Optimal. Leaf size=124 \[ -\frac{\tan (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}-\frac{\csc ^5(e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{5 f}+\frac{\csc ^3(e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}-\frac{3 \csc (e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{f} \]
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Rubi [A] time = 0.121511, antiderivative size = 124, 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, 270} \[ -\frac{\tan (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}-\frac{\csc ^5(e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{5 f}+\frac{\csc ^3(e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}-\frac{3 \csc (e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{f} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3207
Rule 2590
Rule 270
Rubi steps
\begin{align*} \int \cot ^6(e+f x) \sqrt{a-a \sin ^2(e+f x)} \, dx &=\int \sqrt{a \cos ^2(e+f x)} \cot ^6(e+f x) \, dx\\ &=\left (\sqrt{a \cos ^2(e+f x)} \sec (e+f x)\right ) \int \cos (e+f x) \cot ^6(e+f x) \, dx\\ &=-\frac{\left (\sqrt{a \cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{x^6} \, 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^6}-\frac{3}{x^4}+\frac{3}{x^2}\right ) \, dx,x,-\sin (e+f x)\right )}{f}\\ &=-\frac{3 \sqrt{a \cos ^2(e+f x)} \csc (e+f x) \sec (e+f x)}{f}+\frac{\sqrt{a \cos ^2(e+f x)} \csc ^3(e+f x) \sec (e+f x)}{f}-\frac{\sqrt{a \cos ^2(e+f x)} \csc ^5(e+f x) \sec (e+f x)}{5 f}-\frac{\sqrt{a \cos ^2(e+f x)} \tan (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.195662, size = 67, normalized size = 0.54 \[ \frac{(235 \cos (2 (e+f x))-90 \cos (4 (e+f x))+5 \cos (6 (e+f x))-182) \csc ^5(e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{160 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.803, size = 65, normalized size = 0.5 \begin{align*} -{\frac{\cos \left ( fx+e \right ) a \left ( 5\, \left ( \sin \left ( fx+e \right ) \right ) ^{6}+15\, \left ( \sin \left ( fx+e \right ) \right ) ^{4}-5\, \left ( \sin \left ( fx+e \right ) \right ) ^{2}+1 \right ) }{5\, \left ( \sin \left ( fx+e \right ) \right ) ^{5}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 = 1.50161, size = 92, normalized size = 0.74 \begin{align*} -\frac{16 \, \sqrt{a} \tan \left (f x + e\right )^{6} + 8 \, \sqrt{a} \tan \left (f x + e\right )^{4} - 2 \, \sqrt{a} \tan \left (f x + e\right )^{2} + \sqrt{a}}{5 \, \sqrt{\tan \left (f x + e\right )^{2} + 1} f \tan \left (f x + e\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64918, size = 221, normalized size = 1.78 \begin{align*} \frac{{\left (5 \, \cos \left (f x + e\right )^{6} - 30 \, \cos \left (f x + e\right )^{4} + 40 \, \cos \left (f x + e\right )^{2} - 16\right )} \sqrt{a \cos \left (f x + e\right )^{2}}}{5 \,{\left (f \cos \left (f x + e\right )^{5} - 2 \, f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24039, size = 235, normalized size = 1.9 \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 )}^{5} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1\right ) - 20 \,{\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 )}^{3} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1\right ) + 240 \,{\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{320 \, \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}}{160 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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