Optimal. Leaf size=12 \[ -\cot (x) \sqrt {\sin ^2(x)} \]
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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3255, 3286,
2718} \begin {gather*} \sqrt {\sin ^2(x)} (-\cot (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3255
Rule 3286
Rubi steps
\begin {align*} \int \sqrt {1-\cos ^2(x)} \, dx &=\int \sqrt {\sin ^2(x)} \, dx\\ &=\left (\csc (x) \sqrt {\sin ^2(x)}\right ) \int \sin (x) \, dx\\ &=-\cot (x) \sqrt {\sin ^2(x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 12, normalized size = 1.00 \begin {gather*} -\cot (x) \sqrt {\sin ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 13, normalized size = 1.08
method | result | size |
default | \(-\frac {2 \sin \left (x \right ) \cos \left (x \right )}{\sqrt {2-2 \cos \left (2 x \right )}}\) | \(13\) |
risch | \(-\frac {i \sqrt {-\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}\, {\mathrm e}^{2 i x}}{2 \left ({\mathrm e}^{2 i x}-1\right )}-\frac {i \sqrt {-\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}{2 \left ({\mathrm e}^{2 i x}-1\right )}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 10, normalized size = 0.83 \begin {gather*} -\frac {1}{\sqrt {\tan \left (x\right )^{2} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 4, normalized size = 0.33 \begin {gather*} -\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 {1 - \cos ^{2}{\left (x \right )}}\, dx \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. 24 vs.
\(2 (10) = 20\).
time = 0.40, size = 24, normalized size = 2.00 \begin {gather*} -\frac {2 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )\right )}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 10, normalized size = 0.83 \begin {gather*} -\mathrm {cot}\left (x\right )\,\sqrt {{\sin \left (x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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