Integrand size = 25, antiderivative size = 552 \[ \int \frac {\sqrt {a+a \csc (c+d x)}}{\csc ^{\frac {4}{3}}(c+d x)} \, dx=-\frac {15 a \cot (c+d x)}{8 d \left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {a+a \csc (c+d x)}}-\frac {3 a \cos (c+d x)}{4 d \sqrt [3]{\csc (c+d x)} \sqrt {a+a \csc (c+d x)}}-\frac {15 a \cos (c+d x) \csc ^{\frac {2}{3}}(c+d x)}{8 d \sqrt {a+a \csc (c+d x)}}+\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {1+\sqrt [3]{\csc (c+d x)}+\csc ^{\frac {2}{3}}(c+d x)}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}-\sqrt [3]{\csc (c+d x)}}{1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}}\right )|-7-4 \sqrt {3}\right )}{16 d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} (a-a \csc (c+d x)) \sqrt {a+a \csc (c+d x)}}-\frac {5\ 3^{3/4} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {1+\sqrt [3]{\csc (c+d x)}+\csc ^{\frac {2}{3}}(c+d x)}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}-\sqrt [3]{\csc (c+d x)}}{1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}}\right ),-7-4 \sqrt {3}\right )}{4 \sqrt {2} d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} (a-a \csc (c+d x)) \sqrt {a+a \csc (c+d x)}} \]
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Time = 0.42 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3891, 53, 65, 309, 224, 1891} \[ \int \frac {\sqrt {a+a \csc (c+d x)}}{\csc ^{\frac {4}{3}}(c+d x)} \, dx=-\frac {5\ 3^{3/4} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{4 \sqrt {2} d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt {a \csc (c+d x)+a}}+\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{16 d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt {a \csc (c+d x)+a}}-\frac {15 a \cos (c+d x) \csc ^{\frac {2}{3}}(c+d x)}{8 d \sqrt {a \csc (c+d x)+a}}-\frac {3 a \cos (c+d x)}{4 d \sqrt [3]{\csc (c+d x)} \sqrt {a \csc (c+d x)+a}}-\frac {15 a \cot (c+d x)}{8 d \left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right ) \sqrt {a \csc (c+d x)+a}} \]
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Rule 53
Rule 65
Rule 224
Rule 309
Rule 1891
Rule 3891
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2 \cot (c+d x)\right ) \text {Subst}\left (\int \frac {1}{x^{7/3} \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)}{4 d \sqrt [3]{\csc (c+d x)} \sqrt {a+a \csc (c+d x)}}+\frac {\left (5 a^2 \cot (c+d x)\right ) \text {Subst}\left (\int \frac {1}{x^{4/3} \sqrt {a-a x}} \, dx,x,\csc (c+d x)\right )}{8 d \sqrt {a-a \csc (c+d x)} \sqrt {a+a \csc (c+d x)}} \\ & = -\frac {3 a \cos (c+d x)}{4 d \sqrt [3]{\csc (c+d x)} \sqrt {a+a \csc (c+d x)}}-\frac {15 a \cos (c+d x) \csc ^{\frac {2}{3}}(c+d x)}{8 d \sqrt {a+a \csc (c+d x)}}-\frac {\left (5 a^2 \cot (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{x} \sqrt {a-a x}} \, dx,x,\csc (c+d x)\right )}{16 d \sqrt {a-a \csc (c+d x)} \sqrt {a+a \csc (c+d x)}} \\ & = -\frac {3 a \cos (c+d x)}{4 d \sqrt [3]{\csc (c+d x)} \sqrt {a+a \csc (c+d x)}}-\frac {15 a \cos (c+d x) \csc ^{\frac {2}{3}}(c+d x)}{8 d \sqrt {a+a \csc (c+d x)}}-\frac {\left (15 a^2 \cot (c+d x)\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a-a x^3}} \, dx,x,\sqrt [3]{\csc (c+d x)}\right )}{16 d \sqrt {a-a \csc (c+d x)} \sqrt {a+a \csc (c+d x)}} \\ & = -\frac {3 a \cos (c+d x)}{4 d \sqrt [3]{\csc (c+d x)} \sqrt {a+a \csc (c+d x)}}-\frac {15 a \cos (c+d x) \csc ^{\frac {2}{3}}(c+d x)}{8 d \sqrt {a+a \csc (c+d x)}}+\frac {\left (15 a^2 \cot (c+d x)\right ) \text {Subst}\left (\int \frac {1-\sqrt {3}-x}{\sqrt {a-a x^3}} \, dx,x,\sqrt [3]{\csc (c+d x)}\right )}{16 d \sqrt {a-a \csc (c+d x)} \sqrt {a+a \csc (c+d x)}}-\frac {\left (15 \left (1-\sqrt {3}\right ) 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 )}{16 d \sqrt {a-a \csc (c+d x)} \sqrt {a+a \csc (c+d x)}} \\ & = -\frac {15 a \cot (c+d x)}{8 d \left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {a+a \csc (c+d x)}}-\frac {3 a \cos (c+d x)}{4 d \sqrt [3]{\csc (c+d x)} \sqrt {a+a \csc (c+d x)}}-\frac {15 a \cos (c+d x) \csc ^{\frac {2}{3}}(c+d x)}{8 d \sqrt {a+a \csc (c+d x)}}+\frac {15 \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 (\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 )}{16 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 {5\ 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}} \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 )}{4 \sqrt {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*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 15.88 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.13 \[ \int \frac {\sqrt {a+a \csc (c+d x)}}{\csc ^{\frac {4}{3}}(c+d x)} \, dx=-\frac {a \cos (c+d x) \left (3+5 \csc ^{\frac {4}{3}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {3}{2},1-\csc (c+d x)\right )\right )}{4 d \sqrt [3]{\csc (c+d x)} \sqrt {a (1+\csc (c+d x))}} \]
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\[\int \frac {\sqrt {a +a \csc \left (d x +c \right )}}{\csc \left (d x +c \right )^{\frac {4}{3}}}d x\]
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\[ \int \frac {\sqrt {a+a \csc (c+d x)}}{\csc ^{\frac {4}{3}}(c+d x)} \, dx=\int { \frac {\sqrt {a \csc \left (d x + c\right ) + a}}{\csc \left (d x + c\right )^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {\sqrt {a+a \csc (c+d x)}}{\csc ^{\frac {4}{3}}(c+d x)} \, dx=\int \frac {\sqrt {a \left (\csc {\left (c + d x \right )} + 1\right )}}{\csc ^{\frac {4}{3}}{\left (c + d x \right )}}\, dx \]
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\[ \int \frac {\sqrt {a+a \csc (c+d x)}}{\csc ^{\frac {4}{3}}(c+d x)} \, dx=\int { \frac {\sqrt {a \csc \left (d x + c\right ) + a}}{\csc \left (d x + c\right )^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {\sqrt {a+a \csc (c+d x)}}{\csc ^{\frac {4}{3}}(c+d x)} \, dx=\int { \frac {\sqrt {a \csc \left (d x + c\right ) + a}}{\csc \left (d x + c\right )^{\frac {4}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+a \csc (c+d x)}}{\csc ^{\frac {4}{3}}(c+d x)} \, dx=\int \frac {\sqrt {a+\frac {a}{\sin \left (c+d\,x\right )}}}{{\left (\frac {1}{\sin \left (c+d\,x\right )}\right )}^{4/3}} \,d x \]
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