Optimal. Leaf size=56 \[ \frac{3 c \cos (a+b x) \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{1}{2},\frac{4}{3},\sin ^2(a+b x)\right )}{2 b \sqrt{\cos ^2(a+b x)} (c \csc (a+b x))^{2/3}} \]
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Rubi [A] time = 0.0263249, 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} \[ \frac{3 c \cos (a+b x) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\sin ^2(a+b x)\right )}{2 b \sqrt{\cos ^2(a+b x)} (c \csc (a+b x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \sqrt [3]{c \csc (a+b x)} \, dx &=\sqrt [3]{c \csc (a+b x)} \sqrt [3]{\frac{\sin (a+b x)}{c}} \int \frac{1}{\sqrt [3]{\frac{\sin (a+b x)}{c}}} \, dx\\ &=\frac{3 \cos (a+b x) \sqrt [3]{c \csc (a+b x)} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\sin ^2(a+b x)\right ) \sin (a+b x)}{2 b \sqrt{\cos ^2(a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0697324, size = 59, normalized size = 1.05 \[ -\frac{\sin (a+b x) \cos (a+b x) \sqrt [3]{c \csc (a+b x)} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{2}{3},\frac{3}{2},\cos ^2(a+b x)\right )}{b \sqrt [3]{\sin ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.263, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{c\csc \left ( bx+a \right ) }\, 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{1}{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{1}{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 \sqrt [3]{c \csc{\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 \left (c \csc \left (b x + a\right )\right )^{\frac{1}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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