Optimal. Leaf size=14 \[ -\frac{\cot (x)}{\sqrt{a \csc ^2(x)}} \]
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Rubi [A] time = 0.0134393, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4122, 191} \[ -\frac{\cot (x)}{\sqrt{a \csc ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 4122
Rule 191
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*}
Mathematica [A] time = 0.0064168, size = 14, normalized size = 1. \[ -\frac{\cot (x)}{\sqrt{a \csc ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.132, size = 27, normalized size = 1.9 \begin{align*}{\frac{\sqrt{4}\sin \left ( x \right ) }{-2+2\,\cos \left ( x \right ) }{\frac{1}{\sqrt{-{\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{2}-1}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49698, size = 18, normalized size = 1.29 \begin{align*} -\frac{1}{\sqrt{\tan \left (x\right )^{2} + 1} \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.468744, size = 57, normalized size = 4.07 \begin{align*} -\frac{\sqrt{-\frac{a}{\cos \left (x\right )^{2} - 1}} \cos \left (x\right ) \sin \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.789281, size = 17, normalized size = 1.21 \begin{align*} - \frac{\cot{\left (x \right )}}{\sqrt{a} \sqrt{\csc ^{2}{\left (x \right )}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28229, size = 46, normalized size = 3.29 \begin{align*} \frac{2 \, \mathrm{sgn}\left (\sin \left (x\right )\right )}{\sqrt{a}} + \frac{2}{\sqrt{a}{\left (\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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