Optimal. Leaf size=57 \[ -\frac{\tan (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}-\frac{\csc (e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{f} \]
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Rubi [A] time = 0.111669, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3176, 3207, 2590, 14} \[ -\frac{\tan (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}-\frac{\csc (e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{f} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3207
Rule 2590
Rule 14
Rubi steps
\begin{align*} \int \cot ^2(e+f x) \sqrt{a-a \sin ^2(e+f x)} \, dx &=\int \sqrt{a \cos ^2(e+f x)} \cot ^2(e+f x) \, dx\\ &=\left (\sqrt{a \cos ^2(e+f x)} \sec (e+f x)\right ) \int \cos (e+f x) \cot ^2(e+f x) \, dx\\ &=-\frac{\left (\sqrt{a \cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{x^2} \, dx,x,-\sin (e+f x)\right )}{f}\\ &=-\frac{\left (\sqrt{a \cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \left (-1+\frac{1}{x^2}\right ) \, dx,x,-\sin (e+f x)\right )}{f}\\ &=-\frac{\sqrt{a \cos ^2(e+f x)} \csc (e+f x) \sec (e+f x)}{f}-\frac{\sqrt{a \cos ^2(e+f x)} \tan (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.0753841, size = 35, normalized size = 0.61 \[ -\frac{\tan (e+f x) \left (\csc ^2(e+f x)+1\right ) \sqrt{a \cos ^2(e+f x)}}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.707, size = 43, normalized size = 0.8 \begin{align*} -{\frac{\cos \left ( fx+e \right ) a \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{2}+1 \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 [A] time = 1.48685, size = 57, normalized size = 1. \begin{align*} -\frac{2 \, \sqrt{a} \tan \left (f x + e\right )^{2} + \sqrt{a}}{\sqrt{\tan \left (f x + e\right )^{2} + 1} f \tan \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57825, size = 101, normalized size = 1.77 \begin{align*} \frac{\sqrt{a \cos \left (f x + e\right )^{2}}{\left (\cos \left (f x + e\right )^{2} - 2\right )}}{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 \sqrt{- a \left (\sin{\left (e + f x \right )} - 1\right ) \left (\sin{\left (e + f x \right )} + 1\right )} \cot ^{2}{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19357, size = 122, normalized size = 2.14 \begin{align*} \frac{{\left ({\left (\frac{1}{\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )} + \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1\right ) + \frac{4 \, \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1\right )}{\frac{1}{\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )} + \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}\right )} \sqrt{a}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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