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 = 0.11, antiderivative size = 57, normalized size of antiderivative = 1.00, 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 14
Rule 2590
Rule 3176
Rule 3207
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*}
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Mathematica [A] time = 0.07, 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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fricas [A] time = 0.44, size = 42, normalized size = 0.74 \[ \frac {\sqrt {a \cos \left (f x + e\right )^{2}} {\left (\cos \left (f x + e\right )^{2} - 2\right )}}{f \cos \left (f x + e\right ) \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.93, size = 43, normalized size = 0.75 \[ -\frac {\cos \left (f x +e \right ) a \left (1+\sin ^{2}\left (f x +e \right )\right )}{\sin \left (f x +e \right ) \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 42, normalized size = 0.74 \[ -\frac {2 \, \sqrt {a} \tan \left (f x + e\right )^{2} + \sqrt {a}}{\sqrt {\tan \left (f x + e\right )^{2} + 1} f \tan \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 18.53, size = 88, normalized size = 1.54 \[ \frac {\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,\left (-{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,6{}\mathrm {i}+{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}{f\,\left ({\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )} \cot ^{2}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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