Integrand size = 13, antiderivative size = 269 \[ \int \frac {1}{x^2 \sqrt {-1+x^3}} \, dx=\frac {\sqrt {-1+x^3}}{1-\sqrt {3}-x}+\frac {\sqrt {-1+x^3}}{x}-\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{2 \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {\sqrt {2} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \]
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Time = 0.05 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {331, 310, 225, 1893} \[ \int \frac {1}{x^2 \sqrt {-1+x^3}} \, dx=\frac {\sqrt {2} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{2 \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {\sqrt {x^3-1}}{-x-\sqrt {3}+1}+\frac {\sqrt {x^3-1}}{x} \]
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Rule 225
Rule 310
Rule 331
Rule 1893
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-1+x^3}}{x}-\frac {1}{2} \int \frac {x}{\sqrt {-1+x^3}} \, dx \\ & = \frac {\sqrt {-1+x^3}}{x}+\frac {1}{2} \int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx-\frac {1}{2} \left (1+\sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^3}} \, dx \\ & = \frac {\sqrt {-1+x^3}}{1-\sqrt {3}-x}+\frac {\sqrt {-1+x^3}}{x}-\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{2 \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {\sqrt {2} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\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}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.14 \[ \int \frac {1}{x^2 \sqrt {-1+x^3}} \, dx=-\frac {\sqrt {1-x^3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{2},\frac {2}{3},x^3\right )}{x \sqrt {-1+x^3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.88 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.12
method | result | size |
meijerg | \(-\frac {\sqrt {-\operatorname {signum}\left (x^{3}-1\right )}\, {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {1}{3},\frac {1}{2};\frac {2}{3};x^{3}\right )}{\sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, x}\) | \(33\) |
default | \(\frac {\sqrt {x^{3}-1}}{x}-\frac {\left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) E\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) F\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right )}{\sqrt {x^{3}-1}}\) | \(185\) |
risch | \(\frac {\sqrt {x^{3}-1}}{x}-\frac {\left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) E\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) F\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right )}{\sqrt {x^{3}-1}}\) | \(185\) |
elliptic | \(\frac {\sqrt {x^{3}-1}}{x}-\frac {\left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) E\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) F\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right )}{\sqrt {x^{3}-1}}\) | \(185\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.08 \[ \int \frac {1}{x^2 \sqrt {-1+x^3}} \, dx=\frac {x {\rm weierstrassZeta}\left (0, 4, {\rm weierstrassPInverse}\left (0, 4, x\right )\right ) + \sqrt {x^{3} - 1}}{x} \]
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Time = 0.41 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.11 \[ \int \frac {1}{x^2 \sqrt {-1+x^3}} \, dx=- \frac {i \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{2} \\ \frac {2}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} \]
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\[ \int \frac {1}{x^2 \sqrt {-1+x^3}} \, dx=\int { \frac {1}{\sqrt {x^{3} - 1} x^{2}} \,d x } \]
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\[ \int \frac {1}{x^2 \sqrt {-1+x^3}} \, dx=\int { \frac {1}{\sqrt {x^{3} - 1} x^{2}} \,d x } \]
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Time = 5.44 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^2 \sqrt {-1+x^3}} \, dx=\frac {\sqrt {x^3-1}}{x}+\frac {\left (\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )-\left (-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\mathrm {E}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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