Optimal. Leaf size=53 \[ \frac{3 \cos (a+b x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{5}{6},\frac{11}{6},\sin ^2(a+b x)\right )}{5 b \sqrt{\cos ^2(a+b x)} \csc ^{\frac{5}{3}}(a+b x)} \]
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Rubi [A] time = 0.0212965, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3772, 2643} \[ \frac{3 \cos (a+b x) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\sin ^2(a+b x)\right )}{5 b \sqrt{\cos ^2(a+b x)} \csc ^{\frac{5}{3}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \frac{1}{\csc ^{\frac{2}{3}}(a+b x)} \, dx &=\sqrt [3]{\csc (a+b x)} \sqrt [3]{\sin (a+b x)} \int \sin ^{\frac{2}{3}}(a+b x) \, dx\\ &=\frac{3 \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)} \csc ^{\frac{5}{3}}(a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0934955, size = 51, normalized size = 0.96 \[ -\frac{\cos (a+b x) \text{Hypergeometric2F1}\left (\frac{1}{6},\frac{1}{2},\frac{3}{2},\cos ^2(a+b x)\right )}{b \sin ^2(a+b x)^{5/6} \csc ^{\frac{5}{3}}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.142, size = 0, normalized size = 0. \begin{align*} \int \left ( \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}{\csc \left (b x + a\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{1}{\csc \left (b x + a\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 \frac{1}{\csc ^{\frac{2}{3}}{\left (a + b x \right )}}\, 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}{\csc \left (b x + a\right )^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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