3.4.23 \(\int \frac {2+x^2}{(-1+x^2) \sqrt {-1-x^2+x^4}} \, dx\) [323]

3.4.23.1 Optimal result

Integrand size = 27, antiderivative size = 28 \[ \int \frac {2+x^2}{\left (-1+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx=-\arctan \left (\frac {\sqrt {-1-x^2+x^4}}{x \left (-2+x^2\right )}\right ) \]

output

-arctan((x^4-x^2-1)^(1/2)/x/(x^2-2))
 

3.4.23.2 Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {2+x^2}{\left (-1+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx=-\arctan \left (\frac {\sqrt {-1-x^2+x^4}}{x \left (-2+x^2\right )}\right ) \]

input

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

output

-ArcTan[Sqrt[-1 - x^2 + x^4]/(x*(-2 + x^2))]
 

3.4.23.3 Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.86 (sec) , antiderivative size = 534, normalized size of antiderivative = 19.07, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {2228, 25, 1411, 1538, 25, 1411, 1786, 27, 415, 323, 27, 321, 413, 27, 412}

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.

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

\(\Big \downarrow \) 2228

\(\displaystyle \int \frac {1}{\sqrt {x^4-x^2-1}}dx+3 \int -\frac {1}{\left (1-x^2\right ) \sqrt {x^4-x^2-1}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle \int \frac {1}{\sqrt {x^4-x^2-1}}dx-3 \int \frac {1}{\left (1-x^2\right ) \sqrt {x^4-x^2-1}}dx\)

\(\Big \downarrow \) 1411

\(\displaystyle \frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-3 \int \frac {1}{\left (1-x^2\right ) \sqrt {x^4-x^2-1}}dx\)

\(\Big \downarrow \) 1538

\(\displaystyle \frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-3 \left (\frac {2 \int \frac {1}{\sqrt {x^4-x^2-1}}dx}{1-\sqrt {5}}+\frac {\int -\frac {-2 x^2+\sqrt {5}+1}{\left (1-x^2\right ) \sqrt {x^4-x^2-1}}dx}{1-\sqrt {5}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-3 \left (\frac {2 \int \frac {1}{\sqrt {x^4-x^2-1}}dx}{1-\sqrt {5}}-\frac {\int \frac {-2 x^2+\sqrt {5}+1}{\left (1-x^2\right ) \sqrt {x^4-x^2-1}}dx}{1-\sqrt {5}}\right )\)

\(\Big \downarrow \) 1411

\(\displaystyle \frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-3 \left (\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {\int \frac {-2 x^2+\sqrt {5}+1}{\left (1-x^2\right ) \sqrt {x^4-x^2-1}}dx}{1-\sqrt {5}}\right )\)

\(\Big \downarrow \) 1786

\(\displaystyle \frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-3 \left (\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {\sqrt {-2 x^2+\sqrt {5}+1} \sqrt {-x^2-\frac {2}{1+\sqrt {5}}} \int \frac {\sqrt {2} \sqrt {-2 x^2+\sqrt {5}+1}}{\left (1-x^2\right ) \sqrt {-x^2-\frac {2}{1+\sqrt {5}}}}dx}{\sqrt {2} \left (1-\sqrt {5}\right ) \sqrt {x^4-x^2-1}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-3 \left (\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {\sqrt {-2 x^2+\sqrt {5}+1} \sqrt {-x^2-\frac {2}{1+\sqrt {5}}} \int \frac {\sqrt {-2 x^2+\sqrt {5}+1}}{\left (1-x^2\right ) \sqrt {-x^2-\frac {2}{1+\sqrt {5}}}}dx}{\left (1-\sqrt {5}\right ) \sqrt {x^4-x^2-1}}\right )\)

\(\Big \downarrow \) 415

\(\displaystyle \frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-3 \left (\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {\sqrt {-2 x^2+\sqrt {5}+1} \sqrt {-x^2-\frac {2}{1+\sqrt {5}}} \left (2 \int \frac {1}{\sqrt {-2 x^2+\sqrt {5}+1} \sqrt {-x^2-\frac {2}{1+\sqrt {5}}}}dx-\left (1-\sqrt {5}\right ) \int \frac {1}{\sqrt {-2 x^2+\sqrt {5}+1} \left (1-x^2\right ) \sqrt {-x^2-\frac {2}{1+\sqrt {5}}}}dx\right )}{\left (1-\sqrt {5}\right ) \sqrt {x^4-x^2-1}}\right )\)

\(\Big \downarrow \) 323

\(\displaystyle \frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-3 \left (\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {\sqrt {-2 x^2+\sqrt {5}+1} \sqrt {-x^2-\frac {2}{1+\sqrt {5}}} \left (\frac {\sqrt {2} \sqrt {\left (1+\sqrt {5}\right ) x^2+2} \int \frac {\sqrt {2}}{\sqrt {-2 x^2+\sqrt {5}+1} \sqrt {\left (1+\sqrt {5}\right ) x^2+2}}dx}{\sqrt {-x^2-\frac {2}{1+\sqrt {5}}}}-\left (1-\sqrt {5}\right ) \int \frac {1}{\sqrt {-2 x^2+\sqrt {5}+1} \left (1-x^2\right ) \sqrt {-x^2-\frac {2}{1+\sqrt {5}}}}dx\right )}{\left (1-\sqrt {5}\right ) \sqrt {x^4-x^2-1}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-3 \left (\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {\sqrt {-2 x^2+\sqrt {5}+1} \sqrt {-x^2-\frac {2}{1+\sqrt {5}}} \left (\frac {2 \sqrt {\left (1+\sqrt {5}\right ) x^2+2} \int \frac {1}{\sqrt {-2 x^2+\sqrt {5}+1} \sqrt {\left (1+\sqrt {5}\right ) x^2+2}}dx}{\sqrt {-x^2-\frac {2}{1+\sqrt {5}}}}-\left (1-\sqrt {5}\right ) \int \frac {1}{\sqrt {-2 x^2+\sqrt {5}+1} \left (1-x^2\right ) \sqrt {-x^2-\frac {2}{1+\sqrt {5}}}}dx\right )}{\left (1-\sqrt {5}\right ) \sqrt {x^4-x^2-1}}\right )\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-3 \left (\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {\sqrt {-2 x^2+\sqrt {5}+1} \sqrt {-x^2-\frac {2}{1+\sqrt {5}}} \left (\frac {\sqrt {\left (1+\sqrt {5}\right ) x^2+2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {-x^2-\frac {2}{1+\sqrt {5}}}}-\left (1-\sqrt {5}\right ) \int \frac {1}{\sqrt {-2 x^2+\sqrt {5}+1} \left (1-x^2\right ) \sqrt {-x^2-\frac {2}{1+\sqrt {5}}}}dx\right )}{\left (1-\sqrt {5}\right ) \sqrt {x^4-x^2-1}}\right )\)

\(\Big \downarrow \) 413

\(\displaystyle \frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-3 \left (\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {\sqrt {-2 x^2+\sqrt {5}+1} \sqrt {-x^2-\frac {2}{1+\sqrt {5}}} \left (\frac {\sqrt {\left (1+\sqrt {5}\right ) x^2+2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {-x^2-\frac {2}{1+\sqrt {5}}}}-\frac {\left (1-\sqrt {5}\right ) \sqrt {\left (1+\sqrt {5}\right ) x^2+2} \int \frac {\sqrt {2}}{\sqrt {-2 x^2+\sqrt {5}+1} \left (1-x^2\right ) \sqrt {\left (1+\sqrt {5}\right ) x^2+2}}dx}{\sqrt {2} \sqrt {-x^2-\frac {2}{1+\sqrt {5}}}}\right )}{\left (1-\sqrt {5}\right ) \sqrt {x^4-x^2-1}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-3 \left (\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {\sqrt {-2 x^2+\sqrt {5}+1} \sqrt {-x^2-\frac {2}{1+\sqrt {5}}} \left (\frac {\sqrt {\left (1+\sqrt {5}\right ) x^2+2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {-x^2-\frac {2}{1+\sqrt {5}}}}-\frac {\left (1-\sqrt {5}\right ) \sqrt {\left (1+\sqrt {5}\right ) x^2+2} \int \frac {1}{\sqrt {-2 x^2+\sqrt {5}+1} \left (1-x^2\right ) \sqrt {\left (1+\sqrt {5}\right ) x^2+2}}dx}{\sqrt {-x^2-\frac {2}{1+\sqrt {5}}}}\right )}{\left (1-\sqrt {5}\right ) \sqrt {x^4-x^2-1}}\right )\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-3 \left (\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x^2+2}{\left (1-\sqrt {5}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x^2\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x^2+2}} \sqrt {x^4-x^2-1}}-\frac {\sqrt {-2 x^2+\sqrt {5}+1} \sqrt {-x^2-\frac {2}{1+\sqrt {5}}} \left (\frac {\sqrt {\left (1+\sqrt {5}\right ) x^2+2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {-x^2-\frac {2}{1+\sqrt {5}}}}-\frac {\left (1-\sqrt {5}\right ) \sqrt {\left (1+\sqrt {5}\right ) x^2+2} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1+\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {-x^2-\frac {2}{1+\sqrt {5}}}}\right )}{\left (1-\sqrt {5}\right ) \sqrt {x^4-x^2-1}}\right )\)

input

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

output

(Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[ 
5])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2 
]], (5 - Sqrt[5])/10])/(2*5^(1/4)*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[ 
-1 - x^2 + x^4]) - 3*((Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5] 
)*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[ 
-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(5^(1/4)*(1 - Sqrt[5])*Sqrt[( 
2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 + x^4]) - (Sqrt[1 + Sqrt[5] - 2 
*x^2]*Sqrt[-2/(1 + Sqrt[5]) - x^2]*((Sqrt[2 + (1 + Sqrt[5])*x^2]*EllipticF 
[ArcSin[Sqrt[2/(1 + Sqrt[5])]*x], (-3 - Sqrt[5])/2])/Sqrt[-2/(1 + Sqrt[5]) 
 - x^2] - ((1 - Sqrt[5])*Sqrt[2 + (1 + Sqrt[5])*x^2]*EllipticPi[(1 + Sqrt[ 
5])/2, ArcSin[Sqrt[2/(1 + Sqrt[5])]*x], (-3 - Sqrt[5])/2])/(2*Sqrt[-2/(1 + 
 Sqrt[5]) - x^2])))/((1 - Sqrt[5])*Sqrt[-1 - x^2 + x^4]))
 

3.4.23.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 323

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( 
d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 413

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/((a + 
 b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] &&  !GtQ[c, 0]
 

rule 415

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_) 
^2]), x_Symbol] :> Simp[d/b   Int[1/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], 
x] + Simp[(b*c - a*d)/b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2 
]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NegQ[d/c]
 

rule 1411

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b 
^2 - 4*a*c, 2]}, Simp[Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]*(Sqrt[( 
2*a + (b + q)*x^2)/q]/(2*Sqrt[a + b*x^2 + c*x^4]*Sqrt[a/(2*a + (b + q)*x^2) 
]))*EllipticF[ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b + q)/(2*q)], x] 
] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]
 

rule 1538

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_S 
ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/(2*c*d - e*(b - q)))   I 
nt[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[e/(2*c*d - e*(b - q))   Int[(b 
- q + 2*c*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, 
b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 !LtQ[c, 0]
 

rule 1786

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + ( 
b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[(a + b*x^n + c*x 
^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p]) 
   Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + (c/e)*x^n)^p, x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, n, p, q, r}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c 
, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p]
 

rule 2228

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> Simp[B/e   Int[1/Sqrt[a + b*x^2 + c*x^4], x], 
 x] + Simp[(e*A - d*B)/e   Int[1/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], 
 x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && NegQ[c/a 
]
 

3.4.23.4 Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 2.16 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.43

method	result	size
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \sqrt {x^{4}-x^{2}-1}\, x^{3}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}-4 x \sqrt {x^{4}-x^{2}-1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (x -1\right )^{3} \left (1+x \right )^{3}}\right )}{2}\)	\(96\)
default	\(\frac {2 \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-2-2 \sqrt {5}}}{2}, \frac {i \sqrt {5}}{2}-\frac {i}{2}\right )}{\sqrt {-2-2 \sqrt {5}}\, \sqrt {x^{4}-x^{2}-1}}-\frac {3 \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) x^{2}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}\, x , \frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}}{\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}}\right )}{\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}\, \sqrt {x^{4}-x^{2}-1}}\)	\(174\)
elliptic	\(\frac {2 \sqrt {1+\frac {x^{2}}{2}+\frac {\sqrt {5}\, x^{2}}{2}}\, \sqrt {1-\frac {\sqrt {5}\, x^{2}}{2}+\frac {x^{2}}{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-2-2 \sqrt {5}}}{2}, \frac {i \sqrt {5}}{2}-\frac {i}{2}\right )}{\sqrt {-2-2 \sqrt {5}}\, \sqrt {x^{4}-x^{2}-1}}-\frac {3 \sqrt {1+\frac {x^{2}}{2}+\frac {\sqrt {5}\, x^{2}}{2}}\, \sqrt {1-\frac {\sqrt {5}\, x^{2}}{2}+\frac {x^{2}}{2}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}\, x , \frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}}{\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}}\right )}{\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}\, \sqrt {x^{4}-x^{2}-1}}\)	\(178\)

input

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

output

1/2*RootOf(_Z^2+1)*ln(-(-RootOf(_Z^2+1)*x^6+5*RootOf(_Z^2+1)*x^4+2*(x^4-x^ 
2-1)^(1/2)*x^3-5*RootOf(_Z^2+1)*x^2-4*x*(x^4-x^2-1)^(1/2)-RootOf(_Z^2+1))/ 
(x-1)^3/(1+x)^3)
 

3.4.23.5 Fricas [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {2+x^2}{\left (-1+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx=-\frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {x^{4} - x^{2} - 1} {\left (x^{3} - 2 \, x\right )}}{x^{6} - 5 \, x^{4} + 5 \, x^{2} + 1}\right ) \]

input

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

output

-1/2*arctan(2*sqrt(x^4 - x^2 - 1)*(x^3 - 2*x)/(x^6 - 5*x^4 + 5*x^2 + 1))
 

3.4.23.6 Sympy [F]

\[ \int \frac {2+x^2}{\left (-1+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx=\int \frac {x^{2} + 2}{\left (x - 1\right ) \left (x + 1\right ) \sqrt {x^{4} - x^{2} - 1}}\, dx \]

input

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

output

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

3.4.23.7 Maxima [F]

\[ \int \frac {2+x^2}{\left (-1+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx=\int { \frac {x^{2} + 2}{\sqrt {x^{4} - x^{2} - 1} {\left (x^{2} - 1\right )}} \,d x } \]

input

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

output

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

3.4.23.8 Giac [F]

\[ \int \frac {2+x^2}{\left (-1+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx=\int { \frac {x^{2} + 2}{\sqrt {x^{4} - x^{2} - 1} {\left (x^{2} - 1\right )}} \,d x } \]

input

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

output

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

3.4.23.9 Mupad [F(-1)]

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

input

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

output

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