Optimal. Leaf size=21 \[ \log \left (1+\sqrt {-4+x^2}+\sqrt {-1+x^2}\right ) \]
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Rubi [A]
time = 0.19, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 85, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {6816}
\begin {gather*} \log \left (\sqrt {x^2-4}+\sqrt {x^2-1}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 6816
Rubi steps
\begin {align*} \int \frac {x \left (-\sqrt {-4+x^2}+x^2 \sqrt {-4+x^2}-4 \sqrt {-1+x^2}+x^2 \sqrt {-1+x^2}\right )}{\left (4-5 x^2+x^4\right ) \left (1+\sqrt {-4+x^2}+\sqrt {-1+x^2}\right )} \, dx &=\log \left (1+\sqrt {-4+x^2}+\sqrt {-1+x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.62, size = 31, normalized size = 1.48 \begin {gather*} 2 \tanh ^{-1}\left (1-\frac {2}{3} \sqrt {-4+x^2}+\frac {2}{3} \sqrt {-1+x^2}\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. \(1087\) vs.
\(2(17)=34\).
time = 0.20, size = 1088, normalized size = 51.81
method | result | size |
elliptic | \(\frac {\sqrt {\left (x^{2}-4\right ) \left (x^{2}-1\right )}\, \left (\frac {\ln \left (x^{2}-5\right )}{4}+\frac {\ln \left (-\frac {5}{2}+x^{2}+\sqrt {x^{4}-5 x^{2}+4}\right )}{4}+\frac {\arctanh \left (\frac {5 x^{2}-17}{4 \sqrt {\left (x^{2}-5\right )^{2}+5 x^{2}-21}}\right )}{4}+\frac {\arctanh \left (\frac {8+2 \sqrt {5}\, \left (x -\sqrt {5}\right )}{4 \sqrt {\left (x -\sqrt {5}\right )^{2}+2 \sqrt {5}\, \left (x -\sqrt {5}\right )+4}}\right )}{4}+\frac {\arctanh \left (\frac {8-2 \sqrt {5}\, \left (x +\sqrt {5}\right )}{4 \sqrt {\left (x +\sqrt {5}\right )^{2}-2 \sqrt {5}\, \left (x +\sqrt {5}\right )+4}}\right )}{4}-\frac {\arctanh \left (\frac {2+2 \sqrt {5}\, \left (x -\sqrt {5}\right )}{2 \sqrt {\left (x -\sqrt {5}\right )^{2}+2 \sqrt {5}\, \left (x -\sqrt {5}\right )+1}}\right )}{4}-\frac {\arctanh \left (\frac {2-2 \sqrt {5}\, \left (x +\sqrt {5}\right )}{2 \sqrt {\left (x +\sqrt {5}\right )^{2}-2 \sqrt {5}\, \left (x +\sqrt {5}\right )+1}}\right )}{4}\right )}{\sqrt {x^{2}-4}\, \sqrt {x^{2}-1}}\) | \(250\) |
default | \(\text {Expression too large to display}\) | \(1088\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 171 vs.
\(2 (17) = 34\).
time = 2.53, size = 171, normalized size = 8.14 \begin {gather*} \frac {1}{4} \, \log \left (x + 1\right ) + \frac {3}{8} \, \log \left (x - 1\right ) + \frac {1}{8} \, \log \left (x - 2\right ) + \frac {1}{4} \, \log \left (\frac {2 \, x^{4} + 4 \, {\left (x^{2} - 3\right )} \sqrt {x + 1} \sqrt {x - 1} - 7 \, x^{2} + 2 \, {\left ({\left (x^{2} - 1\right )} \sqrt {x + 1} \sqrt {x - 1} \sqrt {x - 2} + {\left (2 \, x^{2} - 3\right )} \sqrt {x - 2}\right )} \sqrt {x + 2} + 3}{2 \, {\left ({\left (x^{2} - 1\right )} \sqrt {x + 1} \sqrt {x - 1} \sqrt {x - 2} + {\left (2 \, x^{2} - 3\right )} \sqrt {x - 2}\right )}}\right ) + \frac {1}{4} \, \log \left (\frac {{\left (x^{2} - 1\right )} \sqrt {x + 1} \sqrt {x - 1} + 2 \, x^{2} - 3}{{\left (x^{2} - 1\right )} \sqrt {x - 1}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 162 vs.
\(2 (17) = 34\).
time = 0.83, size = 162, normalized size = 7.71 \begin {gather*} -\frac {1}{4} \, \log \left (4 \, x^{4} - {\left (4 \, x^{2} - 11\right )} \sqrt {x^{2} - 1} \sqrt {x^{2} - 4} - 21 \, x^{2} + 23\right ) - \frac {1}{4} \, \log \left (x^{2} - \sqrt {x^{2} - 1} {\left (x + 2\right )} + 2 \, x - 1\right ) + \frac {1}{4} \, \log \left (x^{2} - \sqrt {x^{2} - 4} {\left (x + 1\right )} + x - 4\right ) - \frac {1}{4} \, \log \left (x^{2} - \sqrt {x^{2} - 4} {\left (x - 1\right )} - x - 4\right ) + \frac {1}{4} \, \log \left (x^{2} - \sqrt {x^{2} - 1} {\left (x - 2\right )} - 2 \, x - 1\right ) + \frac {1}{4} \, \log \left (x^{2} - 5\right ) + \frac {1}{4} \, \log \left (-x^{2} + \sqrt {x^{2} - 1} \sqrt {x^{2} - 4} + 7\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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. 76 vs.
\(2 (17) = 34\).
time = 0.44, size = 76, normalized size = 3.62 \begin {gather*} -\frac {1}{2} \, \log \left (\sqrt {x^{2} - 1} - \sqrt {x^{2} - 4} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {x^{2} - 1} - \sqrt {x^{2} - 4}\right ) + \frac {1}{2} \, \log \left (\sqrt {x^{2} - 1} + 2\right ) + \frac {1}{2} \, \log \left ({\left | -\sqrt {x^{2} - 1} + \sqrt {x^{2} - 4} - 3 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.37, size = 172, normalized size = 8.19 \begin {gather*} \frac {\ln \left (x-\sqrt {5}\right )}{4}-\mathrm {atanh}\left (\frac {\sqrt {3}-\sqrt {x^2-1}}{\sqrt {x^2-4}}\right )+\frac {\mathrm {atanh}\left (\frac {\sqrt {x^2-1}}{2}\right )}{2}+\frac {\ln \left (x+\sqrt {5}\right )}{4}-\frac {7\,\mathrm {atanh}\left (\frac {4\,\left (\sqrt {3}-\sqrt {x^2-1}\right )}{\sqrt {x^2-4}\,\left (\frac {{\left (\sqrt {3}-\sqrt {x^2-1}\right )}^2}{x^2-4}+1\right )}\right )}{4}+\frac {5\,\mathrm {atanh}\left (\frac {12150\,\left (\sqrt {3}-\sqrt {x^2-1}\right )}{\sqrt {x^2-4}\,\left (\frac {6075\,{\left (\sqrt {3}-\sqrt {x^2-1}\right )}^2}{2\,\left (x^2-4\right )}+\frac {6075}{2}\right )}\right )}{4}-\frac {\mathrm {atanh}\left (\sqrt {x^2-4}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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