Optimal. Leaf size=56 \[ \frac{3 c \cos (a+b x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{5}{6},\frac{11}{6},\sin ^2(a+b x)\right )}{5 b \sqrt{\cos ^2(a+b x)} (c \csc (a+b x))^{5/3}} \]
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Rubi [A] time = 0.0271572, antiderivative size = 56, normalized size of antiderivative = 1., 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 3772
Rule 2643
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*}
Mathematica [A] time = 0.0642439, size = 59, normalized size = 1.05 \[ -\frac{\sin (a+b x) \cos (a+b x) \text{Hypergeometric2F1}\left (\frac{1}{6},\frac{1}{2},\frac{3}{2},\cos ^2(a+b x)\right )}{b \sin ^2(a+b x)^{5/6} (c \csc (a+b x))^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.288, size = 0, normalized size = 0. \begin{align*} \int \left ( c\csc \left ( bx+a \right ) \right ) ^{-{\frac{2}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \csc \left (b x + a\right )\right )^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \csc{\left (a + b x \right )}\right )^{\frac{2}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \csc \left (b x + a\right )\right )^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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