Optimal. Leaf size=16 \[ \tan ^{-1}\left (\sqrt {x-1} \sqrt {x+1}\right ) \]
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Rubi [A] time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {92, 203} \[ \tan ^{-1}\left (\sqrt {x-1} \sqrt {x+1}\right ) \]
Antiderivative was successfully verified.
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Rule 92
Rule 203
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+x} x \sqrt {1+x}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x} \sqrt {1+x}\right )\\ &=\tan ^{-1}\left (\sqrt {-1+x} \sqrt {1+x}\right )\\ \end {align*}
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Mathematica [B] time = 0.01, size = 34, normalized size = 2.12 \[ \frac {\sqrt {x^2-1} \tan ^{-1}\left (\sqrt {x^2-1}\right )}{\sqrt {x-1} \sqrt {x+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 18, normalized size = 1.12 \[ 2 \, \arctan \left (\sqrt {x + 1} \sqrt {x - 1} - x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.25, size = 20, normalized size = 1.25 \[ -2 \, \arctan \left (\frac {1}{2} \, {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 28, normalized size = 1.75 \[ -\frac {\sqrt {x -1}\, \sqrt {x +1}\, \arctan \left (\frac {1}{\sqrt {x^{2}-1}}\right )}{\sqrt {x^{2}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 7, normalized size = 0.44 \[ -\arcsin \left (\frac {1}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.04, size = 49, normalized size = 3.06 \[ -\ln \left (\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}+1\right )\,1{}\mathrm {i}+\ln \left (\frac {\sqrt {x-1}-\mathrm {i}}{\sqrt {x+1}-1}\right )\,1{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.32, size = 56, normalized size = 3.50 \[ - \frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, 1 & 1, 1, \frac {3}{2} \\\frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2} & 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} + \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 1 & \\\frac {1}{4}, \frac {3}{4} & 0, \frac {1}{2}, \frac {1}{2}, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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