Optimal. Leaf size=13 \[ 2 \sqrt {\cos (x) \cot (x)} \tan (x) \]
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Rubi [A]
time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4482, 4485,
2669} \begin {gather*} 2 \tan (x) \sqrt {\cos (x) \cot (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2669
Rule 4482
Rule 4485
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\int \sqrt {\cos (x) \cot (x)} \, dx\\ &=\frac {\sqrt {\cos (x) \cot (x)} \int \sqrt {\cos (x)} \sqrt {\cot (x)} \, dx}{\sqrt {\cos (x)} \sqrt {\cot (x)}}\\ &=2 \sqrt {\cos (x) \cot (x)} \tan (x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} 2 \sqrt {\cos (x) \cot (x)} \tan (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 12, normalized size = 0.92
method | result | size |
default | \(2 \sqrt {\cos \left (x \right ) \cot \left (x \right )}\, \tan \left (x \right )\) | \(12\) |
risch | \(-\frac {i \sqrt {2}\, \sqrt {\frac {i \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-i x}}{{\mathrm e}^{2 i x}-1}}\, \left ({\mathrm e}^{2 i x}-1\right )}{{\mathrm e}^{2 i x}+1}\) | \(51\) |
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. 188 vs.
\(2 (11) = 22\).
time = 0.60, size = 188, normalized size = 14.46 \begin {gather*} \frac {{\left ({\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) + \sin \left (\frac {3}{2} \, x\right ) + \sin \left (\frac {1}{2} \, x\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right ) - {\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) - \sin \left (\frac {3}{2} \, x\right ) - \sin \left (\frac {1}{2} \, x\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right )\right ) - {\left ({\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) - \sin \left (\frac {3}{2} \, x\right ) - \sin \left (\frac {1}{2} \, x\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right ) + {\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) + \sin \left (\frac {3}{2} \, x\right ) + \sin \left (\frac {1}{2} \, x\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right )\right )}{{\left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right )}^{\frac {1}{4}} {\left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right )}^{\frac {1}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.60, size = 19, normalized size = 1.46 \begin {gather*} \frac {2 \, \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} \sin \left (x\right )}{\cos \left (x\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 \sqrt {- \sin {\left (x \right )} + \csc {\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.00, size = 15, normalized size = 1.15 \begin {gather*} \frac {2\,\left |\cos \left (x\right )\right |}{\cos \left (x\right )\,\sqrt {\frac {1}{\sin \left (x\right )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}
Warning: Unable to verify antiderivative.
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