Optimal. Leaf size=54 \[ \frac{3 c \cos (a+b x) \text{Hypergeometric2F1}\left (\frac{1}{6},\frac{1}{2},\frac{7}{6},\sin ^2(a+b x)\right )}{b \sqrt{\cos ^2(a+b x)} \sqrt [3]{c \csc (a+b x)}} \]
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Rubi [A] time = 0.0271586, antiderivative size = 54, 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} \[ \frac{3 c \cos (a+b x) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2(a+b x)\right )}{b \sqrt{\cos ^2(a+b x)} \sqrt [3]{c \csc (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int (c \csc (a+b x))^{2/3} \, dx &=(c \csc (a+b x))^{2/3} \left (\frac{\sin (a+b x)}{c}\right )^{2/3} \int \frac{1}{\left (\frac{\sin (a+b x)}{c}\right )^{2/3}} \, dx\\ &=\frac{3 \cos (a+b x) (c \csc (a+b x))^{2/3} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2(a+b x)\right ) \sin (a+b x)}{b \sqrt{\cos ^2(a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0684013, size = 59, normalized size = 1.09 \[ -\frac{\sin (a+b x) \cos (a+b x) (c \csc (a+b x))^{2/3} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{5}{6},\frac{3}{2},\cos ^2(a+b x)\right )}{b \sqrt [6]{\sin ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.146, 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 \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 (\left (c \csc \left (b x + a\right )\right )^{\frac{2}{3}}, 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 \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 \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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