqrt {4-x^2+4 x^4}}\right )+\frac {1}{4} \text {arctanh}\left (\frac {x}{\sqrt {4-x^2+4 x^4}}\right ) \]
arctan(x/(4*x^4-x^2+4)^(1/2))-3/4*3^(1/2)*arctan(3^(1/2)*x/(4*x^4-x^2+4)^( 1/2))+1/4*arctanh(x/(4*x^4-x^2+4)^(1/2))
Time = 0.56 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^4\right ) \left (1-x^2+x^4\right ) \sqrt {4-x^2+4 x^4}}{\left (1+x^4\right ) \left (4+7 x^4+4 x^8\right )} \, dx=\arctan \left (\frac {x}{\sqrt {4-x^2+4 x^4}}\right )-\frac {3}{4} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{\sqrt {4-x^2+4 x^4}}\right )+\frac {1}{4} \text {arctanh}\left (\frac {x}{\sqrt {4-x^2+4 x^4}}\right ) \]
Integrate[((-1 + x^4)*(1 - x^2 + x^4)*Sqrt[4 - x^2 + 4*x^4])/((1 + x^4)*(4 + 7*x^4 + 4*x^8)),x]
ArcTan[x/Sqrt[4 - x^2 + 4*x^4]] - (3*Sqrt[3]*ArcTan[(Sqrt[3]*x)/Sqrt[4 - x ^2 + 4*x^4]])/4 + ArcTanh[x/Sqrt[4 - x^2 + 4*x^4]]/4
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.89 (sec) , antiderivative size = 860, normalized size of antiderivative = 11.32, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {7276, 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.
| \(\displaystyle \int \frac {\left (x^4-1\right ) \left (x^4-x^2+1\right ) \sqrt {4 x^4-x^2+4}}{\left (x^4+1\right ) \left (4 x^8+7 x^4+4\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {2 \sqrt {4 x^4-x^2+4} x^2}{x^4+1}+\frac {\left (1-4 x^2\right ) \sqrt {4 x^4-x^2+4}}{4 \left (2 x^4-x^2+2\right )}-\frac {3 \left (4 x^2+1\right ) \sqrt {4 x^4-x^2+4}}{4 \left (2 x^4+x^2+2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \arctan \left (\frac {x}{\sqrt {4 x^4-x^2+4}}\right )-\frac {9 \left (i \sqrt {3}+\sqrt {5}\right ) \arctan \left (\frac {\sqrt {3} x}{\sqrt {4 x^4-x^2+4}}\right )}{8 \left (3 i+\sqrt {15}\right )}-\frac {9 \left (i \sqrt {3}-\sqrt {5}\right ) \arctan \left (\frac {\sqrt {3} x}{\sqrt {4 x^4-x^2+4}}\right )}{8 \left (3 i-\sqrt {15}\right )}+\frac {1}{4} \text {arctanh}\left (\frac {x}{\sqrt {4 x^4-x^2+4}}\right )+\frac {3 \left (3 i+2 \sqrt {15}\right ) \left (x^2+1\right ) \sqrt {\frac {4 x^4-x^2+4}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {9}{16}\right )}{4 \left (3 i-\sqrt {15}\right ) \sqrt {4 x^4-x^2+4}}-\frac {\left (i+2 \sqrt {15}\right ) \left (x^2+1\right ) \sqrt {\frac {4 x^4-x^2+4}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {9}{16}\right )}{4 \left (5 i+\sqrt {15}\right ) \sqrt {4 x^4-x^2+4}}+\frac {3 \left (3 i-2 \sqrt {15}\right ) \left (x^2+1\right ) \sqrt {\frac {4 x^4-x^2+4}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {9}{16}\right )}{4 \left (3 i+\sqrt {15}\right ) \sqrt {4 x^4-x^2+4}}-\frac {\left (i-2 \sqrt {15}\right ) \left (x^2+1\right ) \sqrt {\frac {4 x^4-x^2+4}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {9}{16}\right )}{4 \left (5 i-\sqrt {15}\right ) \sqrt {4 x^4-x^2+4}}+\frac {7 \left (x^2+1\right ) \sqrt {\frac {4 x^4-x^2+4}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {9}{16}\right )}{4 \sqrt {4 x^4-x^2+4}}-\frac {9 \left (5 i+\sqrt {15}\right ) \left (x^2+1\right ) \sqrt {\frac {4 x^4-x^2+4}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {3}{8},2 \arctan (x),\frac {9}{16}\right )}{32 \left (3 i-\sqrt {15}\right ) \sqrt {4 x^4-x^2+4}}-\frac {9 \left (5 i-\sqrt {15}\right ) \left (x^2+1\right ) \sqrt {\frac {4 x^4-x^2+4}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {3}{8},2 \arctan (x),\frac {9}{16}\right )}{32 \left (3 i+\sqrt {15}\right ) \sqrt {4 x^4-x^2+4}}+\frac {\left (3 i-\sqrt {15}\right ) \left (x^2+1\right ) \sqrt {\frac {4 x^4-x^2+4}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {5}{8},2 \arctan (x),\frac {9}{16}\right )}{32 \left (5 i+\sqrt {15}\right ) \sqrt {4 x^4-x^2+4}}+\frac {\left (3 i+\sqrt {15}\right ) \left (x^2+1\right ) \sqrt {\frac {4 x^4-x^2+4}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {5}{8},2 \arctan (x),\frac {9}{16}\right )}{32 \left (5 i-\sqrt {15}\right ) \sqrt {4 x^4-x^2+4}}\) |
ArcTan[x/Sqrt[4 - x^2 + 4*x^4]] - (9*(I*Sqrt[3] - Sqrt[5])*ArcTan[(Sqrt[3] *x)/Sqrt[4 - x^2 + 4*x^4]])/(8*(3*I - Sqrt[15])) - (9*(I*Sqrt[3] + Sqrt[5] )*ArcTan[(Sqrt[3]*x)/Sqrt[4 - x^2 + 4*x^4]])/(8*(3*I + Sqrt[15])) + ArcTan h[x/Sqrt[4 - x^2 + 4*x^4]]/4 + (7*(1 + x^2)*Sqrt[(4 - x^2 + 4*x^4)/(1 + x^ 2)^2]*EllipticF[2*ArcTan[x], 9/16])/(4*Sqrt[4 - x^2 + 4*x^4]) - ((I - 2*Sq rt[15])*(1 + x^2)*Sqrt[(4 - x^2 + 4*x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x ], 9/16])/(4*(5*I - Sqrt[15])*Sqrt[4 - x^2 + 4*x^4]) + (3*(3*I - 2*Sqrt[15 ])*(1 + x^2)*Sqrt[(4 - x^2 + 4*x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 9/ 16])/(4*(3*I + Sqrt[15])*Sqrt[4 - x^2 + 4*x^4]) - ((I + 2*Sqrt[15])*(1 + x ^2)*Sqrt[(4 - x^2 + 4*x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 9/16])/(4*( 5*I + Sqrt[15])*Sqrt[4 - x^2 + 4*x^4]) + (3*(3*I + 2*Sqrt[15])*(1 + x^2)*S qrt[(4 - x^2 + 4*x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 9/16])/(4*(3*I - Sqrt[15])*Sqrt[4 - x^2 + 4*x^4]) - (9*(5*I - Sqrt[15])*(1 + x^2)*Sqrt[(4 - x^2 + 4*x^4)/(1 + x^2)^2]*EllipticPi[3/8, 2*ArcTan[x], 9/16])/(32*(3*I + Sqrt[15])*Sqrt[4 - x^2 + 4*x^4]) - (9*(5*I + Sqrt[15])*(1 + x^2)*Sqrt[(4 - x^2 + 4*x^4)/(1 + x^2)^2]*EllipticPi[3/8, 2*ArcTan[x], 9/16])/(32*(3*I - Sqrt[15])*Sqrt[4 - x^2 + 4*x^4]) + ((3*I + Sqrt[15])*(1 + x^2)*Sqrt[(4 - x^2 + 4*x^4)/(1 + x^2)^2]*EllipticPi[5/8, 2*ArcTan[x], 9/16])/(32*(5*I - S qrt[15])*Sqrt[4 - x^2 + 4*x^4]) + ((3*I - Sqrt[15])*(1 + x^2)*Sqrt[(4 - x^ 2 + 4*x^4)/(1 + x^2)^2]*EllipticPi[5/8, 2*ArcTan[x], 9/16])/(32*(5*I + ...
3.11.7.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 17.76 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.13
| method | result | size |
| elliptic | \(\frac {\left (\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {4 x^{4}-x^{2}+4}}{x}\right )}{4}-\sqrt {2}\, \arctan \left (\frac {\sqrt {4 x^{4}-x^{2}+4}}{x}\right )+\frac {3 \sqrt {6}\, \arctan \left (\frac {\sqrt {6}\, \sqrt {4 x^{4}-x^{2}+4}\, \sqrt {2}}{6 x}\right )}{4}\right ) \sqrt {2}}{2}\) | \(86\) |
| trager | \(\frac {\ln \left (-\frac {2 x^{4}+x \sqrt {4 x^{4}-x^{2}+4}+2}{2 x^{4}-x^{2}+2}\right )}{8}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{2}-3 x \sqrt {4 x^{4}-x^{2}+4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{2 x^{4}+x^{2}+2}\right )}{8}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+x \sqrt {4 x^{4}-x^{2}+4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}+1}\right )}{2}\) | \(178\) |
| default | \(\frac {3 \sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (\left (x^{2}+1\right ) \sqrt {2}\, \sqrt {3}-\frac {9 x}{2}\right )}{3 \sqrt {4 x^{4}-x^{2}+4}}\right )}{8}-\frac {3 \sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (\left (x^{2}+1\right ) \sqrt {2}\, \sqrt {3}+\frac {9 x}{2}\right )}{3 \sqrt {4 x^{4}-x^{2}+4}}\right )}{8}-\frac {\arctan \left (\frac {4 \sqrt {2}\, x^{2}+4 \sqrt {2}-9 x}{\sqrt {4 x^{4}-x^{2}+4}}\right )}{2}+\frac {\arctan \left (\frac {9 x +4 \sqrt {2}\, x^{2}+4 \sqrt {2}}{\sqrt {4 x^{4}-x^{2}+4}}\right )}{2}-\frac {\operatorname {arctanh}\left (\frac {9 x +2 \sqrt {10}\, x^{2}+2 \sqrt {10}}{\sqrt {4 x^{4}-x^{2}+4}}\right )}{8}+\frac {\operatorname {arctanh}\left (\frac {2 \sqrt {10}\, x^{2}+2 \sqrt {10}-9 x}{\sqrt {4 x^{4}-x^{2}+4}}\right )}{8}\) | \(224\) |
| pseudoelliptic | \(\frac {3 \sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (\left (x^{2}+1\right ) \sqrt {2}\, \sqrt {3}-\frac {9 x}{2}\right )}{3 \sqrt {4 x^{4}-x^{2}+4}}\right )}{8}-\frac {3 \sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (\left (x^{2}+1\right ) \sqrt {2}\, \sqrt {3}+\frac {9 x}{2}\right )}{3 \sqrt {4 x^{4}-x^{2}+4}}\right )}{8}-\frac {\arctan \left (\frac {4 \sqrt {2}\, x^{2}+4 \sqrt {2}-9 x}{\sqrt {4 x^{4}-x^{2}+4}}\right )}{2}+\frac {\arctan \left (\frac {9 x +4 \sqrt {2}\, x^{2}+4 \sqrt {2}}{\sqrt {4 x^{4}-x^{2}+4}}\right )}{2}-\frac {\operatorname {arctanh}\left (\frac {9 x +2 \sqrt {10}\, x^{2}+2 \sqrt {10}}{\sqrt {4 x^{4}-x^{2}+4}}\right )}{8}+\frac {\operatorname {arctanh}\left (\frac {2 \sqrt {10}\, x^{2}+2 \sqrt {10}-9 x}{\sqrt {4 x^{4}-x^{2}+4}}\right )}{8}\) | \(224\) |
1/2*(1/4*2^(1/2)*arctanh(1/x*(4*x^4-x^2+4)^(1/2))-2^(1/2)*arctan(1/x*(4*x^ 4-x^2+4)^(1/2))+3/4*6^(1/2)*arctan(1/6*6^(1/2)*(4*x^4-x^2+4)^(1/2)*2^(1/2) /x))*2^(1/2)
Time = 0.33 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.50 \[ \int \frac {\left (-1+x^4\right ) \left (1-x^2+x^4\right ) \sqrt {4-x^2+4 x^4}}{\left (1+x^4\right ) \left (4+7 x^4+4 x^8\right )} \, dx=-\frac {3}{8} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {4 \, x^{4} - x^{2} + 4} x}{2 \, {\left (x^{4} - x^{2} + 1\right )}}\right ) + \frac {1}{2} \, \arctan \left (\frac {\sqrt {4 \, x^{4} - x^{2} + 4} x}{2 \, x^{4} - x^{2} + 2}\right ) + \frac {1}{8} \, \log \left (-\frac {2 \, x^{4} + \sqrt {4 \, x^{4} - x^{2} + 4} x + 2}{2 \, x^{4} - x^{2} + 2}\right ) \]
integrate((x^4-1)*(x^4-x^2+1)*(4*x^4-x^2+4)^(1/2)/(x^4+1)/(4*x^8+7*x^4+4), x, algorithm="fricas")
-3/8*sqrt(3)*arctan(1/2*sqrt(3)*sqrt(4*x^4 - x^2 + 4)*x/(x^4 - x^2 + 1)) + 1/2*arctan(sqrt(4*x^4 - x^2 + 4)*x/(2*x^4 - x^2 + 2)) + 1/8*log(-(2*x^4 + sqrt(4*x^4 - x^2 + 4)*x + 2)/(2*x^4 - x^2 + 2))
Timed out. \[ \int \frac {\left (-1+x^4\right ) \left (1-x^2+x^4\right ) \sqrt {4-x^2+4 x^4}}{\left (1+x^4\right ) \left (4+7 x^4+4 x^8\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-1+x^4\right ) \left (1-x^2+x^4\right ) \sqrt {4-x^2+4 x^4}}{\left (1+x^4\right ) \left (4+7 x^4+4 x^8\right )} \, dx=\int { \frac {\sqrt {4 \, x^{4} - x^{2} + 4} {\left (x^{4} - x^{2} + 1\right )} {\left (x^{4} - 1\right )}}{{\left (4 \, x^{8} + 7 \, x^{4} + 4\right )} {\left (x^{4} + 1\right )}} \,d x } \]
integrate((x^4-1)*(x^4-x^2+1)*(4*x^4-x^2+4)^(1/2)/(x^4+1)/(4*x^8+7*x^4+4), x, algorithm="maxima")
integrate(sqrt(4*x^4 - x^2 + 4)*(x^4 - x^2 + 1)*(x^4 - 1)/((4*x^8 + 7*x^4 + 4)*(x^4 + 1)), x)
\[ \int \frac {\left (-1+x^4\right ) \left (1-x^2+x^4\right ) \sqrt {4-x^2+4 x^4}}{\left (1+x^4\right ) \left (4+7 x^4+4 x^8\right )} \, dx=\int { \frac {\sqrt {4 \, x^{4} - x^{2} + 4} {\left (x^{4} - x^{2} + 1\right )} {\left (x^{4} - 1\right )}}{{\left (4 \, x^{8} + 7 \, x^{4} + 4\right )} {\left (x^{4} + 1\right )}} \,d x } \]
integrate((x^4-1)*(x^4-x^2+1)*(4*x^4-x^2+4)^(1/2)/(x^4+1)/(4*x^8+7*x^4+4), x, algorithm="giac")
integrate(sqrt(4*x^4 - x^2 + 4)*(x^4 - x^2 + 1)*(x^4 - 1)/((4*x^8 + 7*x^4 + 4)*(x^4 + 1)), x)
Timed out. \[ \int \frac {\left (-1+x^4\right ) \left (1-x^2+x^4\right ) \sqrt {4-x^2+4 x^4}}{\left (1+x^4\right ) \left (4+7 x^4+4 x^8\right )} \, dx=\int \frac {\left (x^4-1\right )\,\left (x^4-x^2+1\right )\,\sqrt {4\,x^4-x^2+4}}{\left (x^4+1\right )\,\left (4\,x^8+7\,x^4+4\right )} \,d x \]