Integrand size = 25, antiderivative size = 213 \[ \int \sqrt [3]{\csc (c+d x)} \sqrt {a+a \csc (c+d x)} \, dx=-\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {1+\sqrt [3]{\csc (c+d x)}+\csc ^{\frac {2}{3}}(c+d x)}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}-\sqrt [3]{\csc (c+d x)}}{1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}}\right ),-7-4 \sqrt {3}\right )}{d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} (a-a \csc (c+d x)) \sqrt {a+a \csc (c+d x)}} \]
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Time = 0.18 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3891, 65, 224} \[ \int \sqrt [3]{\csc (c+d x)} \sqrt {a+a \csc (c+d x)} \, dx=-\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt {a \csc (c+d x)+a}} \]
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Rule 65
Rule 224
Rule 3891
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2 \cot (c+d x)\right ) \text {Subst}\left (\int \frac {1}{x^{2/3} \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 {\left (3 a^2 \cot (c+d x)\right ) \text {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 {2\ 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}} \operatorname {EllipticF}\left (\arcsin \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*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 3.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.22 \[ \int \sqrt [3]{\csc (c+d x)} \sqrt {a+a \csc (c+d x)} \, dx=-\frac {2 a \cot (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {3}{2},1-\csc (c+d x)\right )}{d \sqrt {a (1+\csc (c+d x))}} \]
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\[\int \csc \left (d x +c \right )^{\frac {1}{3}} \sqrt {a +a \csc \left (d x +c \right )}d x\]
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\[ \int \sqrt [3]{\csc (c+d x)} \sqrt {a+a \csc (c+d x)} \, dx=\int { \sqrt {a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{\frac {1}{3}} \,d x } \]
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\[ \int \sqrt [3]{\csc (c+d x)} \sqrt {a+a \csc (c+d x)} \, dx=\int \sqrt {a \left (\csc {\left (c + d x \right )} + 1\right )} \sqrt [3]{\csc {\left (c + d x \right )}}\, dx \]
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\[ \int \sqrt [3]{\csc (c+d x)} \sqrt {a+a \csc (c+d x)} \, dx=\int { \sqrt {a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{\frac {1}{3}} \,d x } \]
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\[ \int \sqrt [3]{\csc (c+d x)} \sqrt {a+a \csc (c+d x)} \, dx=\int { \sqrt {a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{\frac {1}{3}} \,d x } \]
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Timed out. \[ \int \sqrt [3]{\csc (c+d x)} \sqrt {a+a \csc (c+d x)} \, dx=\int \sqrt {a+\frac {a}{\sin \left (c+d\,x\right )}}\,{\left (\frac {1}{\sin \left (c+d\,x\right )}\right )}^{1/3} \,d x \]
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