Integrand size = 27, antiderivative size = 27 \[ \int \frac {-1+2 x}{(1+x) \sqrt {-x-x^2+x^3}} \, dx=2 \arctan \left (\frac {\sqrt {-x-x^2+x^3}}{1+2 x}\right ) \]
Result contains complex when optimal does not.
Time = 4.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.48 \[ \int \frac {-1+2 x}{(1+x) \sqrt {-x-x^2+x^3}} \, dx=-\frac {i \sqrt {x} \sqrt {-1-x+x^2} \log \left (\frac {-i-2 i x+\sqrt {x} \sqrt {-1-x+x^2}}{-i-2 i x-\sqrt {x} \sqrt {-1-x+x^2}}\right )}{\sqrt {x \left (-1-x+x^2\right )}} \]
((-I)*Sqrt[x]*Sqrt[-1 - x + x^2]*Log[(-I - (2*I)*x + Sqrt[x]*Sqrt[-1 - x + x^2])/(-I - (2*I)*x - Sqrt[x]*Sqrt[-1 - x + x^2])])/Sqrt[x*(-1 - x + x^2) ]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.05 (sec) , antiderivative size = 541, normalized size of antiderivative = 20.04, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {2467, 25, 2035, 2228, 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.
| \(\displaystyle \int \frac {2 x-1}{(x+1) \sqrt {x^3-x^2-x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {x^2-x-1} \int -\frac {1-2 x}{\sqrt {x} (x+1) \sqrt {x^2-x-1}}dx}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {x^2-x-1} \int \frac {1-2 x}{\sqrt {x} (x+1) \sqrt {x^2-x-1}}dx}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \int \frac {1-2 x}{(x+1) \sqrt {x^2-x-1}}d\sqrt {x}}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 2228 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \left (3 \int \frac {1}{(x+1) \sqrt {x^2-x-1}}d\sqrt {x}-2 \int \frac {1}{\sqrt {x^2-x-1}}d\sqrt {x}\right )}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 1411 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \left (3 \int \frac {1}{(x+1) \sqrt {x^2-x-1}}d\sqrt {x}-\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 1538 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \left (3 \left (\frac {2 \int \frac {1}{\sqrt {x^2-x-1}}d\sqrt {x}}{3+\sqrt {5}}-\frac {\int -\frac {-2 x+\sqrt {5}+1}{(x+1) \sqrt {x^2-x-1}}d\sqrt {x}}{3+\sqrt {5}}\right )-\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \left (3 \left (\frac {2 \int \frac {1}{\sqrt {x^2-x-1}}d\sqrt {x}}{3+\sqrt {5}}+\frac {\int \frac {-2 x+\sqrt {5}+1}{(x+1) \sqrt {x^2-x-1}}d\sqrt {x}}{3+\sqrt {5}}\right )-\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 1411 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \left (3 \left (\frac {\int \frac {-2 x+\sqrt {5}+1}{(x+1) \sqrt {x^2-x-1}}d\sqrt {x}}{3+\sqrt {5}}+\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )-\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 1786 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \left (3 \left (\frac {\sqrt {-2 x+\sqrt {5}+1} \sqrt {-x-\frac {2}{1+\sqrt {5}}} \int \frac {\sqrt {2} \sqrt {-2 x+\sqrt {5}+1}}{\sqrt {-x-\frac {2}{1+\sqrt {5}}} (x+1)}d\sqrt {x}}{\sqrt {2} \left (3+\sqrt {5}\right ) \sqrt {x^2-x-1}}+\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )-\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \left (3 \left (\frac {\sqrt {-2 x+\sqrt {5}+1} \sqrt {-x-\frac {2}{1+\sqrt {5}}} \int \frac {\sqrt {-2 x+\sqrt {5}+1}}{\sqrt {-x-\frac {2}{1+\sqrt {5}}} (x+1)}d\sqrt {x}}{\left (3+\sqrt {5}\right ) \sqrt {x^2-x-1}}+\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )-\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 415 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \left (3 \left (\frac {\sqrt {-2 x+\sqrt {5}+1} \sqrt {-x-\frac {2}{1+\sqrt {5}}} \left (\left (3+\sqrt {5}\right ) \int \frac {1}{\sqrt {-2 x+\sqrt {5}+1} \sqrt {-x-\frac {2}{1+\sqrt {5}}} (x+1)}d\sqrt {x}-2 \int \frac {1}{\sqrt {-2 x+\sqrt {5}+1} \sqrt {-x-\frac {2}{1+\sqrt {5}}}}d\sqrt {x}\right )}{\left (3+\sqrt {5}\right ) \sqrt {x^2-x-1}}+\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )-\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \left (3 \left (\frac {\sqrt {-2 x+\sqrt {5}+1} \sqrt {-x-\frac {2}{1+\sqrt {5}}} \left (\left (3+\sqrt {5}\right ) \int \frac {1}{\sqrt {-2 x+\sqrt {5}+1} \sqrt {-x-\frac {2}{1+\sqrt {5}}} (x+1)}d\sqrt {x}-\frac {\sqrt {2} \sqrt {\left (1+\sqrt {5}\right ) x+2} \int \frac {\sqrt {2}}{\sqrt {-2 x+\sqrt {5}+1} \sqrt {\left (1+\sqrt {5}\right ) x+2}}d\sqrt {x}}{\sqrt {-x-\frac {2}{1+\sqrt {5}}}}\right )}{\left (3+\sqrt {5}\right ) \sqrt {x^2-x-1}}+\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )-\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \left (3 \left (\frac {\sqrt {-2 x+\sqrt {5}+1} \sqrt {-x-\frac {2}{1+\sqrt {5}}} \left (\left (3+\sqrt {5}\right ) \int \frac {1}{\sqrt {-2 x+\sqrt {5}+1} \sqrt {-x-\frac {2}{1+\sqrt {5}}} (x+1)}d\sqrt {x}-\frac {2 \sqrt {\left (1+\sqrt {5}\right ) x+2} \int \frac {1}{\sqrt {-2 x+\sqrt {5}+1} \sqrt {\left (1+\sqrt {5}\right ) x+2}}d\sqrt {x}}{\sqrt {-x-\frac {2}{1+\sqrt {5}}}}\right )}{\left (3+\sqrt {5}\right ) \sqrt {x^2-x-1}}+\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )-\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \left (3 \left (\frac {\sqrt {-2 x+\sqrt {5}+1} \sqrt {-x-\frac {2}{1+\sqrt {5}}} \left (\left (3+\sqrt {5}\right ) \int \frac {1}{\sqrt {-2 x+\sqrt {5}+1} \sqrt {-x-\frac {2}{1+\sqrt {5}}} (x+1)}d\sqrt {x}-\frac {\sqrt {\left (1+\sqrt {5}\right ) x+2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {-x-\frac {2}{1+\sqrt {5}}}}\right )}{\left (3+\sqrt {5}\right ) \sqrt {x^2-x-1}}+\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )-\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \left (3 \left (\frac {\sqrt {-2 x+\sqrt {5}+1} \sqrt {-x-\frac {2}{1+\sqrt {5}}} \left (\frac {\left (3+\sqrt {5}\right ) \sqrt {\left (1+\sqrt {5}\right ) x+2} \int \frac {\sqrt {2}}{\sqrt {-2 x+\sqrt {5}+1} (x+1) \sqrt {\left (1+\sqrt {5}\right ) x+2}}d\sqrt {x}}{\sqrt {2} \sqrt {-x-\frac {2}{1+\sqrt {5}}}}-\frac {\sqrt {\left (1+\sqrt {5}\right ) x+2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {-x-\frac {2}{1+\sqrt {5}}}}\right )}{\left (3+\sqrt {5}\right ) \sqrt {x^2-x-1}}+\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )-\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \left (3 \left (\frac {\sqrt {-2 x+\sqrt {5}+1} \sqrt {-x-\frac {2}{1+\sqrt {5}}} \left (\frac {\left (3+\sqrt {5}\right ) \sqrt {\left (1+\sqrt {5}\right ) x+2} \int \frac {1}{\sqrt {-2 x+\sqrt {5}+1} (x+1) \sqrt {\left (1+\sqrt {5}\right ) x+2}}d\sqrt {x}}{\sqrt {-x-\frac {2}{1+\sqrt {5}}}}-\frac {\sqrt {\left (1+\sqrt {5}\right ) x+2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {-x-\frac {2}{1+\sqrt {5}}}}\right )}{\left (3+\sqrt {5}\right ) \sqrt {x^2-x-1}}+\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )-\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \left (3 \left (\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}+\frac {\sqrt {-2 x+\sqrt {5}+1} \sqrt {-x-\frac {2}{1+\sqrt {5}}} \left (\frac {\left (3+\sqrt {5}\right ) \sqrt {\left (1+\sqrt {5}\right ) x+2} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-1-\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {-x-\frac {2}{1+\sqrt {5}}}}-\frac {\sqrt {\left (1+\sqrt {5}\right ) x+2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {-x-\frac {2}{1+\sqrt {5}}}}\right )}{\left (3+\sqrt {5}\right ) \sqrt {x^2-x-1}}\right )-\frac {\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}\right )}{\sqrt {x^3-x^2-x}}\) |
(-2*Sqrt[x]*Sqrt[-1 - x + x^2]*(-((Sqrt[-2 - (1 - Sqrt[5])*x]*Sqrt[(2 + (1 + Sqrt[5])*x)/(2 + (1 - Sqrt[5])*x)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*Sq rt[x])/Sqrt[-2 - (1 - Sqrt[5])*x]], (5 - Sqrt[5])/10])/(5^(1/4)*Sqrt[(2 + (1 - Sqrt[5])*x)^(-1)]*Sqrt[-1 - x + x^2])) + 3*((Sqrt[-2 - (1 - Sqrt[5])* x]*Sqrt[(2 + (1 + Sqrt[5])*x)/(2 + (1 - Sqrt[5])*x)]*EllipticF[ArcSin[(Sqr t[2]*5^(1/4)*Sqrt[x])/Sqrt[-2 - (1 - Sqrt[5])*x]], (5 - Sqrt[5])/10])/(5^( 1/4)*(3 + Sqrt[5])*Sqrt[(2 + (1 - Sqrt[5])*x)^(-1)]*Sqrt[-1 - x + x^2]) + (Sqrt[1 + Sqrt[5] - 2*x]*Sqrt[-2/(1 + Sqrt[5]) - x]*(-((Sqrt[2 + (1 + Sqrt [5])*x]*EllipticF[ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2] )/Sqrt[-2/(1 + Sqrt[5]) - x]) + ((3 + Sqrt[5])*Sqrt[2 + (1 + Sqrt[5])*x]*E llipticPi[(-1 - Sqrt[5])/2, ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - S qrt[5])/2])/(2*Sqrt[-2/(1 + Sqrt[5]) - x])))/((3 + Sqrt[5])*Sqrt[-1 - x + x^2]))))/Sqrt[-x - x^2 + x^3]
3.3.100.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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])
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]
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]
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]
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]
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]
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
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 ]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.26
| method | result | size |
| trager | \(-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +4 \sqrt {x^{3}-x^{2}-x}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+2 \sqrt {x^{3}-x^{2}-x}}{\left (1+x \right )^{3}}\right )\) | \(88\) |
| default | \(\frac {4 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \sqrt {-5 \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}}-\frac {6 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \sqrt {-5 \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {3}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {3}{2}-\frac {\sqrt {5}}{2}\right )}\) | \(250\) |
| elliptic | \(\frac {4 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \sqrt {-5 \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}}-\frac {6 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \sqrt {-5 \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {3}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {3}{2}-\frac {\sqrt {5}}{2}\right )}\) | \(250\) |
-RootOf(_Z^2+1)*ln((RootOf(_Z^2+1)*x^3-5*RootOf(_Z^2+1)*x^2-5*RootOf(_Z^2+ 1)*x+4*(x^3-x^2-x)^(1/2)*x-RootOf(_Z^2+1)+2*(x^3-x^2-x)^(1/2))/(1+x)^3)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (25) = 50\).
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {-1+2 x}{(1+x) \sqrt {-x-x^2+x^3}} \, dx=\arctan \left (\frac {\sqrt {x^{3} - x^{2} - x} {\left (x^{3} - 5 \, x^{2} - 5 \, x - 1\right )}}{2 \, {\left (2 \, x^{4} - x^{3} - 3 \, x^{2} - x\right )}}\right ) \]
\[ \int \frac {-1+2 x}{(1+x) \sqrt {-x-x^2+x^3}} \, dx=\int \frac {2 x - 1}{\sqrt {x \left (x^{2} - x - 1\right )} \left (x + 1\right )}\, dx \]
\[ \int \frac {-1+2 x}{(1+x) \sqrt {-x-x^2+x^3}} \, dx=\int { \frac {2 \, x - 1}{\sqrt {x^{3} - x^{2} - x} {\left (x + 1\right )}} \,d x } \]
\[ \int \frac {-1+2 x}{(1+x) \sqrt {-x-x^2+x^3}} \, dx=\int { \frac {2 \, x - 1}{\sqrt {x^{3} - x^{2} - x} {\left (x + 1\right )}} \,d x } \]
Time = 5.39 (sec) , antiderivative size = 167, normalized size of antiderivative = 6.19 \[ \int \frac {-1+2 x}{(1+x) \sqrt {-x-x^2+x^3}} \, dx=\frac {\left (2\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )-3\,\Pi \left (-\frac {\sqrt {5}}{2}-\frac {1}{2};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\left (\sqrt {5}+1\right )\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}}{\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}} \]
((2*ellipticF(asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/ 2)/2 - 1/2)) - 3*ellipticPi(- 5^(1/2)/2 - 1/2, asin((x/(5^(1/2)/2 + 1/2))^ (1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/2 - 1/2)))*(x/(5^(1/2)/2 + 1/2))^(1/2) *((x + 5^(1/2)/2 - 1/2)/(5^(1/2)/2 - 1/2))^(1/2)*(5^(1/2) + 1)*((5^(1/2)/2 - x + 1/2)/(5^(1/2)/2 + 1/2))^(1/2))/(x^3 - x^2 - x*(5^(1/2)/2 - 1/2)*(5^ (1/2)/2 + 1/2))^(1/2)