Optimal. Leaf size=254 \[ -\frac{3^{3/4} \sqrt{2+\sqrt{3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1}\right ),-7-4 \sqrt{3}\right )}{2 d \sqrt{\frac{1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}-\frac{3 a \cos (c+d x) \sqrt [3]{\csc (c+d x)}}{2 d \sqrt{a \csc (c+d x)+a}} \]
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Rubi [A] time = 0.144931, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3806, 51, 63, 218} \[ -\frac{3^{3/4} \sqrt{2+\sqrt{3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2 d \sqrt{\frac{1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}-\frac{3 a \cos (c+d x) \sqrt [3]{\csc (c+d x)}}{2 d \sqrt{a \csc (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3806
Rule 51
Rule 63
Rule 218
Rubi steps
\begin{align*} \int \frac{\sqrt{a+a \csc (c+d x)}}{\csc ^{\frac{2}{3}}(c+d x)} \, dx &=\frac{\left (a^2 \cot (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{x^{5/3} \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{3 a \cos (c+d x) \sqrt [3]{\csc (c+d x)}}{2 d \sqrt{a+a \csc (c+d x)}}+\frac{\left (a^2 \cot (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{x^{2/3} \sqrt{a-a x}} \, dx,x,\csc (c+d x)\right )}{4 d \sqrt{a-a \csc (c+d x)} \sqrt{a+a \csc (c+d x)}}\\ &=-\frac{3 a \cos (c+d x) \sqrt [3]{\csc (c+d x)}}{2 d \sqrt{a+a \csc (c+d x)}}+\frac{\left (3 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 )}{4 d \sqrt{a-a \csc (c+d x)} \sqrt{a+a \csc (c+d x)}}\\ &=-\frac{3 a \cos (c+d x) \sqrt [3]{\csc (c+d x)}}{2 d \sqrt{a+a \csc (c+d x)}}-\frac{3^{3/4} \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}} 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 )}{2 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 = 0.44718, size = 110, normalized size = 0.43 \[ -\frac{\sqrt{a (\csc (c+d x)+1)} \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \left (\csc ^{\frac{2}{3}}(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{2}{3},\frac{3}{2},1-\csc (c+d x)\right )+3\right )}{2 d \csc ^{\frac{2}{3}}(c+d x) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.448, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{a+a\csc \left ( dx+c \right ) } \left ( \csc \left ( dx+c \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{\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 (\frac{\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 \frac{\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 \frac{\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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