Optimal. Leaf size=25 \[ -\frac{\cot (e+f x)}{f \sqrt{a \cos ^2(e+f x)}} \]
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Rubi [A] time = 0.101829, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3176, 3207, 2606, 8} \[ -\frac{\cot (e+f x)}{f \sqrt{a \cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3207
Rule 2606
Rule 8
Rubi steps
\begin{align*} \int \frac{\cot ^2(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx &=\int \frac{\cot ^2(e+f x)}{\sqrt{a \cos ^2(e+f x)}} \, dx\\ &=\frac{\cos (e+f x) \int \cot (e+f x) \csc (e+f x) \, dx}{\sqrt{a \cos ^2(e+f x)}}\\ &=-\frac{\cos (e+f x) \operatorname{Subst}(\int 1 \, dx,x,\csc (e+f x))}{f \sqrt{a \cos ^2(e+f x)}}\\ &=-\frac{\cot (e+f x)}{f \sqrt{a \cos ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.0291194, size = 25, normalized size = 1. \[ -\frac{\cot (e+f x)}{f \sqrt{a \cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.246, size = 32, normalized size = 1.3 \begin{align*} -{\frac{\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) f}{\frac{1}{\sqrt{a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.72, size = 122, normalized size = 4.88 \begin{align*} -\frac{2 \,{\left (\cos \left (f x + e\right ) \sin \left (2 \, f x + 2 \, e\right ) - \cos \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + \sin \left (f x + e\right )\right )} \sqrt{a}}{{\left (a \cos \left (2 \, f x + 2 \, e\right )^{2} + a \sin \left (2 \, f x + 2 \, e\right )^{2} - 2 \, a \cos \left (2 \, f x + 2 \, e\right ) + a\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54424, size = 77, normalized size = 3.08 \begin{align*} -\frac{\sqrt{a \cos \left (f x + e\right )^{2}}}{a f \cos \left (f x + e\right ) \sin \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot ^{2}{\left (e + f x \right )}}{\sqrt{- a \left (\sin{\left (e + f x \right )} - 1\right ) \left (\sin{\left (e + f x \right )} + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2792, size = 90, normalized size = 3.6 \begin{align*} \frac{\frac{\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1\right )} + \frac{1}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1\right ) \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}}{2 \, \sqrt{a} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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