Optimal. Leaf size=58 \[ -\frac{3 \sqrt{\cos ^2(e+f x)} \sec (e+f x) \, _2F_1\left (-\frac{1}{3},\frac{3}{2};\frac{2}{3};\sin ^2(e+f x)\right )}{2 b f (b \sin (e+f x))^{2/3}} \]
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Rubi [A] time = 0.0455026, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2577} \[ -\frac{3 \sqrt{\cos ^2(e+f x)} \sec (e+f x) \, _2F_1\left (-\frac{1}{3},\frac{3}{2};\frac{2}{3};\sin ^2(e+f x)\right )}{2 b f (b \sin (e+f x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 2577
Rubi steps
\begin{align*} \int \frac{\sec ^2(e+f x)}{(b \sin (e+f x))^{5/3}} \, dx &=-\frac{3 \sqrt{\cos ^2(e+f x)} \, _2F_1\left (-\frac{1}{3},\frac{3}{2};\frac{2}{3};\sin ^2(e+f x)\right ) \sec (e+f x)}{2 b f (b \sin (e+f x))^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0424613, size = 55, normalized size = 0.95 \[ -\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) \, _2F_1\left (-\frac{1}{3},\frac{3}{2};\frac{2}{3};\sin ^2(e+f x)\right )}{2 f (b \sin (e+f x))^{5/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sec \left ( fx+e \right ) \right ) ^{2} \left ( b\sin \left ( fx+e \right ) \right ) ^{-{\frac{5}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\left (b \sin \left (f x + e\right )\right )^{\frac{1}{3}} \sec \left (f x + e\right )^{2}}{b^{2} \cos \left (f x + e\right )^{2} - b^{2}}, x\right ) \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{\sec \left (f x + e\right )^{2}}{\left (b \sin \left (f x + e\right )\right )^{\frac{5}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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