Integrand size = 10, antiderivative size = 65 \[ \int \left (a \csc ^2(x)\right )^{5/2} \, dx=-\frac {3}{8} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {3}{8} a^2 \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{4} a \cot (x) \left (a \csc ^2(x)\right )^{3/2} \]
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Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4207, 201, 223, 212} \[ \int \left (a \csc ^2(x)\right )^{5/2} \, dx=-\frac {3}{8} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {3}{8} a^2 \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{4} a \cot (x) \left (a \csc ^2(x)\right )^{3/2} \]
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Rule 201
Rule 212
Rule 223
Rule 4207
Rubi steps \begin{align*} \text {integral}& = -\left (a \text {Subst}\left (\int \left (a+a x^2\right )^{3/2} \, dx,x,\cot (x)\right )\right ) \\ & = -\frac {1}{4} a \cot (x) \left (a \csc ^2(x)\right )^{3/2}-\frac {1}{4} \left (3 a^2\right ) \text {Subst}\left (\int \sqrt {a+a x^2} \, dx,x,\cot (x)\right ) \\ & = -\frac {3}{8} a^2 \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{4} a \cot (x) \left (a \csc ^2(x)\right )^{3/2}-\frac {1}{8} \left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+a x^2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {3}{8} a^2 \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{4} a \cot (x) \left (a \csc ^2(x)\right )^{3/2}-\frac {1}{8} \left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\cot (x)}{\sqrt {a \csc ^2(x)}}\right ) \\ & = -\frac {3}{8} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {3}{8} a^2 \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{4} a \cot (x) \left (a \csc ^2(x)\right )^{3/2} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.78 \[ \int \left (a \csc ^2(x)\right )^{5/2} \, dx=\frac {1}{64} \left (a \csc ^2(x)\right )^{5/2} \sin (x) \left (-22 \cos (x)+6 \left (\cos (3 x)+4 \left (-\log \left (\cos \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )\right )\right ) \sin ^4(x)\right )\right ) \]
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Time = 0.53 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.68
method | result | size |
default | \(\frac {a^{2} \sqrt {a \csc \left (x \right )^{2}}\, \left (3 \sin \left (x \right ) \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+3 \cot \left (x \right )^{3}-5 \csc \left (x \right )^{2} \cot \left (x \right )\right ) \sqrt {4}}{16}\) | \(44\) |
risch | \(-\frac {i a^{2} \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left (3 \,{\mathrm e}^{6 i x}-11 \,{\mathrm e}^{4 i x}-11 \,{\mathrm e}^{2 i x}+3\right )}{4 \left ({\mathrm e}^{2 i x}-1\right )^{3}}-\frac {3 a^{2} \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )}{4}+\frac {3 a^{2} \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )}{4}\) | \(127\) |
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none
Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.23 \[ \int \left (a \csc ^2(x)\right )^{5/2} \, dx=-\frac {{\left (6 \, a^{2} \cos \left (x\right )^{3} - 10 \, a^{2} \cos \left (x\right ) + 3 \, {\left (a^{2} \cos \left (x\right )^{4} - 2 \, a^{2} \cos \left (x\right )^{2} + a^{2}\right )} \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right )\right )} \sqrt {-\frac {a}{\cos \left (x\right )^{2} - 1}}}{16 \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \]
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\[ \int \left (a \csc ^2(x)\right )^{5/2} \, dx=\int \left (a \csc ^{2}{\left (x \right )}\right )^{\frac {5}{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 1113 vs. \(2 (49) = 98\).
Time = 0.46 (sec) , antiderivative size = 1113, normalized size of antiderivative = 17.12 \[ \int \left (a \csc ^2(x)\right )^{5/2} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (49) = 98\).
Time = 0.28 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.91 \[ \int \left (a \csc ^2(x)\right )^{5/2} \, dx=\frac {1}{64} \, {\left (12 \, a^{2} \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) - \frac {8 \, a^{2} {\left (\cos \left (x\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1} + \frac {a^{2} {\left (\cos \left (x\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (x\right )\right )}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {{\left (a^{2} \mathrm {sgn}\left (\sin \left (x\right )\right ) - \frac {8 \, a^{2} {\left (\cos \left (x\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1} + \frac {18 \, a^{2} {\left (\cos \left (x\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (x\right )\right )}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (x\right ) + 1\right )}^{2}}{{\left (\cos \left (x\right ) - 1\right )}^{2}}\right )} \sqrt {a} \]
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Timed out. \[ \int \left (a \csc ^2(x)\right )^{5/2} \, dx=\int {\left (\frac {a}{{\sin \left (x\right )}^2}\right )}^{5/2} \,d x \]
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