Integrand size = 26, antiderivative size = 19 \[ \int \frac {-2-2 x+x^2}{\left (1+x+x^2\right ) \sqrt {-1+x^3}} \, dx=-\frac {2 \sqrt {-1+x^3}}{1+x+x^2} \]
[Out]
Result contains higher order function than in optimal. Order 4 vs. order 2 in optimal.
Time = 1.48 (sec) , antiderivative size = 719, normalized size of antiderivative = 37.84, number of steps used = 27, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6860, 225, 2161, 2167, 2138, 21, 425, 435, 455, 37} \[ \int \frac {-2-2 x+x^2}{\left (1+x+x^2\right ) \sqrt {-1+x^3}} \, dx=\frac {2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (\sqrt {3}+i\right ) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} (1-x) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\left (3+(2+i) \sqrt {3}\right ) \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {2 \sqrt [4]{3} \left (-\sqrt {3}+i\right ) \sqrt {2-\sqrt {3}} \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} (1-x) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\left (3+(2-i) \sqrt {3}\right ) \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {2 \sqrt {2-\sqrt {3}} \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} (1-x) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {6 \sqrt [4]{3} \left (\sqrt {3}+(-2+i)\right ) \left (\sqrt {3}+i\right ) \sqrt {26+15 \sqrt {3}} \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} (1-x) E\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\left (3+(2+i) \sqrt {3}\right )^3 \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {6 \sqrt [4]{3} \left (1+i \sqrt {3}\right ) \left (\sqrt {3}+(-2-i)\right ) \sqrt {26+15 \sqrt {3}} \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} (1-x) E\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\left (3 i+(1+2 i) \sqrt {3}\right )^3 \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {2 (1-x)}{\sqrt {x^3-1}} \]
[In]
[Out]
Rule 21
Rule 37
Rule 225
Rule 425
Rule 435
Rule 455
Rule 2138
Rule 2161
Rule 2167
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {-1+x^3}}-\frac {3 (1+x)}{\left (1+x+x^2\right ) \sqrt {-1+x^3}}\right ) \, dx \\ & = -\left (3 \int \frac {1+x}{\left (1+x+x^2\right ) \sqrt {-1+x^3}} \, dx\right )+\int \frac {1}{\sqrt {-1+x^3}} \, dx \\ & = -\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-3 \int \left (\frac {1-\frac {i}{\sqrt {3}}}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {-1+x^3}}+\frac {1+\frac {i}{\sqrt {3}}}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {-1+x^3}}\right ) \, dx \\ & = -\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\left (3-i \sqrt {3}\right ) \int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx-\left (3+i \sqrt {3}\right ) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx \\ & = -\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {\left (3 i+\sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^3}} \, dx}{3 i+(1+2 i) \sqrt {3}}-\frac {\left (2 \left (3 i+\sqrt {3}\right )\right ) \int \frac {1+\sqrt {3}-x}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx}{3 i+(1+2 i) \sqrt {3}}-\frac {\left (3+i \sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^3}} \, dx}{3+(2+i) \sqrt {3}}-\frac {\left (2 \left (3+i \sqrt {3}\right )\right ) \int \frac {1+\sqrt {3}-x}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx}{3+(2+i) \sqrt {3}} \\ & = -\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+i \sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\left (3 i+(1+2 i) \sqrt {3}\right ) \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (i+\sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\left (3+(2+i) \sqrt {3}\right ) \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {\left (8 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (3 i+\sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 \left (1-\sqrt {3}\right )+\left (1-i \sqrt {3}+2 \left (1+\sqrt {3}\right )\right ) x\right ) \sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx,x,\frac {-1+\sqrt {3}+x}{1+\sqrt {3}-x}\right )}{\left (3 i+(1+2 i) \sqrt {3}\right ) \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {\left (8 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (3+i \sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 \left (1-\sqrt {3}\right )+\left (1+i \sqrt {3}+2 \left (1+\sqrt {3}\right )\right ) x\right ) \sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx,x,\frac {-1+\sqrt {3}+x}{1+\sqrt {3}-x}\right )}{\left (3+(2+i) \sqrt {3}\right ) \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \\ & = -\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+i \sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\left (3 i+(1+2 i) \sqrt {3}\right ) \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (i+\sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\left (3+(2+i) \sqrt {3}\right ) \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {\left (8 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (3+i \sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (1+i \sqrt {3}+2 \left (1-\sqrt {3}\right )\right )^2-\left (1+i \sqrt {3}+2 \left (1+\sqrt {3}\right )\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}+x}{1+\sqrt {3}-x}\right )}{\sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {\left (8 \sqrt [4]{3} \left (3-(2+i) \sqrt {3}\right ) \sqrt {2-\sqrt {3}} \left (3 i+\sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (1-i \sqrt {3}+2 \left (1-\sqrt {3}\right )\right )^2-\left (1-i \sqrt {3}+2 \left (1+\sqrt {3}\right )\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}+x}{1+\sqrt {3}-x}\right )}{\left (3 i+(1+2 i) \sqrt {3}\right ) \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {\left (8 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (3 i+\sqrt {3}\right ) \left (3+(2-i) \sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (1-i \sqrt {3}+2 \left (1-\sqrt {3}\right )\right )^2-\left (1-i \sqrt {3}+2 \left (1+\sqrt {3}\right )\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}+x}{1+\sqrt {3}-x}\right )}{\left (3 i+(1+2 i) \sqrt {3}\right ) \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {\left (8 \sqrt [4]{3} \left (3-(2-i) \sqrt {3}\right ) \sqrt {2-\sqrt {3}} \left (3+i \sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (1+i \sqrt {3}+2 \left (1-\sqrt {3}\right )\right )^2-\left (1+i \sqrt {3}+2 \left (1+\sqrt {3}\right )\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}+x}{1+\sqrt {3}-x}\right )}{\left (3+(2+i) \sqrt {3}\right ) \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \\ & = -\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+i \sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\left (3 i+(1+2 i) \sqrt {3}\right ) \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (i+\sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\left (3+(2+i) \sqrt {3}\right ) \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {\left (8 \sqrt [4]{3} \left (3-(2+i) \sqrt {3}\right ) \sqrt {2-\sqrt {3}} \left (3 i+\sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left (7-4 \sqrt {3}+x^2\right )^{3/2}} \, dx,x,\frac {-1+\sqrt {3}+x}{1+\sqrt {3}-x}\right )}{\left (3 i+(1+2 i) \sqrt {3}\right )^3 \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {\left (8 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (3 i+\sqrt {3}\right ) \left (3+(2-i) \sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \left (7-4 \sqrt {3}+x^2\right )^{3/2}} \, dx,x,\frac {-1+\sqrt {3}+x}{1+\sqrt {3}-x}\right )}{\left (3 i+(1+2 i) \sqrt {3}\right )^3 \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {\left (8 \sqrt [4]{3} \left (3-(2-i) \sqrt {3}\right ) \sqrt {2-\sqrt {3}} \left (3+i \sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left (7-4 \sqrt {3}+x^2\right )^{3/2}} \, dx,x,\frac {-1+\sqrt {3}+x}{1+\sqrt {3}-x}\right )}{\left (3+(2+i) \sqrt {3}\right )^3 \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {\left (8 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (3+i \sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \left (7-4 \sqrt {3}+x^2\right )^{3/2}} \, dx,x,\frac {-1+\sqrt {3}+x}{1+\sqrt {3}-x}\right )}{\left (3+(2+i) \sqrt {3}\right )^2 \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \\ & = -\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+i \sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\left (3 i+(1+2 i) \sqrt {3}\right ) \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (i+\sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\left (3+(2+i) \sqrt {3}\right ) \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {\left (4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (3 i+\sqrt {3}\right ) \left (3+(2-i) \sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \left (7-4 \sqrt {3}+x\right )^{3/2}} \, dx,x,\frac {\left (-1+\sqrt {3}+x\right )^2}{\left (1+\sqrt {3}-x\right )^2}\right )}{\left (3 i+(1+2 i) \sqrt {3}\right )^3 \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {\left (4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (3+i \sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \left (7-4 \sqrt {3}+x\right )^{3/2}} \, dx,x,\frac {\left (-1+\sqrt {3}+x\right )^2}{\left (1+\sqrt {3}-x\right )^2}\right )}{\left (3+(2+i) \sqrt {3}\right )^2 \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {\left (2 \sqrt [4]{3} \left (3-(2+i) \sqrt {3}\right ) \sqrt {2-\sqrt {3}} \left (3 i+\sqrt {3}\right ) \left (26+15 \sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {-7+4 \sqrt {3}-x^2}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx,x,\frac {-1+\sqrt {3}+x}{1+\sqrt {3}-x}\right )}{\left (3 i+(1+2 i) \sqrt {3}\right )^3 \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {\left (2 \sqrt [4]{3} \left (3-(2-i) \sqrt {3}\right ) \sqrt {2-\sqrt {3}} \left (3+i \sqrt {3}\right ) \left (26+15 \sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {-7+4 \sqrt {3}-x^2}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx,x,\frac {-1+\sqrt {3}+x}{1+\sqrt {3}-x}\right )}{\left (3+(2+i) \sqrt {3}\right )^3 \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \\ & = \frac {1-x}{\sqrt {-1+x^3}}+\frac {6 \left (i+\sqrt {3}\right ) (1-x)}{\left (2-\sqrt {3}\right ) \left (3+(2+i) \sqrt {3}\right )^2 \sqrt {-1+x^3}}-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+i \sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\left (3 i+(1+2 i) \sqrt {3}\right ) \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (i+\sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\left (3+(2+i) \sqrt {3}\right ) \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {\left (2 \sqrt [4]{3} \left (3-(2+i) \sqrt {3}\right ) \sqrt {2-\sqrt {3}} \left (3 i+\sqrt {3}\right ) \left (26+15 \sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {\sqrt {7-4 \sqrt {3}+x^2}}{\sqrt {1-x^2}} \, dx,x,\frac {-1+\sqrt {3}+x}{1+\sqrt {3}-x}\right )}{\left (3 i+(1+2 i) \sqrt {3}\right )^3 \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {\left (2 \sqrt [4]{3} \left (3-(2-i) \sqrt {3}\right ) \sqrt {2-\sqrt {3}} \left (3+i \sqrt {3}\right ) \left (26+15 \sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {\sqrt {7-4 \sqrt {3}+x^2}}{\sqrt {1-x^2}} \, dx,x,\frac {-1+\sqrt {3}+x}{1+\sqrt {3}-x}\right )}{\left (3+(2+i) \sqrt {3}\right )^3 \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \\ & = \frac {1-x}{\sqrt {-1+x^3}}+\frac {6 \left (i+\sqrt {3}\right ) (1-x)}{\left (2-\sqrt {3}\right ) \left (3+(2+i) \sqrt {3}\right )^2 \sqrt {-1+x^3}}+\frac {i \sqrt [4]{3} \sqrt {\frac {2}{3}-\frac {1}{\sqrt {3}}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {6 \sqrt [4]{3} \left ((-2+i)+\sqrt {3}\right ) \left (i+\sqrt {3}\right ) \sqrt {26+15 \sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right )|-7-4 \sqrt {3}\right )}{\left (3+(2+i) \sqrt {3}\right )^3 \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+i \sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\left (3 i+(1+2 i) \sqrt {3}\right ) \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (i+\sqrt {3}\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\left (3+(2+i) \sqrt {3}\right ) \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \\ \end{align*}
Time = 0.64 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-2-2 x+x^2}{\left (1+x+x^2\right ) \sqrt {-1+x^3}} \, dx=-\frac {2 \sqrt {-1+x^3}}{1+x+x^2} \]
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Time = 3.67 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68
method | result | size |
gosper | \(-\frac {2 \left (x -1\right )}{\sqrt {x^{3}-1}}\) | \(13\) |
risch | \(-\frac {2 \left (x -1\right )}{\sqrt {x^{3}-1}}\) | \(13\) |
default | \(-\frac {2 \left (x -1\right )}{\sqrt {\left (x -1\right ) \left (x^{2}+x +1\right )}}\) | \(18\) |
trager | \(-\frac {2 \sqrt {x^{3}-1}}{x^{2}+x +1}\) | \(18\) |
elliptic | \(-\frac {2 \left (x -1\right )}{\sqrt {\left (x -1\right ) \left (x^{2}+x +1\right )}}\) | \(18\) |
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Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {-2-2 x+x^2}{\left (1+x+x^2\right ) \sqrt {-1+x^3}} \, dx=-\frac {2 \, \sqrt {x^{3} - 1}}{x^{2} + x + 1} \]
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\[ \int \frac {-2-2 x+x^2}{\left (1+x+x^2\right ) \sqrt {-1+x^3}} \, dx=\int \frac {x^{2} - 2 x - 2}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + x + 1\right )}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-2-2 x+x^2}{\left (1+x+x^2\right ) \sqrt {-1+x^3}} \, dx=-\frac {2 \, \sqrt {x - 1}}{\sqrt {x^{2} + x + 1}} \]
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\[ \int \frac {-2-2 x+x^2}{\left (1+x+x^2\right ) \sqrt {-1+x^3}} \, dx=\int { \frac {x^{2} - 2 \, x - 2}{\sqrt {x^{3} - 1} {\left (x^{2} + x + 1\right )}} \,d x } \]
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Time = 4.98 (sec) , antiderivative size = 276, normalized size of antiderivative = 14.53 \[ \int \frac {-2-2 x+x^2}{\left (1+x+x^2\right ) \sqrt {-1+x^3}} \, dx=\frac {\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (6+9\,\sin \left (2\,\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\right )\,\sqrt {\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}+1}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}-6\,x+\sqrt {3}\,x\,2{}\mathrm {i}+\sqrt {3}\,2{}\mathrm {i}-\sqrt {3}\,x^2\,4{}\mathrm {i}-\sqrt {3}\,\sin \left (2\,\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\right )\,\sqrt {\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}+1}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,3{}\mathrm {i}\right )}{6\,\sqrt {1-\frac {x-1}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}+1}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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