Optimal. Leaf size=38 \[ \frac{a}{f \sqrt{a \cos ^2(e+f x)}}+\frac{\sqrt{a \cos ^2(e+f x)}}{f} \]
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Rubi [A] time = 0.104688, antiderivative size = 38, 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, 3205, 16, 43} \[ \frac{a}{f \sqrt{a \cos ^2(e+f x)}}+\frac{\sqrt{a \cos ^2(e+f x)}}{f} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3205
Rule 16
Rule 43
Rubi steps
\begin{align*} \int \sqrt{a-a \sin ^2(e+f x)} \tan ^3(e+f x) \, dx &=\int \sqrt{a \cos ^2(e+f x)} \tan ^3(e+f x) \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(1-x) \sqrt{a x}}{x^2} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac{a^2 \operatorname{Subst}\left (\int \frac{1-x}{(a x)^{3/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac{a^2 \operatorname{Subst}\left (\int \left (\frac{1}{(a x)^{3/2}}-\frac{1}{a \sqrt{a x}}\right ) \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=\frac{a}{f \sqrt{a \cos ^2(e+f x)}}+\frac{\sqrt{a \cos ^2(e+f x)}}{f}\\ \end{align*}
Mathematica [A] time = 0.081039, size = 29, normalized size = 0.76 \[ \frac{a \left (\cos ^2(e+f x)+1\right )}{f \sqrt{a \cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.115, size = 35, normalized size = 0.9 \begin{align*}{\frac{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}+1}{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}f}\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 = 1.00257, size = 62, normalized size = 1.63 \begin{align*} \frac{\sqrt{-a \sin \left (f x + e\right )^{2} + a} a^{2} + \frac{a^{3}}{\sqrt{-a \sin \left (f x + e\right )^{2} + a}}}{a^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59576, size = 86, normalized size = 2.26 \begin{align*} \frac{\sqrt{a \cos \left (f x + e\right )^{2}}{\left (\cos \left (f x + e\right )^{2} + 1\right )}}{f \cos \left (f x + e\right )^{2}} \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 ^{3}{\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 = 1.50308, size = 53, normalized size = 1.39 \begin{align*} \frac{4 \, \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 )^{4} - 1\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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