Optimal. Leaf size=252 \[ -\frac{3 \sqrt{3-i \sqrt{3}} \sec (a+b x) \sqrt{1-\frac{(c \sin (a+b x))^{2/3}}{c^{2/3}}} \sqrt{\frac{2 (c \sin (a+b x))^{2/3}}{\left (3-i \sqrt{3}\right ) c^{2/3}}+\frac{\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt{\frac{2 (c \sin (a+b x))^{2/3}}{\left (3+i \sqrt{3}\right ) c^{2/3}}+\frac{-\sqrt{3}+i}{-\sqrt{3}+3 i}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{1-\frac{(c \sin (a+b x))^{2/3}}{c^{2/3}}}}{\sqrt{3-i \sqrt{3}}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{2} b \sqrt [3]{c}} \]
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Rubi [C] time = 0.0165006, antiderivative size = 58, normalized size of antiderivative = 0.23, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2643} \[ \frac{3 \cos (a+b x) (c \sin (a+b x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\sin ^2(a+b x)\right )}{2 b c \sqrt{\cos ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [3]{c \sin (a+b x)}} \, dx &=\frac{3 \cos (a+b x) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\sin ^2(a+b x)\right ) (c \sin (a+b x))^{2/3}}{2 b c \sqrt{\cos ^2(a+b x)}}\\ \end{align*}
Mathematica [C] time = 0.0452701, size = 55, normalized size = 0.22 \[ \frac{3 \sqrt{\cos ^2(a+b x)} \tan (a+b x) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\sin ^2(a+b x)\right )}{2 b \sqrt [3]{c \sin (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.058, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [3]{c\sin \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 \frac{1}{\left (c \sin \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 (\frac{\left (c \sin \left (b x + a\right )\right )^{\frac{2}{3}}}{c \sin \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}{\sqrt [3]{c \sin{\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}{\left (c \sin \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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