Integrand size = 29, antiderivative size = 24 \[ \int \frac {\sqrt {-1+\frac {1}{x}} \sqrt {\frac {1}{x}} \sqrt {x}}{\sqrt {1+x}} \, dx=-\frac {2 \sqrt {-x} E\left (\left .\arcsin \left (\sqrt {-x}\right )\right |-1\right )}{\sqrt {x}} \]
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Leaf count is larger than twice the leaf count of optimal. \(49\) vs. \(2(24)=48\).
Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {15, 446, 112, 111} \[ \int \frac {\sqrt {-1+\frac {1}{x}} \sqrt {\frac {1}{x}} \sqrt {x}}{\sqrt {1+x}} \, dx=-\frac {2 \sqrt {\frac {1}{x}-1} \sqrt {\frac {1}{x}} \sqrt {-x} \sqrt {x} E\left (\left .\arcsin \left (\sqrt {-x}\right )\right |-1\right )}{\sqrt {1-x}} \]
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Rule 15
Rule 111
Rule 112
Rule 446
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\frac {1}{x}} \sqrt {x}\right ) \int \frac {\sqrt {-1+\frac {1}{x}}}{\sqrt {1+x}} \, dx \\ & = \frac {\sqrt {-1+\frac {1}{x}} \int \frac {\sqrt {1-x}}{\sqrt {x} \sqrt {1+x}} \, dx}{\sqrt {1-x} \sqrt {\frac {1}{x}}} \\ & = \frac {\left (\sqrt {-1+\frac {1}{x}} \sqrt {-x}\right ) \int \frac {\sqrt {1-x}}{\sqrt {-x} \sqrt {1+x}} \, dx}{\sqrt {1-x} \sqrt {\frac {1}{x}} \sqrt {x}} \\ & = -\frac {2 \sqrt {-1+\frac {1}{x}} \sqrt {-x} E\left (\left .\sin ^{-1}\left (\sqrt {-x}\right )\right |-1\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}} \sqrt {x}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.75 \[ \int \frac {\sqrt {-1+\frac {1}{x}} \sqrt {\frac {1}{x}} \sqrt {x}}{\sqrt {1+x}} \, dx=-\frac {2 \sqrt {\frac {x}{1+x}} \sqrt {1-x^2} \left (-3 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},x^2\right )+x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},x^2\right )\right )}{3 \sqrt {1-x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs. \(2(18)=36\).
Time = 1.79 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.04
method | result | size |
default | \(-\frac {2 \sqrt {\frac {1}{x}}\, \sqrt {x}\, \sqrt {-\frac {-1+x}{x}}\, E\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right ) \sqrt {-x}\, \sqrt {2-2 x}}{-1+x}\) | \(49\) |
derivativedivides | \(\frac {2 x^{\frac {5}{2}} \left (\frac {1}{x}\right )^{\frac {5}{2}} \sqrt {\left (\frac {1}{x}+1\right ) x}\, \sqrt {-1+\frac {1}{x}}\, \left (\sqrt {\frac {1}{x}+1}\, \sqrt {2}\, \sqrt {1-\frac {1}{x}}\, \sqrt {-\frac {1}{x}}\, E\left (\sqrt {\frac {1}{x}+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {\frac {1}{x}+1}\, \sqrt {2}\, \sqrt {1-\frac {1}{x}}\, \sqrt {-\frac {1}{x}}\, F\left (\sqrt {\frac {1}{x}+1}, \frac {\sqrt {2}}{2}\right )+\frac {1}{x^{2}}-1\right )}{\frac {1}{x^{2}}-1}\) | \(120\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {-1+\frac {1}{x}} \sqrt {\frac {1}{x}} \sqrt {x}}{\sqrt {1+x}} \, dx=-2 i \, {\rm weierstrassPInverse}\left (4, 0, x\right ) - 2 i \, {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, x\right )\right ) \]
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Exception generated. \[ \int \frac {\sqrt {-1+\frac {1}{x}} \sqrt {\frac {1}{x}} \sqrt {x}}{\sqrt {1+x}} \, dx=\text {Exception raised: RecursionError} \]
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\[ \int \frac {\sqrt {-1+\frac {1}{x}} \sqrt {\frac {1}{x}} \sqrt {x}}{\sqrt {1+x}} \, dx=\int { \frac {\sqrt {\frac {1}{x} - 1}}{\sqrt {x + 1}} \,d x } \]
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\[ \int \frac {\sqrt {-1+\frac {1}{x}} \sqrt {\frac {1}{x}} \sqrt {x}}{\sqrt {1+x}} \, dx=\int { \frac {\sqrt {\frac {1}{x} - 1}}{\sqrt {x + 1}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {-1+\frac {1}{x}} \sqrt {\frac {1}{x}} \sqrt {x}}{\sqrt {1+x}} \, dx=\int \frac {\sqrt {x}\,\sqrt {\frac {1}{x}-1}\,\sqrt {\frac {1}{x}}}{\sqrt {x+1}} \,d x \]
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