Integrand size = 10, antiderivative size = 65 \[ \int (a+a \csc (x))^{5/2} \, dx=-2 a^{5/2} \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a+a \csc (x)}}\right )-\frac {14 a^3 \cot (x)}{3 \sqrt {a+a \csc (x)}}-\frac {2}{3} a^2 \cot (x) \sqrt {a+a \csc (x)} \]
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Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3860, 4000, 3859, 209, 3877} \[ \int (a+a \csc (x))^{5/2} \, dx=-2 a^{5/2} \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right )-\frac {14 a^3 \cot (x)}{3 \sqrt {a \csc (x)+a}}-\frac {2}{3} a^2 \cot (x) \sqrt {a \csc (x)+a} \]
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Rule 209
Rule 3859
Rule 3860
Rule 3877
Rule 4000
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3} a^2 \cot (x) \sqrt {a+a \csc (x)}+\frac {1}{3} (2 a) \int \sqrt {a+a \csc (x)} \left (\frac {3 a}{2}+\frac {7}{2} a \csc (x)\right ) \, dx \\ & = -\frac {2}{3} a^2 \cot (x) \sqrt {a+a \csc (x)}+a^2 \int \sqrt {a+a \csc (x)} \, dx+\frac {1}{3} \left (7 a^2\right ) \int \csc (x) \sqrt {a+a \csc (x)} \, dx \\ & = -\frac {14 a^3 \cot (x)}{3 \sqrt {a+a \csc (x)}}-\frac {2}{3} a^2 \cot (x) \sqrt {a+a \csc (x)}-\left (2 a^3\right ) \text {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,\frac {a \cot (x)}{\sqrt {a+a \csc (x)}}\right ) \\ & = -2 a^{5/2} \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a+a \csc (x)}}\right )-\frac {14 a^3 \cot (x)}{3 \sqrt {a+a \csc (x)}}-\frac {2}{3} a^2 \cot (x) \sqrt {a+a \csc (x)} \\ \end{align*}
Time = 2.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.23 \[ \int (a+a \csc (x))^{5/2} \, dx=-\frac {2 a^2 \sqrt {a (1+\csc (x))} \left (3 \arctan \left (\sqrt {-1+\csc (x)}\right )+\sqrt {-1+\csc (x)} (8+\csc (x))\right ) \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )}{3 \sqrt {-1+\csc (x)} \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(274\) vs. \(2(51)=102\).
Time = 0.88 (sec) , antiderivative size = 275, normalized size of antiderivative = 4.23
method | result | size |
default | \(\frac {\csc \left (x \right ) {\left (\frac {a \left (\csc \left (x \right ) \left (1-\cos \left (x \right )\right )^{2}+2-2 \cos \left (x \right )+\sin \left (x \right )\right )}{1-\cos \left (x \right )}\right )}^{\frac {5}{2}} \left (1-\cos \left (x \right )\right ) \left (2 \csc \left (x \right )^{3} \left (1-\cos \left (x \right )\right )^{3}+3 \sqrt {2}\, \left (\csc \left (x \right )-\cot \left (x \right )\right )^{\frac {3}{2}} \ln \left (-\frac {\csc \left (x \right )-\cot \left (x \right )+\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+1}{\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}-\csc \left (x \right )+\cot \left (x \right )-1}\right )+12 \sqrt {2}\, \left (\csc \left (x \right )-\cot \left (x \right )\right )^{\frac {3}{2}} \arctan \left (\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+1\right )+12 \sqrt {2}\, \left (\csc \left (x \right )-\cot \left (x \right )\right )^{\frac {3}{2}} \arctan \left (\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}-1\right )+3 \sqrt {2}\, \left (\csc \left (x \right )-\cot \left (x \right )\right )^{\frac {3}{2}} \ln \left (-\frac {\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}-\csc \left (x \right )+\cot \left (x \right )-1}{\csc \left (x \right )-\cot \left (x \right )+\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+1}\right )+30 \csc \left (x \right )^{2} \left (1-\cos \left (x \right )\right )^{2}-30 \csc \left (x \right )+30 \cot \left (x \right )-2\right ) \sqrt {2}}{12 \left (\csc \left (x \right )-\cot \left (x \right )+1\right )^{5}}\) | \(275\) |
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Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (51) = 102\).
Time = 0.27 (sec) , antiderivative size = 318, normalized size of antiderivative = 4.89 \[ \int (a+a \csc (x))^{5/2} \, dx=\left [\frac {3 \, {\left (a^{2} \cos \left (x\right )^{2} - a^{2} - {\left (a^{2} \cos \left (x\right ) + a^{2}\right )} \sin \left (x\right )\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (x\right )^{2} - 2 \, {\left (\cos \left (x\right )^{2} + {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )} \sqrt {-a} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}} + a \cos \left (x\right ) - {\left (2 \, a \cos \left (x\right ) + a\right )} \sin \left (x\right ) - a}{\cos \left (x\right ) + \sin \left (x\right ) + 1}\right ) + 2 \, {\left (8 \, a^{2} \cos \left (x\right )^{2} + a^{2} \cos \left (x\right ) - 7 \, a^{2} + {\left (8 \, a^{2} \cos \left (x\right ) + 7 \, a^{2}\right )} \sin \left (x\right )\right )} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}}}{3 \, {\left (\cos \left (x\right )^{2} - {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )}}, \frac {2 \, {\left (3 \, {\left (a^{2} \cos \left (x\right )^{2} - a^{2} - {\left (a^{2} \cos \left (x\right ) + a^{2}\right )} \sin \left (x\right )\right )} \sqrt {a} \arctan \left (-\frac {\sqrt {a} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}} {\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )}}{a \cos \left (x\right ) + a \sin \left (x\right ) + a}\right ) + {\left (8 \, a^{2} \cos \left (x\right )^{2} + a^{2} \cos \left (x\right ) - 7 \, a^{2} + {\left (8 \, a^{2} \cos \left (x\right ) + 7 \, a^{2}\right )} \sin \left (x\right )\right )} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}}\right )}}{3 \, {\left (\cos \left (x\right )^{2} - {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )}}\right ] \]
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\[ \int (a+a \csc (x))^{5/2} \, dx=\int \left (a \csc {\left (x \right )} + a\right )^{\frac {5}{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (51) = 102\).
Time = 0.35 (sec) , antiderivative size = 417, normalized size of antiderivative = 6.42 \[ \int (a+a \csc (x))^{5/2} \, dx=\frac {1}{22} \, \sqrt {2} a^{\frac {5}{2}} \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )^{\frac {11}{2}} + \frac {5}{18} \, \sqrt {2} a^{\frac {5}{2}} \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )^{\frac {9}{2}} + \frac {9}{14} \, \sqrt {2} a^{\frac {5}{2}} \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )^{\frac {7}{2}} + \frac {1}{2} \, \sqrt {2} a^{\frac {5}{2}} \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )^{\frac {5}{2}} - \frac {2}{3} \, \sqrt {2} a^{\frac {5}{2}} \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )^{\frac {3}{2}} + \sqrt {2} {\left (\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right )}\right ) + \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right )}\right )\right )} a^{\frac {5}{2}} - 2 \, \sqrt {2} a^{\frac {5}{2}} \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}} - \frac {\frac {693 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {1155 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {1386 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {990 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {385 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {63 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}}}{1386 \, \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}} - \frac {\frac {7 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {105 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {210 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {70 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac {21 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac {3 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}}}{42 \, \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )^{\frac {5}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (51) = 102\).
Time = 0.46 (sec) , antiderivative size = 250, normalized size of antiderivative = 3.85 \[ \int (a+a \csc (x))^{5/2} \, dx={\left (a^{2} \sqrt {{\left | a \right |}} + a {\left | a \right |}^{\frac {3}{2}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} + 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right ) + {\left (a^{2} \sqrt {{\left | a \right |}} + a {\left | a \right |}^{\frac {3}{2}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} - 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right ) + \frac {1}{2} \, {\left (a^{2} \sqrt {{\left | a \right |}} - a {\left | a \right |}^{\frac {3}{2}}\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) + \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right ) - \frac {1}{2} \, {\left (a^{2} \sqrt {{\left | a \right |}} - a {\left | a \right |}^{\frac {3}{2}}\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) - \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right ) + \frac {1}{6} \, \sqrt {2} {\left (\sqrt {a \tan \left (\frac {1}{2} \, x\right )} a^{2} \tan \left (\frac {1}{2} \, x\right ) + 15 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )} a^{2}\right )} - \frac {\sqrt {2} {\left (15 \, a^{4} \tan \left (\frac {1}{2} \, x\right ) + a^{4}\right )}}{6 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )} a \tan \left (\frac {1}{2} \, x\right )} \]
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Timed out. \[ \int (a+a \csc (x))^{5/2} \, dx=\int {\left (a+\frac {a}{\sin \left (x\right )}\right )}^{5/2} \,d x \]
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