Optimal. Leaf size=470 \[ \frac{2 \sqrt{2} 3^{3/4} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1}\right ),-7-4 \sqrt{3}\right )}{d \sqrt{\frac{1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{d \sqrt{\frac{1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}+\frac{6 a \cot (c+d x)}{d \left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right ) \sqrt{a \csc (c+d x)+a}} \]
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Rubi [A] time = 0.253267, antiderivative size = 470, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3806, 63, 303, 218, 1877} \[ \frac{2 \sqrt{2} 3^{3/4} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{d \sqrt{\frac{1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{d \sqrt{\frac{1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}+\frac{6 a \cot (c+d x)}{d \left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right ) \sqrt{a \csc (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3806
Rule 63
Rule 303
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \csc ^{\frac{2}{3}}(c+d x) \sqrt{a+a \csc (c+d x)} \, dx &=\frac{\left (a^2 \cot (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{x} \sqrt{a-a x}} \, dx,x,\csc (c+d x)\right )}{d \sqrt{a-a \csc (c+d x)} \sqrt{a+a \csc (c+d x)}}\\ &=\frac{\left (3 a^2 \cot (c+d x)\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a-a x^3}} \, dx,x,\sqrt [3]{\csc (c+d x)}\right )}{d \sqrt{a-a \csc (c+d x)} \sqrt{a+a \csc (c+d x)}}\\ &=-\frac{\left (3 a^2 \cot (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1-\sqrt{3}-x}{\sqrt{a-a x^3}} \, dx,x,\sqrt [3]{\csc (c+d x)}\right )}{d \sqrt{a-a \csc (c+d x)} \sqrt{a+a \csc (c+d x)}}-\frac{\left (3 \sqrt{2 \left (2-\sqrt{3}\right )} a^2 \cot (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x^3}} \, dx,x,\sqrt [3]{\csc (c+d x)}\right )}{d \sqrt{a-a \csc (c+d x)} \sqrt{a+a \csc (c+d x)}}\\ &=\frac{6 a \cot (c+d x)}{d \left (1+\sqrt{3}-\sqrt [3]{\csc (c+d x)}\right ) \sqrt{a+a \csc (c+d x)}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt{\frac{1+\sqrt [3]{\csc (c+d x)}+\csc ^{\frac{2}{3}}(c+d x)}{\left (1+\sqrt{3}-\sqrt [3]{\csc (c+d x)}\right )^2}} E\left (\sin ^{-1}\left (\frac{1-\sqrt{3}-\sqrt [3]{\csc (c+d x)}}{1+\sqrt{3}-\sqrt [3]{\csc (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{d \sqrt{\frac{1-\sqrt [3]{\csc (c+d x)}}{\left (1+\sqrt{3}-\sqrt [3]{\csc (c+d x)}\right )^2}} (a-a \csc (c+d x)) \sqrt{a+a \csc (c+d x)}}+\frac{2 \sqrt{2} 3^{3/4} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt{\frac{1+\sqrt [3]{\csc (c+d x)}+\csc ^{\frac{2}{3}}(c+d x)}{\left (1+\sqrt{3}-\sqrt [3]{\csc (c+d x)}\right )^2}} F\left (\sin ^{-1}\left (\frac{1-\sqrt{3}-\sqrt [3]{\csc (c+d x)}}{1+\sqrt{3}-\sqrt [3]{\csc (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{d \sqrt{\frac{1-\sqrt [3]{\csc (c+d x)}}{\left (1+\sqrt{3}-\sqrt [3]{\csc (c+d x)}\right )^2}} (a-a \csc (c+d x)) \sqrt{a+a \csc (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.985294, size = 109, normalized size = 0.23 \[ \frac{2 \sqrt{a (\csc (c+d x)+1)} \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \left (2 \sqrt [3]{\csc (c+d x)} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{4}{3},\frac{3}{2},1-\csc (c+d x)\right )-3\right )}{d \sqrt [3]{\csc (c+d x)} \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.325, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}\sqrt{a+a\csc \left ( dx+c \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 \sqrt{a \csc \left (d x + c\right ) + a} \csc \left (d x + c\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 (\sqrt{a \csc \left (d x + c\right ) + a} \csc \left (d x + c\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 \sqrt{a \left (\csc{\left (c + d x \right )} + 1\right )} \csc ^{\frac{2}{3}}{\left (c + d 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 \sqrt{a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{\frac{2}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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