Optimal. Leaf size=271 \[ \frac {3^{3/4} \sec (a+b x) \sqrt [3]{c \sin (a+b x)} \left (c^{2/3}-(c \sin (a+b x))^{2/3}\right ) \sqrt {\frac {c^{4/3} \left (\frac {(c \sin (a+b x))^{4/3}}{c^{4/3}}+\frac {(c \sin (a+b x))^{2/3}}{c^{2/3}}+1\right )}{\left (c^{2/3}-\left (1+\sqrt {3}\right ) (c \sin (a+b x))^{2/3}\right )^2}} F\left (\cos ^{-1}\left (\frac {c^{2/3}-\left (1-\sqrt {3}\right ) (c \sin (a+b x))^{2/3}}{c^{2/3}-\left (1+\sqrt {3}\right ) (c \sin (a+b x))^{2/3}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 b c^{5/3} \sqrt {-\frac {(c \sin (a+b x))^{2/3} \left (c^{2/3}-(c \sin (a+b x))^{2/3}\right )}{\left (c^{2/3}-\left (1+\sqrt {3}\right ) (c \sin (a+b x))^{2/3}\right )^2}}} \]
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Rubi [C] time = 0.02, antiderivative size = 56, normalized size of antiderivative = 0.21, 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} \[ \frac {3 \cos (a+b x) \sqrt [3]{c \sin (a+b x)} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\sin ^2(a+b x)\right )}{b c \sqrt {\cos ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rubi steps
\begin {align*} \int \frac {1}{(c \sin (a+b x))^{2/3}} \, dx &=\frac {3 \cos (a+b x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\sin ^2(a+b x)\right ) \sqrt [3]{c \sin (a+b x)}}{b c \sqrt {\cos ^2(a+b x)}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 53, normalized size = 0.20 \[ \frac {3 \sqrt {\cos ^2(a+b x)} \tan (a+b x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\sin ^2(a+b x)\right )}{b (c \sin (a+b x))^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (c \sin \left (b x + a\right )\right )^{\frac {1}{3}}}{c \sin \left (b x + a\right )}, x\right ) \]
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 \[ \int \frac {1}{\left (c \sin \left (b x + a\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \sin \left (b x +a \right )\right )^{\frac {2}{3}}}\, dx \]
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 \[ \int \frac {1}{\left (c \sin \left (b x + a\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (c\,\sin \left (a+b\,x\right )\right )}^{2/3}} \,d x \]
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 \[ \int \frac {1}{\left (c \sin {\left (a + b x \right )}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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