Optimal. Leaf size=56 \[ \frac {3 c \cos (a+b x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\sin ^2(a+b x)\right )}{5 b \sqrt {\cos ^2(a+b x)} (c \csc (a+b x))^{5/3}} \]
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Rubi [A] time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3772, 2643} \[ \frac {3 c \cos (a+b x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\sin ^2(a+b x)\right )}{5 b \sqrt {\cos ^2(a+b x)} (c \csc (a+b x))^{5/3}} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3772
Rubi steps
\begin {align*} \int \frac {1}{(c \csc (a+b x))^{2/3}} \, dx &=\sqrt [3]{c \csc (a+b x)} \sqrt [3]{\frac {\sin (a+b x)}{c}} \int \left (\frac {\sin (a+b x)}{c}\right )^{2/3} \, dx\\ &=\frac {3 \cos (a+b x) \sqrt [3]{c \csc (a+b x)} \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\sin ^2(a+b x)\right ) \sin ^2(a+b x)}{5 b c \sqrt {\cos ^2(a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 59, normalized size = 1.05 \[ -\frac {\sin (a+b x) \cos (a+b x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {3}{2};\cos ^2(a+b x)\right )}{b \sin ^2(a+b x)^{5/6} (c \csc (a+b x))^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (c \csc \left (b x + a\right )\right )^{\frac {1}{3}}}{c \csc \left (b x + a\right )}, x\right ) \]
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 \[ \int \frac {1}{\left (c \csc \left (b x + a\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.63, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \csc \left (b x +a \right )\right )^{\frac {2}{3}}}\, dx \]
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 \[ \int \frac {1}{\left (c \csc \left (b x + a\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (\frac {c}{\sin \left (a+b\,x\right )}\right )}^{2/3}} \,d x \]
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 \[ \int \frac {1}{\left (c \csc {\left (a + b x \right )}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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