Integrand size = 27, antiderivative size = 53 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2} \, dx=-\frac {x \sqrt {1+x^4}}{2 \left (1+3 x^2+x^4\right )}-\frac {\arctan \left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{2 \sqrt {3}} \]
Time = 0.68 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2} \, dx=-\frac {x \sqrt {1+x^4}}{2 \left (1+3 x^2+x^4\right )}-\frac {\arctan \left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{2 \sqrt {3}} \]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.74 (sec) , antiderivative size = 743, normalized size of antiderivative = 14.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {7293, 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.
| \(\displaystyle \int \frac {\left (x^4-1\right ) \sqrt {x^4+1}}{\left (x^4+3 x^2+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4+1} \left (-3 x^2-2\right )}{\left (x^4+3 x^2+1\right )^2}+\frac {\sqrt {x^4+1}}{x^4+3 x^2+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {3} x}{\sqrt {x^4+1}}\right )}{\sqrt {15} \left (3+\sqrt {5}\right )}-\frac {\arctan \left (\frac {\sqrt {3} x}{\sqrt {x^4+1}}\right )}{\sqrt {15} \left (3-\sqrt {5}\right )}+\frac {2 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{5 \left (3+\sqrt {5}\right ) \sqrt {x^4+1}}+\frac {2 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{5 \left (3-\sqrt {5}\right ) \sqrt {x^4+1}}-\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{5 \sqrt {x^4+1}}-\frac {2 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{2}\right .\right )}{5 \left (3+\sqrt {5}\right ) \sqrt {x^4+1}}-\frac {2 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{2}\right .\right )}{5 \left (3-\sqrt {5}\right ) \sqrt {x^4+1}}+\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{2}\right .\right )}{5 \sqrt {x^4+1}}-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {1}{4},2 \arctan (x),\frac {1}{2}\right )}{2 \left (3+\sqrt {5}\right ) \sqrt {x^4+1}}-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {1}{4},2 \arctan (x),\frac {1}{2}\right )}{2 \left (3-\sqrt {5}\right ) \sqrt {x^4+1}}+\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {1}{4},2 \arctan (x),\frac {1}{2}\right )}{4 \sqrt {x^4+1}}+\frac {2 \sqrt {x^4+1} x}{5 \left (3+\sqrt {5}\right ) \left (x^2+1\right )}+\frac {2 \sqrt {x^4+1} x}{5 \left (3-\sqrt {5}\right ) \left (x^2+1\right )}-\frac {3 \sqrt {x^4+1} x}{5 \left (x^2+1\right )}-\frac {4 \sqrt {x^4+1} x}{5 \left (3-\sqrt {5}\right ) \left (2 x^2-\sqrt {5}+3\right )}+\frac {3 \sqrt {x^4+1} x}{5 \left (2 x^2-\sqrt {5}+3\right )}-\frac {4 \sqrt {x^4+1} x}{5 \left (3+\sqrt {5}\right ) \left (2 x^2+\sqrt {5}+3\right )}+\frac {3 \sqrt {x^4+1} x}{5 \left (2 x^2+\sqrt {5}+3\right )}\) |
(-3*x*Sqrt[1 + x^4])/(5*(1 + x^2)) + (2*x*Sqrt[1 + x^4])/(5*(3 - Sqrt[5])* (1 + x^2)) + (2*x*Sqrt[1 + x^4])/(5*(3 + Sqrt[5])*(1 + x^2)) + (3*x*Sqrt[1 + x^4])/(5*(3 - Sqrt[5] + 2*x^2)) - (4*x*Sqrt[1 + x^4])/(5*(3 - Sqrt[5])* (3 - Sqrt[5] + 2*x^2)) + (3*x*Sqrt[1 + x^4])/(5*(3 + Sqrt[5] + 2*x^2)) - ( 4*x*Sqrt[1 + x^4])/(5*(3 + Sqrt[5])*(3 + Sqrt[5] + 2*x^2)) - ArcTan[(Sqrt[ 3]*x)/Sqrt[1 + x^4]]/(Sqrt[15]*(3 - Sqrt[5])) + ArcTan[(Sqrt[3]*x)/Sqrt[1 + x^4]]/(Sqrt[15]*(3 + Sqrt[5])) + (3*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2 ]*EllipticE[2*ArcTan[x], 1/2])/(5*Sqrt[1 + x^4]) - (2*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/2])/(5*(3 - Sqrt[5])*Sqrt[1 + x ^4]) - (2*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/2 ])/(5*(3 + Sqrt[5])*Sqrt[1 + x^4]) - (3*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2) ^2]*EllipticF[2*ArcTan[x], 1/2])/(5*Sqrt[1 + x^4]) + (2*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(5*(3 - Sqrt[5])*Sqrt[1 + x^4]) + (2*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1 /2])/(5*(3 + Sqrt[5])*Sqrt[1 + x^4]) + (3*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^ 2)^2]*EllipticPi[-1/4, 2*ArcTan[x], 1/2])/(4*Sqrt[1 + x^4]) - ((1 + x^2)*S qrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[-1/4, 2*ArcTan[x], 1/2])/(2*(3 - Sqr t[5])*Sqrt[1 + x^4]) - ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[- 1/4, 2*ArcTan[x], 1/2])/(2*(3 + Sqrt[5])*Sqrt[1 + x^4])
3.7.71.3.1 Defintions of rubi rules used
Time = 4.88 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.85
| method | result | size |
| risch | \(\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \sqrt {x^{4}+1}}{3 x}\right )}{6}-\frac {x \sqrt {x^{4}+1}}{2 \left (x^{4}+3 x^{2}+1\right )}\) | \(45\) |
| default | \(\frac {\sqrt {3}\, \left (x^{4}+3 x^{2}+1\right ) \arctan \left (\frac {\sqrt {3}\, \sqrt {x^{4}+1}}{3 x}\right )-3 \sqrt {x^{4}+1}\, x}{6 x^{4}+18 x^{2}+6}\) | \(57\) |
| pseudoelliptic | \(\frac {\sqrt {3}\, \left (x^{4}+3 x^{2}+1\right ) \arctan \left (\frac {\sqrt {3}\, \sqrt {x^{4}+1}}{3 x}\right )-3 \sqrt {x^{4}+1}\, x}{6 x^{4}+18 x^{2}+6}\) | \(57\) |
| elliptic | \(\frac {\left (-\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{4 x \left (\frac {x^{4}+1}{2 x^{2}}+\frac {3}{2}\right )}+\frac {\sqrt {6}\, \arctan \left (\frac {\sqrt {6}\, \sqrt {2}\, \sqrt {x^{4}+1}}{6 x}\right )}{6}\right ) \sqrt {2}}{2}\) | \(60\) |
| trager | \(-\frac {x \sqrt {x^{4}+1}}{2 \left (x^{4}+3 x^{2}+1\right )}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{4}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{2}+6 \sqrt {x^{4}+1}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{x^{4}+3 x^{2}+1}\right )}{12}\) | \(85\) |
Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.23 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2} \, dx=-\frac {\sqrt {3} {\left (x^{4} + 3 \, x^{2} + 1\right )} \arctan \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 1} x}{x^{4} - 3 \, x^{2} + 1}\right ) + 6 \, \sqrt {x^{4} + 1} x}{12 \, {\left (x^{4} + 3 \, x^{2} + 1\right )}} \]
-1/12*(sqrt(3)*(x^4 + 3*x^2 + 1)*arctan(2*sqrt(3)*sqrt(x^4 + 1)*x/(x^4 - 3 *x^2 + 1)) + 6*sqrt(x^4 + 1)*x)/(x^4 + 3*x^2 + 1)
\[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt {x^{4} + 1}}{\left (x^{4} + 3 x^{2} + 1\right )^{2}}\, dx \]
\[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2} \, dx=\int { \frac {\sqrt {x^{4} + 1} {\left (x^{4} - 1\right )}}{{\left (x^{4} + 3 \, x^{2} + 1\right )}^{2}} \,d x } \]
\[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2} \, dx=\int { \frac {\sqrt {x^{4} + 1} {\left (x^{4} - 1\right )}}{{\left (x^{4} + 3 \, x^{2} + 1\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2} \, dx=\int \frac {\left (x^4-1\right )\,\sqrt {x^4+1}}{{\left (x^4+3\,x^2+1\right )}^2} \,d x \]