Optimal. Leaf size=15 \[ \sqrt {-1+x} \sqrt {1+x} \]
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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {75}
\begin {gather*} \sqrt {x-1} \sqrt {x+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 75
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx &=\sqrt {-1+x} \sqrt {1+x}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 15, normalized size = 1.00 \begin {gather*} \sqrt {-1+x} \sqrt {1+x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 12, normalized size = 0.80
method | result | size |
gosper | \(\sqrt {-1+x}\, \sqrt {1+x}\) | \(12\) |
default | \(\sqrt {-1+x}\, \sqrt {1+x}\) | \(12\) |
risch | \(\sqrt {-1+x}\, \sqrt {1+x}\) | \(12\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 3 vs. order
2.
time = 0.29, size = 7, normalized size = 0.47 \begin {gather*} \sqrt {x^{2} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.51, size = 11, normalized size = 0.73 \begin {gather*} \sqrt {x + 1} \sqrt {x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.58, size = 76, normalized size = 5.07 \begin {gather*} \frac {{G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{4} & 0, 0, \frac {1}{2}, 1 \\- \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{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} -1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 1 & \\- \frac {3}{4}, - \frac {1}{4} & -1, - \frac {1}{2}, - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.98, size = 11, normalized size = 0.73 \begin {gather*} \sqrt {x + 1} \sqrt {x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.65, size = 67, normalized size = 4.47 \begin {gather*} -\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^2\,8{}\mathrm {i}}{{\left (\sqrt {x+1}-1\right )}^2\,\left (1+\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {x+1}-1\right )}^4}-\frac {2\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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