Integrand size = 27, antiderivative size = 186 \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{4+3 x^4+x^8} \, dx=-4 \sqrt [4]{2} \text {RootSum}\left [1-4 \text {$\#$1}^2-122 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (-\sqrt {2}+2^{3/4} x-x^2\right ) \text {$\#$1}+\log \left (\sqrt {2+x^4}+\sqrt {2} \text {$\#$1}-2^{3/4} x \text {$\#$1}+x^2 \text {$\#$1}\right ) \text {$\#$1}-\log \left (-\sqrt {2}+2^{3/4} x-x^2\right ) \text {$\#$1}^3+\log \left (\sqrt {2+x^4}+\sqrt {2} \text {$\#$1}-2^{3/4} x \text {$\#$1}+x^2 \text {$\#$1}\right ) \text {$\#$1}^3}{-1-61 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\&\right ] \]
Time = 0.48 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.18 \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{4+3 x^4+x^8} \, dx=-\frac {1}{2} \arctan \left (\frac {x}{\sqrt {2+x^4}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt {2+x^4}}\right ) \]
Result contains complex when optimal does not.
Time = 2.77 (sec) , antiderivative size = 1559, normalized size of antiderivative = 8.38, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, 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.
| \(\displaystyle \int \frac {\left (x^4-2\right ) \sqrt {x^4+2}}{x^8+3 x^4+4} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\left (1+i \sqrt {7}\right ) \sqrt {x^4+2}}{2 x^4-i \sqrt {7}+3}+\frac {\left (1-i \sqrt {7}\right ) \sqrt {x^4+2}}{2 x^4+i \sqrt {7}+3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {1-i \sqrt {7}} \left (5 i+\sqrt {7}\right ) \arctan \left (\frac {\sqrt {1-i \sqrt {7}} x}{\sqrt [4]{2 \left (-3-i \sqrt {7}\right )} \sqrt {x^4+2}}\right )}{2 \left (3 i-\sqrt {7}\right ) \left (2 \left (-3-i \sqrt {7}\right )\right )^{3/4}}-\frac {\left (5 i-\sqrt {7}\right ) \sqrt {1+i \sqrt {7}} \arctan \left (\frac {\sqrt {1+i \sqrt {7}} x}{\sqrt [4]{2 \left (-3+i \sqrt {7}\right )} \sqrt {x^4+2}}\right )}{2 \left (2 \left (-3+i \sqrt {7}\right )\right )^{3/4} \left (3 i+\sqrt {7}\right )}-\frac {\sqrt {1-i \sqrt {7}} \left (5 i+\sqrt {7}\right ) \text {arctanh}\left (\frac {\sqrt {1-i \sqrt {7}} x}{\sqrt [4]{2 \left (-3-i \sqrt {7}\right )} \sqrt {x^4+2}}\right )}{2 \left (3 i-\sqrt {7}\right ) \left (2 \left (-3-i \sqrt {7}\right )\right )^{3/4}}-\frac {\left (5 i-\sqrt {7}\right ) \sqrt {1+i \sqrt {7}} \text {arctanh}\left (\frac {\sqrt {1+i \sqrt {7}} x}{\sqrt [4]{2 \left (-3+i \sqrt {7}\right )} \sqrt {x^4+2}}\right )}{2 \left (2 \left (-3+i \sqrt {7}\right )\right )^{3/4} \left (3 i+\sqrt {7}\right )}+\frac {\left (1+i \sqrt {7}\right )^2 \left (1+\frac {2}{\sqrt {-3+i \sqrt {7}}}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{2} \left (7-i \sqrt {7}\right ) \sqrt {x^4+2}}+\frac {\left (1+i \sqrt {7}\right )^2 \left (1-\frac {2}{\sqrt {-3+i \sqrt {7}}}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{2} \left (7-i \sqrt {7}\right ) \sqrt {x^4+2}}+\frac {\left (1-i \sqrt {7}\right )^2 \left (1+\frac {2}{\sqrt {-3-i \sqrt {7}}}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{2} \left (7+i \sqrt {7}\right ) \sqrt {x^4+2}}+\frac {\left (1-i \sqrt {7}\right )^2 \left (1-\frac {2}{\sqrt {-3-i \sqrt {7}}}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{2} \left (7+i \sqrt {7}\right ) \sqrt {x^4+2}}+\frac {\left (1+i \sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {x^4+2}}+\frac {\left (1-i \sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {x^4+2}}-\frac {\left (3 i-\sqrt {7}\right ) \left (2+\sqrt {-3-i \sqrt {7}}\right )^2 \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (2-\sqrt {-3-i \sqrt {7}}\right )^2}{8 \sqrt {-3-i \sqrt {7}}},2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{16 \sqrt [4]{2} \left (7 i-5 \sqrt {7}\right ) \sqrt {x^4+2}}-\frac {\left (3 i-\sqrt {7}\right ) \left (2-\sqrt {-3-i \sqrt {7}}\right )^2 \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (2+\sqrt {-3-i \sqrt {7}}\right )^2}{8 \sqrt {-3-i \sqrt {7}}},2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{16 \sqrt [4]{2} \left (7 i-5 \sqrt {7}\right ) \sqrt {x^4+2}}-\frac {\left (3 i+\sqrt {7}\right ) \left (2+\sqrt {-3+i \sqrt {7}}\right )^2 \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (2-\sqrt {-3+i \sqrt {7}}\right )^2}{8 \sqrt {-3+i \sqrt {7}}},2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{16 \sqrt [4]{2} \left (7 i+5 \sqrt {7}\right ) \sqrt {x^4+2}}-\frac {\left (3 i+\sqrt {7}\right ) \left (2-\sqrt {-3+i \sqrt {7}}\right )^2 \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (2+\sqrt {-3+i \sqrt {7}}\right )^2}{8 \sqrt {-3+i \sqrt {7}}},2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{16 \sqrt [4]{2} \left (7 i+5 \sqrt {7}\right ) \sqrt {x^4+2}}\) |
-1/2*(Sqrt[1 - I*Sqrt[7]]*(5*I + Sqrt[7])*ArcTan[(Sqrt[1 - I*Sqrt[7]]*x)/( (2*(-3 - I*Sqrt[7]))^(1/4)*Sqrt[2 + x^4])])/((3*I - Sqrt[7])*(2*(-3 - I*Sq rt[7]))^(3/4)) - ((5*I - Sqrt[7])*Sqrt[1 + I*Sqrt[7]]*ArcTan[(Sqrt[1 + I*S qrt[7]]*x)/((2*(-3 + I*Sqrt[7]))^(1/4)*Sqrt[2 + x^4])])/(2*(2*(-3 + I*Sqrt [7]))^(3/4)*(3*I + Sqrt[7])) - (Sqrt[1 - I*Sqrt[7]]*(5*I + Sqrt[7])*ArcTan h[(Sqrt[1 - I*Sqrt[7]]*x)/((2*(-3 - I*Sqrt[7]))^(1/4)*Sqrt[2 + x^4])])/(2* (3*I - Sqrt[7])*(2*(-3 - I*Sqrt[7]))^(3/4)) - ((5*I - Sqrt[7])*Sqrt[1 + I* Sqrt[7]]*ArcTanh[(Sqrt[1 + I*Sqrt[7]]*x)/((2*(-3 + I*Sqrt[7]))^(1/4)*Sqrt[ 2 + x^4])])/(2*(2*(-3 + I*Sqrt[7]))^(3/4)*(3*I + Sqrt[7])) + ((1 - I*Sqrt[ 7])*(Sqrt[2] + x^2)*Sqrt[(2 + x^4)/(Sqrt[2] + x^2)^2]*EllipticF[2*ArcTan[x /2^(1/4)], 1/2])/(4*2^(1/4)*Sqrt[2 + x^4]) + ((1 + I*Sqrt[7])*(Sqrt[2] + x ^2)*Sqrt[(2 + x^4)/(Sqrt[2] + x^2)^2]*EllipticF[2*ArcTan[x/2^(1/4)], 1/2]) /(4*2^(1/4)*Sqrt[2 + x^4]) + ((1 - I*Sqrt[7])^2*(1 - 2/Sqrt[-3 - I*Sqrt[7] ])*(Sqrt[2] + x^2)*Sqrt[(2 + x^4)/(Sqrt[2] + x^2)^2]*EllipticF[2*ArcTan[x/ 2^(1/4)], 1/2])/(8*2^(1/4)*(7 + I*Sqrt[7])*Sqrt[2 + x^4]) + ((1 - I*Sqrt[7 ])^2*(1 + 2/Sqrt[-3 - I*Sqrt[7]])*(Sqrt[2] + x^2)*Sqrt[(2 + x^4)/(Sqrt[2] + x^2)^2]*EllipticF[2*ArcTan[x/2^(1/4)], 1/2])/(8*2^(1/4)*(7 + I*Sqrt[7])* Sqrt[2 + x^4]) + ((1 + I*Sqrt[7])^2*(1 - 2/Sqrt[-3 + I*Sqrt[7]])*(Sqrt[2] + x^2)*Sqrt[(2 + x^4)/(Sqrt[2] + x^2)^2]*EllipticF[2*ArcTan[x/2^(1/4)], 1/ 2])/(8*2^(1/4)*(7 - I*Sqrt[7])*Sqrt[2 + x^4]) + ((1 + I*Sqrt[7])^2*(1 +...
3.24.50.3.1 Defintions of rubi rules used
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]
Time = 5.33 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.22
| method | result | size |
| elliptic | \(\frac {\left (\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}+2}}{x}\right )}{2}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{4}+2}}{x}\right )}{2}\right ) \sqrt {2}}{2}\) | \(41\) |
| trager | \(-\frac {\ln \left (-\frac {x^{4}+2 x \sqrt {x^{4}+2}+x^{2}+2}{x^{4}-x^{2}+2}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {x^{4}+2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}+x^{2}+2}\right )}{4}\) | \(98\) |
| default | \(\frac {\operatorname {arctanh}\left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {1+2 \sqrt {2}}+2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{4}-\frac {\arctan \left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {-1+2 \sqrt {2}}+2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{4}+\frac {\arctan \left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {-1+2 \sqrt {2}}-2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{4}-\frac {\operatorname {arctanh}\left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {1+2 \sqrt {2}}-2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{4}\) | \(142\) |
| pseudoelliptic | \(\frac {\operatorname {arctanh}\left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {1+2 \sqrt {2}}+2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{4}-\frac {\arctan \left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {-1+2 \sqrt {2}}+2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{4}+\frac {\arctan \left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {-1+2 \sqrt {2}}-2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{4}-\frac {\operatorname {arctanh}\left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {1+2 \sqrt {2}}-2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{4}\) | \(142\) |
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.32 \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{4+3 x^4+x^8} \, dx=-\frac {1}{4} \, \arctan \left (\frac {2 \, \sqrt {x^{4} + 2} x}{x^{4} - x^{2} + 2}\right ) + \frac {1}{4} \, \log \left (\frac {x^{4} + x^{2} - 2 \, \sqrt {x^{4} + 2} x + 2}{x^{4} - x^{2} + 2}\right ) \]
-1/4*arctan(2*sqrt(x^4 + 2)*x/(x^4 - x^2 + 2)) + 1/4*log((x^4 + x^2 - 2*sq rt(x^4 + 2)*x + 2)/(x^4 - x^2 + 2))
Timed out. \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{4+3 x^4+x^8} \, dx=\text {Timed out} \]
Not integrable
Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{4+3 x^4+x^8} \, dx=\int { \frac {\sqrt {x^{4} + 2} {\left (x^{4} - 2\right )}}{x^{8} + 3 \, x^{4} + 4} \,d x } \]
Not integrable
Time = 1.65 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{4+3 x^4+x^8} \, dx=\int { \frac {\sqrt {x^{4} + 2} {\left (x^{4} - 2\right )}}{x^{8} + 3 \, x^{4} + 4} \,d x } \]
Not integrable
Time = 6.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{4+3 x^4+x^8} \, dx=\int \frac {\left (x^4-2\right )\,\sqrt {x^4+2}}{x^8+3\,x^4+4} \,d x \]