Optimal. Leaf size=58 \[ -\frac {3 \cos (e+f x) \, _2F_1\left (-\frac {1}{3},\frac {1}{2};\frac {2}{3};\sin ^2(e+f x)\right )}{2 b f \sqrt {\cos ^2(e+f x)} (b \sin (e+f x))^{2/3}} \]
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Rubi [A] time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, 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} \[ -\frac {3 \cos (e+f x) \, _2F_1\left (-\frac {1}{3},\frac {1}{2};\frac {2}{3};\sin ^2(e+f x)\right )}{2 b f \sqrt {\cos ^2(e+f x)} (b \sin (e+f x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 2643
Rubi steps
\begin {align*} \int \frac {1}{(b \sin (e+f x))^{5/3}} \, dx &=-\frac {3 \cos (e+f x) \, _2F_1\left (-\frac {1}{3},\frac {1}{2};\frac {2}{3};\sin ^2(e+f x)\right )}{2 b f \sqrt {\cos ^2(e+f x)} (b \sin (e+f x))^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 55, normalized size = 0.95 \[ -\frac {3 \sqrt {\cos ^2(e+f x)} \tan (e+f x) \, _2F_1\left (-\frac {1}{3},\frac {1}{2};\frac {2}{3};\sin ^2(e+f x)\right )}{2 f (b \sin (e+f x))^{5/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\left (b \sin \left (f x + e\right )\right )^{\frac {1}{3}}}{b^{2} \cos \left (f x + e\right )^{2} - b^{2}}, x\right ) \]
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 \[ \int \frac {1}{\left (b \sin \left (f x + e\right )\right )^{\frac {5}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \sin \left (f x +e \right )\right )^{\frac {5}{3}}}\, dx \]
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 \[ \int \frac {1}{\left (b \sin \left (f x + e\right )\right )^{\frac {5}{3}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (b\,\sin \left (e+f\,x\right )\right )}^{5/3}} \,d x \]
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 \[ \int \frac {1}{\left (b \sin {\left (e + f x \right )}\right )^{\frac {5}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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