Optimal. Leaf size=42 \[ \frac{a}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}}-\frac{1}{f \sqrt{a \cos ^2(e+f x)}} \]
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Rubi [A] time = 0.107025, antiderivative size = 42, 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} \[ \frac{a}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}}-\frac{1}{f \sqrt{a \cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3205
Rule 16
Rule 43
Rubi steps
\begin{align*} \int \frac{\tan ^3(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx &=\int \frac{\tan ^3(e+f x)}{\sqrt{a \cos ^2(e+f x)}} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1-x}{x^2 \sqrt{a x}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac{a^2 \operatorname{Subst}\left (\int \frac{1-x}{(a x)^{5/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac{a^2 \operatorname{Subst}\left (\int \left (\frac{1}{(a x)^{5/2}}-\frac{1}{a (a x)^{3/2}}\right ) \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=\frac{a}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}}-\frac{1}{f \sqrt{a \cos ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.0638454, size = 31, normalized size = 0.74 \[ \frac{\sec ^2(e+f x)-3}{3 f \sqrt{a \cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.779, size = 41, normalized size = 1. \begin{align*} -{\frac{3\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-1}{3\,a \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 = 1.0294, size = 62, normalized size = 1.48 \begin{align*} \frac{3 \,{\left (a \sin \left (f x + e\right )^{2} - a\right )} a^{2} + a^{3}}{3 \,{\left (-a \sin \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} a^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61233, size = 99, normalized size = 2.36 \begin{align*} -\frac{\sqrt{a \cos \left (f x + e\right )^{2}}{\left (3 \, \cos \left (f x + e\right )^{2} - 1\right )}}{3 \, a 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{3}{\left (e + f x \right )}}{\sqrt{- a \left (\sin{\left (e + f x \right )} - 1\right ) \left (\sin{\left (e + f x \right )} + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.65157, size = 77, normalized size = 1.83 \begin{align*} \frac{4 \,{\left (3 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right )}}{3 \,{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right )}^{3} \sqrt{a} f \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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