Optimal. Leaf size=19 \[ -\frac{\sqrt{a \cos ^2(e+f x)}}{f} \]
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Rubi [A] time = 0.0629348, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3176, 3205, 16, 32} \[ -\frac{\sqrt{a \cos ^2(e+f x)}}{f} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3205
Rule 16
Rule 32
Rubi steps
\begin{align*} \int \sqrt{a-a \sin ^2(e+f x)} \tan (e+f x) \, dx &=\int \sqrt{a \cos ^2(e+f x)} \tan (e+f x) \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a x}}{x} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{1}{\sqrt{a x}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac{\sqrt{a \cos ^2(e+f x)}}{f}\\ \end{align*}
Mathematica [A] time = 0.0397946, size = 19, normalized size = 1. \[ -\frac{\sqrt{a \cos ^2(e+f x)}}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.131, size = 21, normalized size = 1.1 \begin{align*} -{\frac{1}{f}\sqrt{a-a \left ( \sin \left ( fx+e \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6049, size = 36, normalized size = 1.89 \begin{align*} -\frac{\sqrt{a \cos \left (f x + e\right )^{2}}}{f} \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 )} \tan{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1338, size = 53, normalized size = 2.79 \begin{align*} \frac{2 \, \sqrt{a} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1\right )}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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