Optimal. Leaf size=58 \[ \frac{3 \cos (e+f x) (b \sin (e+f x))^{4/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(e+f x)\right )}{4 b f \sqrt{\cos ^2(e+f x)}} \]
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Rubi [A] time = 0.01495, 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} \[ \frac{3 \cos (e+f x) (b \sin (e+f x))^{4/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(e+f x)\right )}{4 b f \sqrt{\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rubi steps
\begin{align*} \int \sqrt [3]{b \sin (e+f x)} \, dx &=\frac{3 \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(e+f x)\right ) (b \sin (e+f x))^{4/3}}{4 b f \sqrt{\cos ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.036219, size = 55, normalized size = 0.95 \[ \frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) \sqrt [3]{b \sin (e+f x)} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(e+f x)\right )}{4 f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.068, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{b\sin \left ( fx+e \right ) }\, 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{1}{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{1}{3}}, x\right ) \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 \sqrt [3]{b \sin{\left (e + f x \right )}}\, dx \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{1}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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