Optimal. Leaf size=21 \[ \frac{1}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.0744949, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3176, 3205, 16, 32} \[ \frac{1}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3205
Rule 16
Rule 32
Rubi steps
\begin{align*} \int \frac{\tan (e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac{\tan (e+f x)}{\left (a \cos ^2(e+f x)\right )^{3/2}} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x (a x)^{3/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{1}{(a x)^{5/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=\frac{1}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0300188, size = 21, normalized size = 1. \[ \frac{1}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.119, size = 21, normalized size = 1. \begin{align*}{\frac{1}{3\,f} \left ( a-a \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56206, size = 69, normalized size = 3.29 \begin{align*} \frac{\sqrt{a \cos \left (f x + e\right )^{2}}}{3 \, a^{2} 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{\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 [B] time = 1.25045, size = 49, normalized size = 2.33 \begin{align*} \frac{\tan \left (f x + e\right )^{2} + 1}{3 \, a 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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