Optimal. Leaf size=58 \[ \frac{3 \cos (e+f x) (b \sin (e+f x))^{8/3} \, _2F_1\left (\frac{1}{2},\frac{4}{3};\frac{7}{3};\sin ^2(e+f x)\right )}{8 b f \sqrt{\cos ^2(e+f x)}} \]
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Rubi [A] time = 0.0141124, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2643} \[ \frac{3 \cos (e+f x) (b \sin (e+f x))^{8/3} \, _2F_1\left (\frac{1}{2},\frac{4}{3};\frac{7}{3};\sin ^2(e+f x)\right )}{8 b f \sqrt{\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rubi steps
\begin{align*} \int (b \sin (e+f x))^{5/3} \, dx &=\frac{3 \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{4}{3};\frac{7}{3};\sin ^2(e+f x)\right ) (b \sin (e+f x))^{8/3}}{8 b f \sqrt{\cos ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.041237, size = 55, normalized size = 0.95 \[ \frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) (b \sin (e+f x))^{5/3} \, _2F_1\left (\frac{1}{2},\frac{4}{3};\frac{7}{3};\sin ^2(e+f x)\right )}{8 f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b \sin \left (f x + e\right )\right )^{\frac{2}{3}} b \sin \left (f x + e\right ), 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 \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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