Optimal. Leaf size=15 \[ -\frac {\tanh ^{-1}(\cos (x)) \sin (x)}{\sqrt {\sin ^2(x)}} \]
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Rubi [A]
time = 0.01, antiderivative size = 15, 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,
3855} \begin {gather*} -\frac {\sin (x) \tanh ^{-1}(\cos (x))}{\sqrt {\sin ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3255
Rule 3286
Rule 3855
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-\cos ^2(x)}} \, dx &=\int \frac {1}{\sqrt {\sin ^2(x)}} \, dx\\ &=\frac {\sin (x) \int \csc (x) \, dx}{\sqrt {\sin ^2(x)}}\\ &=-\frac {\tanh ^{-1}(\cos (x)) \sin (x)}{\sqrt {\sin ^2(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 28, normalized size = 1.87 \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 {\sin ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 14, normalized size = 0.93
method | result | size |
default | \(-\frac {2 \arctanh \left (\cos \left (x \right )\right ) \sin \left (x \right )}{\sqrt {2-2 \cos \left (2 x \right )}}\) | \(14\) |
risch | \(\frac {2 \ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )}{\sqrt {-\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}-\frac {2 \ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )}{\sqrt {-\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}\) | \(62\) |
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. 35 vs.
\(2 (13) = 26\).
time = 0.51, size = 35, normalized size = 2.33 \begin {gather*} \frac {1}{2} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) - \frac {1}{2} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 19, normalized size = 1.27 \begin {gather*} -\frac {1}{2} \, \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + \frac {1}{2} \, \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\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 ^{2}{\left (x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 21, normalized size = 1.40 \begin {gather*} \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int \frac {1}{\sqrt {1-{\cos \left (x\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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