Optimal. Leaf size=56 \[ -\frac {3 \cos (a+b x) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\sin ^2(a+b x)\right )}{b c \sqrt {\cos ^2(a+b x)} \sqrt [3]{c \sin (a+b x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 56, 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} \[ -\frac {3 \cos (a+b x) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\sin ^2(a+b x)\right )}{b c \sqrt {\cos ^2(a+b x)} \sqrt [3]{c \sin (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rubi steps
\begin {align*} \int \frac {1}{(c \sin (a+b x))^{4/3}} \, dx &=-\frac {3 \cos (a+b x) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\sin ^2(a+b x)\right )}{b c \sqrt {\cos ^2(a+b x)} \sqrt [3]{c \sin (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 53, normalized size = 0.95 \[ -\frac {3 \sqrt {\cos ^2(a+b x)} \tan (a+b x) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\sin ^2(a+b x)\right )}{b (c \sin (a+b x))^{4/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\left (c \sin \left (b x + a\right )\right )^{\frac {2}{3}}}{c^{2} \cos \left (b x + a\right )^{2} - c^{2}}, x\right ) \]
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 \[ \int \frac {1}{\left (c \sin \left (b x + a\right )\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \sin \left (b x +a \right )\right )^{\frac {4}{3}}}\, dx \]
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 \[ \int \frac {1}{\left (c \sin \left (b x + a\right )\right )^{\frac {4}{3}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (c\,\sin \left (a+b\,x\right )\right )}^{4/3}} \,d x \]
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 \[ \int \frac {1}{\left (c \sin {\left (a + b x \right )}\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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