Optimal. Leaf size=14 \[ -\frac {\cot (x)}{\sqrt {a \csc ^2(x)}} \]
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Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4122, 191} \[ -\frac {\cot (x)}{\sqrt {a \csc ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a \csc ^2(x)}} \, dx &=-\left (a \operatorname {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\cot (x)\right )\right )\\ &=-\frac {\cot (x)}{\sqrt {a \csc ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 1.00 \[ -\frac {\cot (x)}{\sqrt {a \csc ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 22, normalized size = 1.57 \[ -\frac {\sqrt {-\frac {a}{\cos \relax (x)^{2} - 1}} \cos \relax (x) \sin \relax (x)}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 34, normalized size = 2.43 \[ \frac {2 \, \mathrm {sgn}\left (\sin \relax (x)\right )}{\sqrt {a}} + \frac {2}{\sqrt {a} {\left (\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1} - 1\right )} \mathrm {sgn}\left (\sin \relax (x)\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.62, size = 27, normalized size = 1.93 \[ \frac {\sin \relax (x ) \sqrt {4}}{2 \sqrt {-\frac {a}{-1+\cos ^{2}\relax (x )}}\, \left (-1+\cos \relax (x )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 13, normalized size = 0.93 \[ -\frac {1}{\sqrt {\tan \relax (x)^{2} + 1} \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 15, normalized size = 1.07 \[ -\frac {\sin \left (2\,x\right )}{2\,\sqrt {a}\,\sqrt {{\sin \relax (x)}^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.45, size = 17, normalized size = 1.21 \[ - \frac {\cot {\relax (x )}}{\sqrt {a} \sqrt {\csc ^{2}{\relax (x )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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