Optimal. Leaf size=19 \[ 2 \log \left (\sqrt {x-4}+\sqrt {x-1}+1\right ) \]
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Rubi [A] time = 0.56, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 66, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6688, 1586, 6684} \[ 2 \log \left (\sqrt {x-4}+\sqrt {x-1}+1\right ) \]
Antiderivative was successfully verified.
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Rule 1586
Rule 6684
Rule 6688
Rubi steps
\begin {align*} \int \frac {-\sqrt {-4+x}-4 \sqrt {-1+x}+\sqrt {-4+x} x+\sqrt {-1+x} x}{\left (1+\sqrt {-4+x}+\sqrt {-1+x}\right ) \left (4-5 x+x^2\right )} \, dx &=\int \frac {\sqrt {-1+x} \left (-4+\sqrt {-4+x} \sqrt {-1+x}+x\right )}{\left (1+\sqrt {-4+x}+\sqrt {-1+x}\right ) \left (4-5 x+x^2\right )} \, dx\\ &=\int \frac {-4+\sqrt {-4+x} \sqrt {-1+x}+x}{\left (1+\sqrt {-4+x}+\sqrt {-1+x}\right ) (-4+x) \sqrt {-1+x}} \, dx\\ &=2 \log \left (1+\sqrt {-4+x}+\sqrt {-1+x}\right )\\ \end {align*}
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Mathematica [B] time = 1.38, size = 75, normalized size = 3.95 \[ \frac {1}{2} \log \left (-5 x-4 \sqrt {x-4} \sqrt {x-1}+17\right )+\frac {1}{2} \log \left (-2 x-2 \sqrt {x-4} \sqrt {x-1}+5\right )-\tanh ^{-1}\left (\sqrt {x-4}\right )+\tanh ^{-1}\left (\frac {\sqrt {x-1}}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.97, size = 96, normalized size = 5.05 \[ -\frac {1}{2} \, \log \left (-{\left (4 \, x - 11\right )} \sqrt {x - 1} \sqrt {x - 4} + 4 \, x^{2} - 21 \, x + 23\right ) + \frac {1}{2} \, \log \left (\sqrt {x - 1} \sqrt {x - 4} - x + 7\right ) + \frac {1}{2} \, \log \left (x - 5\right ) + \frac {1}{2} \, \log \left (\sqrt {x - 1} + 2\right ) - \frac {1}{2} \, \log \left (\sqrt {x - 1} - 2\right ) - \frac {1}{2} \, \log \left (\sqrt {x - 4} + 1\right ) + \frac {1}{2} \, \log \left (\sqrt {x - 4} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.62, size = 58, normalized size = 3.05 \[ -\log \left (\sqrt {x - 1} - \sqrt {x - 4} + 1\right ) - \log \left (\sqrt {x - 1} - \sqrt {x - 4}\right ) + \log \left (\sqrt {x - 1} + 2\right ) + \log \left ({\left | -\sqrt {x - 1} + \sqrt {x - 4} - 3 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 147, normalized size = 7.74 \[ \frac {7 \sqrt {x -4}\, \sqrt {x -1}\, \arctanh \left (\frac {5 x -17}{4 \sqrt {x^{2}-5 x +4}}\right )}{4 \sqrt {x^{2}-5 x +4}}+\frac {\ln \left (x -5\right )}{2}-\frac {\ln \left (-2+\sqrt {x -1}\right )}{2}+\frac {\ln \left (-1+\sqrt {x -4}\right )}{2}-\frac {\ln \left (1+\sqrt {x -4}\right )}{2}+\frac {\ln \left (\sqrt {x -1}+2\right )}{2}+\frac {\sqrt {x -4}\, \sqrt {x -1}\, \left (-5 \arctanh \left (\frac {5 x -17}{4 \sqrt {x^{2}-5 x +4}}\right )+2 \ln \left (x -\frac {5}{2}+\sqrt {x^{2}-5 x +4}\right )\right )}{4 \sqrt {x^{2}-5 x +4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.65, size = 94, normalized size = 4.95 \[ \frac {1}{2} \, \log \left (x - 1\right ) + \frac {1}{2} \, \log \left (\frac {2 \, x^{2} + 2 \, {\left ({\left (x - 1\right )} \sqrt {x - 4} + 2 \, x - 6\right )} \sqrt {x - 1} + 2 \, {\left (2 \, x - 3\right )} \sqrt {x - 4} - 7 \, x + 3}{2 \, {\left ({\left (x - 1\right )} \sqrt {x - 4} + 2 \, x - 6\right )}}\right ) + \frac {1}{2} \, \log \left (\frac {{\left (x - 1\right )} \sqrt {x - 4} + 2 \, x - 6}{x - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.10, size = 132, normalized size = 6.95 \[ \frac {\ln \left (x-5\right )}{2}+2\,\mathrm {atanh}\left (\frac {\sqrt {x-1}-\sqrt {3}}{\sqrt {x-4}}\right )+\frac {7\,\mathrm {atanh}\left (\frac {4\,\left (\sqrt {x-1}-\sqrt {3}\right )}{\left (\frac {{\left (\sqrt {x-1}-\sqrt {3}\right )}^2}{x-4}+1\right )\,\sqrt {x-4}}\right )}{2}-\frac {5\,\mathrm {atanh}\left (\frac {194400\,\left (\sqrt {x-1}-\sqrt {3}\right )}{\left (\frac {48600\,{\left (\sqrt {x-1}-\sqrt {3}\right )}^2}{x-4}+48600\right )\,\sqrt {x-4}}\right )}{2}-\mathrm {atanh}\left (\sqrt {x-4}\right )+\mathrm {atanh}\left (\frac {\sqrt {x-1}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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