Optimal. Leaf size=44 \[ \frac{a}{5 f \left (a \cos ^2(e+f x)\right )^{5/2}}-\frac{1}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.119488, antiderivative size = 44, 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} \[ \frac{a}{5 f \left (a \cos ^2(e+f x)\right )^{5/2}}-\frac{1}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]
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)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac{\tan ^3(e+f x)}{\left (a \cos ^2(e+f x)\right )^{3/2}} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1-x}{x^2 (a x)^{3/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac{a^2 \operatorname{Subst}\left (\int \frac{1-x}{(a x)^{7/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac{a^2 \operatorname{Subst}\left (\int \left (\frac{1}{(a x)^{7/2}}-\frac{1}{a (a x)^{5/2}}\right ) \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=\frac{a}{5 f \left (a \cos ^2(e+f x)\right )^{5/2}}-\frac{1}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.110993, size = 34, normalized size = 0.77 \[ \frac{a \left (3-5 \cos ^2(e+f x)\right )}{15 f \left (a \cos ^2(e+f x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.908, size = 41, normalized size = 0.9 \begin{align*} -{\frac{5\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-3}{15\,{a}^{2} \left ( \cos \left ( fx+e \right ) \right ) ^{6}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 = 1.03413, size = 65, normalized size = 1.48 \begin{align*} \frac{5 \,{\left (a \sin \left (f x + e\right )^{2} - a\right )} a^{2} + 3 \, a^{3}}{15 \,{\left (-a \sin \left (f x + e\right )^{2} + a\right )}^{\frac{5}{2}} a^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59149, size = 103, normalized size = 2.34 \begin{align*} -\frac{\sqrt{a \cos \left (f x + e\right )^{2}}{\left (5 \, \cos \left (f x + e\right )^{2} - 3\right )}}{15 \, a^{2} f \cos \left (f x + e\right )^{6}} \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 )}}{\left (- a \left (\sin{\left (e + f x \right )} - 1\right ) \left (\sin{\left (e + f x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30612, size = 78, normalized size = 1.77 \begin{align*} \frac{{\left (\tan \left (f x + e\right )^{2} + 1\right )}^{2}{\left (3 \, a - \frac{5 \, a}{\tan \left (f x + e\right )^{2} + 1}\right )}}{15 \, a^{2} f \sqrt{\frac{a}{\tan \left (f x + e\right )^{2} + 1}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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