Optimal. Leaf size=26 \[ \frac {1}{2} \sqrt {-1+x} x \sqrt {1+x}-\frac {1}{2} \cosh ^{-1}(x) \]
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Rubi [A]
time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {38, 54}
\begin {gather*} \frac {1}{2} \sqrt {x-1} x \sqrt {x+1}-\frac {1}{2} \cosh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 38
Rule 54
Rubi steps
\begin {align*} \int \sqrt {-1+x} \sqrt {1+x} \, dx &=\frac {1}{2} \sqrt {-1+x} x \sqrt {1+x}-\frac {1}{2} \int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx\\ &=\frac {1}{2} \sqrt {-1+x} x \sqrt {1+x}-\frac {1}{2} \cosh ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 36, normalized size = 1.38 \begin {gather*} \frac {1}{2} \sqrt {-1+x} x \sqrt {1+x}-\tanh ^{-1}\left (\sqrt {\frac {-1+x}{1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs.
\(2(18)=36\).
time = 0.10, size = 57, normalized size = 2.19
method | result | size |
risch | \(\frac {x \sqrt {-1+x}\, \sqrt {1+x}}{2}-\frac {\ln \left (x +\sqrt {x^{2}-1}\right ) \sqrt {\left (1+x \right ) \left (-1+x \right )}}{2 \sqrt {-1+x}\, \sqrt {1+x}}\) | \(46\) |
default | \(\frac {\sqrt {-1+x}\, \left (1+x \right )^{\frac {3}{2}}}{2}-\frac {\sqrt {-1+x}\, \sqrt {1+x}}{2}-\frac {\ln \left (x +\sqrt {x^{2}-1}\right ) \sqrt {\left (1+x \right ) \left (-1+x \right )}}{2 \sqrt {-1+x}\, \sqrt {1+x}}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 27, normalized size = 1.04 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} - 1} x - \frac {1}{2} \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.48, size = 32, normalized size = 1.23 \begin {gather*} \frac {1}{2} \, \sqrt {x + 1} \sqrt {x - 1} x + \frac {1}{2} \, \log \left (\sqrt {x + 1} \sqrt {x - 1} - x\right ) \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 = 1.96, size = 131, normalized size = 5.04 \begin {gather*} \begin {cases} - \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {x - 1}} - \frac {3 \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {x - 1}} + \frac {\sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\i \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {i \left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {1 - x}} + \frac {3 i \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {1 - x}} - \frac {i \sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs.
\(2 (18) = 36\).
time = 1.61, size = 41, normalized size = 1.58 \begin {gather*} \frac {1}{2} \, \sqrt {x + 1} \sqrt {x - 1} {\left (x - 2\right )} + \sqrt {x + 1} \sqrt {x - 1} + \log \left (\sqrt {x + 1} - \sqrt {x - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.08, size = 30, normalized size = 1.15 \begin {gather*} \frac {x\,\sqrt {x-1}\,\sqrt {x+1}}{2}-\frac {\ln \left (x+\sqrt {x-1}\,\sqrt {x+1}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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