Optimal. Leaf size=68 \[ \frac{a^2}{7 f \left (a \cos ^2(e+f x)\right )^{7/2}}-\frac{2 a}{5 f \left (a \cos ^2(e+f x)\right )^{5/2}}+\frac{1}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.12691, antiderivative size = 68, 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} \[ \frac{a^2}{7 f \left (a \cos ^2(e+f x)\right )^{7/2}}-\frac{2 a}{5 f \left (a \cos ^2(e+f x)\right )^{5/2}}+\frac{1}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3205
Rule 16
Rule 43
Rubi steps
\begin{align*} \int \frac{\tan ^5(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac{\tan ^5(e+f x)}{\left (a \cos ^2(e+f x)\right )^{3/2}} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(1-x)^2}{x^3 (a x)^{3/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac{a^3 \operatorname{Subst}\left (\int \frac{(1-x)^2}{(a x)^{9/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac{a^3 \operatorname{Subst}\left (\int \left (\frac{1}{(a x)^{9/2}}-\frac{2}{a (a x)^{7/2}}+\frac{1}{a^2 (a x)^{5/2}}\right ) \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=\frac{a^2}{7 f \left (a \cos ^2(e+f x)\right )^{7/2}}-\frac{2 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 = 0.107422, size = 51, normalized size = 0.75 \[ \frac{\left (35 \cos ^4(e+f x)-42 \cos ^2(e+f x)+15\right ) \sec ^4(e+f x)}{105 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.991, size = 51, normalized size = 0.8 \begin{align*}{\frac{35\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}-42\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+15}{105\,{a}^{2} \left ( \cos \left ( fx+e \right ) \right ) ^{8}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 = 1.0354, size = 93, normalized size = 1.37 \begin{align*} \frac{35 \,{\left (a \sin \left (f x + e\right )^{2} - a\right )}^{2} a^{3} + 42 \,{\left (a \sin \left (f x + e\right )^{2} - a\right )} a^{4} + 15 \, a^{5}}{105 \,{\left (-a \sin \left (f x + e\right )^{2} + a\right )}^{\frac{7}{2}} a^{3} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67547, size = 132, normalized size = 1.94 \begin{align*} \frac{{\left (35 \, \cos \left (f x + e\right )^{4} - 42 \, \cos \left (f x + e\right )^{2} + 15\right )} \sqrt{a \cos \left (f x + e\right )^{2}}}{105 \, a^{2} f \cos \left (f x + e\right )^{8}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (f x + e\right )^{5}}{{\left (-a \sin \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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