Optimal. Leaf size=12 \[ -\frac {\cot (x)}{\sqrt {\csc ^2(x)}} \]
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Rubi [A] time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4122, 191} \[ -\frac {\cot (x)}{\sqrt {\csc ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\csc ^2(x)}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{3/2}} \, dx,x,\cot (x)\right )\\ &=-\frac {\cot (x)}{\sqrt {\csc ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.00 \[ -\frac {\cot (x)}{\sqrt {\csc ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 4, normalized size = 0.33 \[ -\cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 11, normalized size = 0.92 \[ -\cos \relax (x) \mathrm {sgn}\left (\sin \relax (x)\right ) + \mathrm {sgn}\left (\sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.58, size = 26, normalized size = 2.17 \[ \frac {\sin \relax (x ) \sqrt {4}}{2 \sqrt {-\frac {1}{-1+\cos ^{2}\relax (x )}}\, \left (-1+\cos \relax (x )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 10, normalized size = 0.83 \[ -\frac {1}{\sqrt {\tan \relax (x)^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 12, normalized size = 1.00 \[ -\frac {\sin \left (2\,x\right )}{2\,\sqrt {{\sin \relax (x)}^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 12, normalized size = 1.00 \[ - \frac {\cot {\relax (x )}}{\sqrt {\csc ^{2}{\relax (x )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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