Optimal. Leaf size=30 \[ -\frac {\tanh ^{-1}\left (\frac {\sin (2 x)}{\sqrt {2} \sqrt {1-\cos (2 x)}}\right )}{\sqrt {2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2728, 212}
\begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sin (2 x)}{\sqrt {2} \sqrt {1-\cos (2 x)}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2728
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-\cos (2 x)}} \, dx &=-\text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\frac {\sin (2 x)}{\sqrt {1-\cos (2 x)}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sin (2 x)}{\sqrt {2} \sqrt {1-\cos (2 x)}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 33, normalized size = 1.10 \begin {gather*} -\frac {\left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right ) \sin (x)}{\sqrt {1-\cos (2 x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 17, normalized size = 0.57
method | result | size |
default | \(-\frac {\sin \left (x \right ) \arctanh \left (\cos \left (x \right )\right ) \sqrt {2}}{\sqrt {2-2 \cos \left (2 x \right )}}\) | \(17\) |
risch | \(\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )}{\sqrt {-\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}-\frac {\sqrt {2}\, \ln \left (1+{\mathrm e}^{i x}\right ) \sin \left (x \right )}{\sqrt {-\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}\) | \(67\) |
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. 101 vs.
\(2 (25) = 50\).
time = 3.44, size = 101, normalized size = 3.37 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right )\right )\right )^{2} + 2 \, \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right )\right )\right ) + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right )\right )\right )^{2} - 2 \, \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right )\right )\right ) + 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. 58 vs.
\(2 (25) = 50\).
time = 1.09, size = 58, normalized size = 1.93 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (-\frac {{\left (\cos \left (2 \, x\right ) + 3\right )} \sin \left (2 \, x\right ) - 2 \, {\left (\sqrt {2} \cos \left (2 \, x\right ) + \sqrt {2}\right )} \sqrt {-\cos \left (2 \, x\right ) + 1}}{{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {1 - \cos {\left (2 x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.08, size = 16, normalized size = 0.53 \begin {gather*} \frac {\sqrt {2} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 28, normalized size = 0.93 \begin {gather*} -\frac {\sqrt {2}\,\sin \left (2\,x\right )\,\mathrm {atanh}\left (\sqrt {{\cos \left (x\right )}^2}\right )}{2\,\sqrt {1-{\cos \left (2\,x\right )}^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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