Optimal. Leaf size=50 \[ \frac {\sqrt {a \cos ^2(e+f x)}}{f}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{f} \]
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Rubi [A] time = 0.08, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3176, 3205, 50, 63, 206} \[ \frac {\sqrt {a \cos ^2(e+f x)}}{f}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 3176
Rule 3205
Rubi steps
\begin {align*} \int \cot (e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx &=\int \sqrt {a \cos ^2(e+f x)} \cot (e+f x) \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a x}}{1-x} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=\frac {\sqrt {a \cos ^2(e+f x)}}{f}-\frac {a \operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {a x}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=\frac {\sqrt {a \cos ^2(e+f x)}}{f}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a \cos ^2(e+f x)}\right )}{f}\\ &=-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{f}+\frac {\sqrt {a \cos ^2(e+f x)}}{f}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 55, normalized size = 1.10 \[ \frac {\sec (e+f x) \sqrt {a \cos ^2(e+f x)} \left (\cos (e+f x)+\log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 57, normalized size = 1.14 \[ \frac {\sqrt {a \cos \left (f x + e\right )^{2}} {\left (2 \, \cos \left (f x + e\right ) - \log \left (-\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1}\right )\right )}}{2 \, f \cos \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 55, normalized size = 1.10 \[ \frac {a {\left (\frac {\arctan \left (\frac {\sqrt {-a \sin \left (f x + e\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \frac {\sqrt {-a \sin \left (f x + e\right )^{2} + a}}{a}\right )}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.64, size = 55, normalized size = 1.10 \[ -\frac {\sqrt {a}\, \ln \left (\frac {2 \sqrt {a}\, \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}+2 a}{\sin \left (f x +e \right )}\right )}{f}+\frac {\sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 70, normalized size = 1.40 \[ -\frac {\sqrt {a} \log \left (\frac {2 \, \sqrt {-a \sin \left (f x + e\right )^{2} + a} \sqrt {a}}{{\left | \sin \left (f x + e\right ) \right |}} + \frac {2 \, a}{{\left | \sin \left (f x + e\right ) \right |}}\right ) - \sqrt {-a \sin \left (f x + e\right )^{2} + a}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {cot}\left (e+f\,x\right )\,\sqrt {a-a\,{\sin \left (e+f\,x\right )}^2} \,d x \]
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 \[ \int \sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )} \cot {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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