Optimal. Leaf size=125 \[ \frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {x^3-x^2-x}}{x^2-x-1}\right )-\frac {1}{4} \sqrt {1-2 i} \tan ^{-1}\left (\frac {\sqrt {1-2 i} \sqrt {x^3-x^2-x}}{x^2-x-1}\right )-\frac {1}{4} \sqrt {1+2 i} \tan ^{-1}\left (\frac {\sqrt {1+2 i} \sqrt {x^3-x^2-x}}{x^2-x-1}\right ) \]
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Rubi [C] time = 5.08, antiderivative size = 1650, normalized size of antiderivative = 13.20, number of steps used = 55, number of rules used = 19, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.792, Rules used = {2056, 6725, 918, 6733, 1716, 1187, 1098, 1184, 1214, 1456, 540, 421, 419, 538, 537, 1712, 1700, 1698, 205}
result too large to display
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 205
Rule 419
Rule 421
Rule 537
Rule 538
Rule 540
Rule 918
Rule 1098
Rule 1184
Rule 1187
Rule 1214
Rule 1456
Rule 1698
Rule 1700
Rule 1712
Rule 1716
Rule 2056
Rule 6725
Rule 6733
Rubi steps
\begin {align*} \int \frac {\sqrt {-x-x^2+x^3}}{-1+x^4} \, dx &=\frac {\sqrt {-x-x^2+x^3} \int \frac {\sqrt {x} \sqrt {-1-x+x^2}}{-1+x^4} \, dx}{\sqrt {x} \sqrt {-1-x+x^2}}\\ &=\frac {\sqrt {-x-x^2+x^3} \int \left (-\frac {\sqrt {x} \sqrt {-1-x+x^2}}{2 \left (1-x^2\right )}-\frac {\sqrt {x} \sqrt {-1-x+x^2}}{2 \left (1+x^2\right )}\right ) \, dx}{\sqrt {x} \sqrt {-1-x+x^2}}\\ &=-\frac {\sqrt {-x-x^2+x^3} \int \frac {\sqrt {x} \sqrt {-1-x+x^2}}{1-x^2} \, dx}{2 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {-x-x^2+x^3} \int \frac {\sqrt {x} \sqrt {-1-x+x^2}}{1+x^2} \, dx}{2 \sqrt {x} \sqrt {-1-x+x^2}}\\ &=-\frac {\sqrt {-x-x^2+x^3} \int \left (\frac {i \sqrt {x} \sqrt {-1-x+x^2}}{2 (i-x)}+\frac {i \sqrt {x} \sqrt {-1-x+x^2}}{2 (i+x)}\right ) \, dx}{2 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {-x-x^2+x^3} \int \left (\frac {\sqrt {x} \sqrt {-1-x+x^2}}{2 (1-x)}+\frac {\sqrt {x} \sqrt {-1-x+x^2}}{2 (1+x)}\right ) \, dx}{2 \sqrt {x} \sqrt {-1-x+x^2}}\\ &=-\frac {\left (i \sqrt {-x-x^2+x^3}\right ) \int \frac {\sqrt {x} \sqrt {-1-x+x^2}}{i-x} \, dx}{4 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\left (i \sqrt {-x-x^2+x^3}\right ) \int \frac {\sqrt {x} \sqrt {-1-x+x^2}}{i+x} \, dx}{4 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {-x-x^2+x^3} \int \frac {\sqrt {x} \sqrt {-1-x+x^2}}{1-x} \, dx}{4 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {-x-x^2+x^3} \int \frac {\sqrt {x} \sqrt {-1-x+x^2}}{1+x} \, dx}{4 \sqrt {x} \sqrt {-1-x+x^2}}\\ &=-\frac {\left (i \sqrt {-x-x^2+x^3}\right ) \int \frac {-i-(2+2 i) x-(1-3 i) x^2}{(i-x) \sqrt {x} \sqrt {-1-x+x^2}} \, dx}{12 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\left (i \sqrt {-x-x^2+x^3}\right ) \int \frac {-i+(2-2 i) x+(1+3 i) x^2}{\sqrt {x} (i+x) \sqrt {-1-x+x^2}} \, dx}{12 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {-x-x^2+x^3} \int \frac {-1-4 x+2 x^2}{(1-x) \sqrt {x} \sqrt {-1-x+x^2}} \, dx}{12 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \int \frac {-1+4 x^2}{\sqrt {x} (1+x) \sqrt {-1-x+x^2}} \, dx}{12 \sqrt {x} \sqrt {-1-x+x^2}}\\ &=-\frac {\left (i \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {-i-(2+2 i) x^2-(1-3 i) x^4}{\left (i-x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\left (i \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {-i+(2-2 i) x^2+(1+3 i) x^4}{\left (i+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {-x-x^2+x^3} \operatorname {Subst}\left (\int \frac {-1-4 x^2+2 x^4}{\left (1-x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \operatorname {Subst}\left (\int \frac {-1+4 x^4}{\left (1+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}}\\ &=-\frac {\left (i \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {(1-4 i)-(2+2 i) x^2}{\left (i-x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\left (i \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {(-1-4 i)+(2-2 i) x^2}{\left (i+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {-x-x^2+x^3} \operatorname {Subst}\left (\int \frac {4-4 x^2}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \operatorname {Subst}\left (\int \frac {-2+2 x^2}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\left (\left (\frac {1}{2}-\frac {i}{6}\right ) \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {i-x^2}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}-\frac {\left (\left (\frac {1}{2}+\frac {i}{6}\right ) \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {i+x^2}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}\\ &=-\frac {\left (\left (\frac {1}{4}-\frac {i}{12}\right ) \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {-1-\sqrt {5}+2 x^2}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}+-\frac {\left (\left (\frac {1}{4}+\frac {i}{2}\right ) \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {i-x^2}{\left (i+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}--\frac {\left (\left (\frac {1}{12}+\frac {i}{6}\right ) \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}+\frac {\left (\left (\frac {1}{12}-\frac {i}{6}\right ) \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \operatorname {Subst}\left (\int \frac {-1-\sqrt {5}+2 x^2}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\left (\left (\frac {1}{4}+\frac {i}{12}\right ) \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {-1-\sqrt {5}+2 x^2}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}-\frac {\left (\left (\frac {1}{4}-\frac {i}{2}\right ) \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {i+x^2}{\left (i-x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \operatorname {Subst}\left (\int \frac {-1-\sqrt {5}+2 x^2}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\left (\left (\frac {1}{4}-\frac {i}{12}\right ) \left ((-1+2 i)-\sqrt {5}\right ) \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \operatorname {Subst}\left (\int \frac {-1-\sqrt {5}+2 x^2}{\left (1-x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-\sqrt {5}\right ) \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\left (1-\sqrt {5}\right ) \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\left (\left (1-\sqrt {5}\right ) \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\left (\left (1-\sqrt {5}\right ) \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\left (\left (\frac {1}{4}+\frac {i}{12}\right ) \left ((1+2 i)+\sqrt {5}\right ) \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {-x-x^2+x^3} \operatorname {Subst}\left (\int \frac {-1-\sqrt {5}+2 x^2}{\left (1+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (3+\sqrt {5}\right ) \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\left (3+\sqrt {5}\right ) \sqrt {x} \sqrt {-1-x+x^2}}\\ &=-\frac {\sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{12 \sqrt [4]{5} \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}-\frac {\sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}-\frac {\left (\frac {1}{24}+\frac {i}{8}\right ) \left ((2+i)+i \sqrt {5}\right ) \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}+\frac {\left (\frac {1}{8}+\frac {i}{24}\right ) \left ((1+2 i)+\sqrt {5}\right ) \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}-\frac {\sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}+\frac {\left (\sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-1-\sqrt {5}+2 x^2}}{\left (1-x^2\right ) \sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}}} \, dx,x,\sqrt {x}\right )}{4 \left (1-\sqrt {5}\right ) \sqrt {x} \left (-1-x+x^2\right )}-\frac {\left (\sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-1-\sqrt {5}+2 x^2}}{\sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \left (1+x^2\right )} \, dx,x,\sqrt {x}\right )}{4 \left (3+\sqrt {5}\right ) \sqrt {x} \left (-1-x+x^2\right )}+\frac {\left (\left (\frac {1}{2}-\frac {i}{4}\right ) \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{i-(2-i) x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {-1-x+x^2}}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}-\frac {\left (\left (\frac {1}{2}+\frac {i}{4}\right ) \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{i+(2+i) x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {-1-x+x^2}}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}\\ &=-\frac {\sqrt {1-2 i} \sqrt {-x-x^2+x^3} \tan ^{-1}\left (\frac {\sqrt {1-2 i} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )}{4 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {1+2 i} \sqrt {-x-x^2+x^3} \tan ^{-1}\left (\frac {\sqrt {1+2 i} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )}{4 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{12 \sqrt [4]{5} \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}-\frac {\sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}-\frac {\left (\frac {1}{24}+\frac {i}{8}\right ) \left ((2+i)+i \sqrt {5}\right ) \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}+\frac {\left (\frac {1}{8}+\frac {i}{24}\right ) \left ((1+2 i)+\sqrt {5}\right ) \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}-\frac {\sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}+\frac {\left (\sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {-1-\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{4 \sqrt {x} \left (-1-x+x^2\right )}-\frac {\left (\sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {-1-\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{2 \left (1-\sqrt {5}\right ) \sqrt {x} \left (-1-x+x^2\right )}-\frac {\left (\sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {-1-\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{2 \left (3+\sqrt {5}\right ) \sqrt {x} \left (-1-x+x^2\right )}-\frac {\left (\left (-3-\sqrt {5}\right ) \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \left (1+x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{4 \left (3+\sqrt {5}\right ) \sqrt {x} \left (-1-x+x^2\right )}\\ &=-\frac {\sqrt {1-2 i} \sqrt {-x-x^2+x^3} \tan ^{-1}\left (\frac {\sqrt {1-2 i} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )}{4 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {1+2 i} \sqrt {-x-x^2+x^3} \tan ^{-1}\left (\frac {\sqrt {1+2 i} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )}{4 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{12 \sqrt [4]{5} \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}-\frac {\sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}-\frac {\left (\frac {1}{24}+\frac {i}{8}\right ) \left ((2+i)+i \sqrt {5}\right ) \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}+\frac {\left (\frac {1}{8}+\frac {i}{24}\right ) \left ((1+2 i)+\sqrt {5}\right ) \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}-\frac {\sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}+\frac {\left (\sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{4 \sqrt {x} \left (-1-x+x^2\right )}-\frac {\left (\sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{2 \left (1-\sqrt {5}\right ) \sqrt {x} \left (-1-x+x^2\right )}-\frac {\left (\sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{2 \left (3+\sqrt {5}\right ) \sqrt {x} \left (-1-x+x^2\right )}-\frac {\left (\left (-3-\sqrt {5}\right ) \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \sqrt {-x-x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \left (1+x^2\right ) \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{4 \left (3+\sqrt {5}\right ) \sqrt {x} \left (-1-x+x^2\right )}\\ &=-\frac {\sqrt {1-2 i} \sqrt {-x-x^2+x^3} \tan ^{-1}\left (\frac {\sqrt {1-2 i} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )}{4 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {1+2 i} \sqrt {-x-x^2+x^3} \tan ^{-1}\left (\frac {\sqrt {1+2 i} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )}{4 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {2} \left (1-\sqrt {5}\right ) \sqrt {x} \left (1+x-x^2\right )}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {2} \left (3+\sqrt {5}\right ) \sqrt {x} \left (1+x-x^2\right )}-\frac {\sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{12 \sqrt [4]{5} \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}-\frac {\sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}-\frac {\left (\frac {1}{24}+\frac {i}{8}\right ) \left ((2+i)+i \sqrt {5}\right ) \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}+\frac {\left (\frac {1}{8}+\frac {i}{24}\right ) \left ((1+2 i)+\sqrt {5}\right ) \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}-\frac {\sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {x} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \left (1+x-x^2\right )}-\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \sqrt {-x-x^2+x^3} \Pi \left (\frac {1}{2} \left (-1-\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \sqrt {2} \sqrt {x} \left (1+x-x^2\right )}-\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \sqrt {-x-x^2+x^3} \Pi \left (\frac {1}{2} \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{4 \sqrt {2} \sqrt {x} \left (1+x-x^2\right )}\\ \end {align*}
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Mathematica [C] time = 13.09, size = 2469, normalized size = 19.75 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.23, size = 125, normalized size = 1.00 \begin {gather*} \frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )-\frac {1}{4} \sqrt {1-2 i} \tan ^{-1}\left (\frac {\sqrt {1-2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )-\frac {1}{4} \sqrt {1+2 i} \tan ^{-1}\left (\frac {\sqrt {1+2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.50, size = 2484, normalized size = 19.87
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{3} - x^{2} - x}}{x^{4} - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.32, size = 917, normalized size = 7.34
method | result | size |
trager | \(-\frac {\RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right ) \ln \left (\frac {3935232 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{5} x^{2}-5902848 x \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{5}-3935232 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{5}-1338592 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{3} x^{2}+962592 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{3} x +65024 \sqrt {x^{3}-x^{2}-x}\, \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2}+1338592 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{3}-160080 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right ) x^{2}+693680 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right ) x -286015 \sqrt {x^{3}-x^{2}-x}+160080 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )}{\left (16 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2} x -32 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2}+5 x \right )^{2}}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+16 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2}+2\right ) \ln \left (-\frac {-983808 \RootOf \left (\textit {\_Z}^{2}+16 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2}+2\right ) \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{4} x^{2}+1475712 \RootOf \left (\textit {\_Z}^{2}+16 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2}+2\right ) \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{4} x +983808 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{4} \RootOf \left (\textit {\_Z}^{2}+16 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2}+2\right )-580600 \RootOf \left (\textit {\_Z}^{2}+16 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2}+2\right ) \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2} x^{2}+609576 \RootOf \left (\textit {\_Z}^{2}+16 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2}+2\right ) \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2} x +65024 \sqrt {x^{3}-x^{2}-x}\, \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2}+580600 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2} \RootOf \left (\textit {\_Z}^{2}+16 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2}+2\right )-17183 \RootOf \left (\textit {\_Z}^{2}+16 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2}+2\right ) x^{2}-120281 \RootOf \left (\textit {\_Z}^{2}+16 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2}+2\right ) x +294143 \sqrt {x^{3}-x^{2}-x}+17183 \RootOf \left (\textit {\_Z}^{2}+16 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2}+2\right )}{\left (16 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2} x -32 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2}-3 x -4\right )^{2}}\right )}{8}-2 \ln \left (\frac {-16 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2} x^{2}+32 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2} x +16 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2}-x^{2}+4 \sqrt {x^{3}-x^{2}-x}+2 x +1}{\left (-1+x \right ) \left (1+x \right )}\right ) \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2}-\frac {\ln \left (\frac {-16 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2} x^{2}+32 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2} x +16 \RootOf \left (256 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+5\right )^{2}-x^{2}+4 \sqrt {x^{3}-x^{2}-x}+2 x +1}{\left (-1+x \right ) \left (1+x \right )}\right )}{8}\) | \(917\) |
elliptic | \(\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, -\frac {i \sqrt {5}}{5}+\frac {1}{2}-\frac {\sqrt {5}}{10}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{10 \sqrt {x^{3}-x^{2}-x}}+\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, -\frac {i \sqrt {5}}{5}+\frac {1}{2}-\frac {\sqrt {5}}{10}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{20 \sqrt {x^{3}-x^{2}-x}}-\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, -\frac {i \sqrt {5}}{5}+\frac {1}{2}-\frac {\sqrt {5}}{10}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{20 \sqrt {x^{3}-x^{2}-x}}+\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {i \sqrt {5}}{5}+\frac {1}{2}-\frac {\sqrt {5}}{10}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{10 \sqrt {x^{3}-x^{2}-x}}-\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {i \sqrt {5}}{5}+\frac {1}{2}-\frac {\sqrt {5}}{10}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{20 \sqrt {x^{3}-x^{2}-x}}+\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {i \sqrt {5}}{5}+\frac {1}{2}-\frac {\sqrt {5}}{10}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{20 \sqrt {x^{3}-x^{2}-x}}+\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \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 ) \sqrt {5}}{20 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {3}{2}-\frac {\sqrt {5}}{2}\right )}-\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \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 )}{20 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {3}{2}-\frac {\sqrt {5}}{2}\right )}-\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \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 {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{20 \sqrt {x^{3}-x^{2}-x}\, \left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}+\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \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 {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{20 \sqrt {x^{3}-x^{2}-x}\, \left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}\) | \(1430\) |
default | \(\text {Expression too large to display}\) | \(2289\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{3} - x^{2} - x}}{x^{4} - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 537, normalized size = 4.30 \begin {gather*} \frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\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}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\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 )}{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}}+\frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\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}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\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 )}{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}}+\frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\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}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\Pi \left (-\frac {\sqrt {5}\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i};\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 )\,\left (-\frac {1}{2}+1{}\mathrm {i}\right )}{\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}}+\frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\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}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\Pi \left (\frac {\sqrt {5}\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i};\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 )\,\left (-\frac {1}{2}-\mathrm {i}\right )}{\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (x^{2} - x - 1\right )}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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