Optimal. Leaf size=58 \[ \frac {3 \sqrt {\cos ^2(e+f x)} \sec (e+f x) (b \sin (e+f x))^{4/3} \, _2F_1\left (\frac {2}{3},\frac {3}{2};\frac {5}{3};\sin ^2(e+f x)\right )}{4 b f} \]
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Rubi [A] time = 0.04, antiderivative size = 58, normalized size of antiderivative = 1.00, 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} \[ \frac {3 \sqrt {\cos ^2(e+f x)} \sec (e+f x) (b \sin (e+f x))^{4/3} \, _2F_1\left (\frac {2}{3},\frac {3}{2};\frac {5}{3};\sin ^2(e+f x)\right )}{4 b f} \]
Antiderivative was successfully verified.
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Rule 2577
Rubi steps
\begin {align*} \int \sec ^2(e+f x) \sqrt [3]{b \sin (e+f x)} \, dx &=\frac {3 \sqrt {\cos ^2(e+f x)} \, _2F_1\left (\frac {2}{3},\frac {3}{2};\frac {5}{3};\sin ^2(e+f x)\right ) \sec (e+f x) (b \sin (e+f x))^{4/3}}{4 b f}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 55, normalized size = 0.95 \[ \frac {3 \sqrt {\cos ^2(e+f x)} \tan (e+f x) \sqrt [3]{b \sin (e+f x)} \, _2F_1\left (\frac {2}{3},\frac {3}{2};\frac {5}{3};\sin ^2(e+f x)\right )}{4 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \sin \left (f x + e\right )\right )^{\frac {1}{3}} \sec \left (f x + e\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \sin \left (f x + e\right )\right )^{\frac {1}{3}} \sec \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{2}\left (f x +e \right )\right ) \left (b \sin \left (f x +e \right )\right )^{\frac {1}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \sin \left (f x + e\right )\right )^{\frac {1}{3}} \sec \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (b\,\sin \left (e+f\,x\right )\right )}^{1/3}}{{\cos \left (e+f\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt [3]{b \sin {\left (e + f x \right )}} \sec ^{2}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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