Optimal. Leaf size=64 \[ \frac{a^2}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}}-\frac{2 a}{f \sqrt{a \cos ^2(e+f x)}}-\frac{\sqrt{a \cos ^2(e+f x)}}{f} \]
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Rubi [A] time = 0.109609, antiderivative size = 64, 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} \[ \frac{a^2}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}}-\frac{2 a}{f \sqrt{a \cos ^2(e+f x)}}-\frac{\sqrt{a \cos ^2(e+f x)}}{f} \]
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 ^5(e+f x) \, dx &=\int \sqrt{a \cos ^2(e+f x)} \tan ^5(e+f x) \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(1-x)^2 \sqrt{a x}}{x^3} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac{a^3 \operatorname{Subst}\left (\int \frac{(1-x)^2}{(a x)^{5/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac{a^3 \operatorname{Subst}\left (\int \left (\frac{1}{(a x)^{5/2}}-\frac{2}{a (a x)^{3/2}}+\frac{1}{a^2 \sqrt{a x}}\right ) \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=\frac{a^2}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}}-\frac{2 a}{f \sqrt{a \cos ^2(e+f x)}}-\frac{\sqrt{a \cos ^2(e+f x)}}{f}\\ \end{align*}
Mathematica [A] time = 0.0937716, size = 51, normalized size = 0.8 \[ -\frac{\left (3 \cos ^4(e+f x)+6 \cos ^2(e+f x)-1\right ) \sec ^4(e+f x) \sqrt{a \cos ^2(e+f x)}}{3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.165, size = 48, normalized size = 0.8 \begin{align*} -{\frac{3\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+6\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-1}{3\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}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 = 1.01499, size = 93, normalized size = 1.45 \begin{align*} -\frac{3 \, \sqrt{-a \sin \left (f x + e\right )^{2} + a} a^{3} - \frac{6 \,{\left (a \sin \left (f x + e\right )^{2} - a\right )} a^{4} + a^{5}}{{\left (-a \sin \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}}}}{3 \, a^{3} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65234, size = 122, normalized size = 1.91 \begin{align*} -\frac{{\left (3 \, \cos \left (f x + e\right )^{4} + 6 \, \cos \left (f x + e\right )^{2} - 1\right )} \sqrt{a \cos \left (f x + e\right )^{2}}}{3 \, f \cos \left (f x + e\right )^{4}} \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 [B] time = 2.74912, size = 184, normalized size = 2.88 \begin{align*} \frac{2 \, \sqrt{a}{\left (\frac{3 \, \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1\right )}{\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1} - \frac{3 \, \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1\right ) \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 12 \, \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1\right ) \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 5 \, \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 )}^{3}}\right )}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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