Integrand size = 10, antiderivative size = 46 \[ \int \left (a \csc ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {1}{2} a \cot (x) \sqrt {a \csc ^2(x)} \]
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Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4207, 201, 223, 212} \[ \int \left (a \csc ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {1}{2} a \cot (x) \sqrt {a \csc ^2(x)} \]
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Rule 201
Rule 212
Rule 223
Rule 4207
Rubi steps \begin{align*} \text {integral}& = -\left (a \text {Subst}\left (\int \sqrt {a+a x^2} \, dx,x,\cot (x)\right )\right ) \\ & = -\frac {1}{2} a \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{2} a^2 \text {Subst}\left (\int \frac {1}{\sqrt {a+a x^2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {1}{2} a \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{2} a^2 \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\cot (x)}{\sqrt {a \csc ^2(x)}}\right ) \\ & = -\frac {1}{2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {1}{2} a \cot (x) \sqrt {a \csc ^2(x)} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \left (a \csc ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} a \sqrt {a \csc ^2(x)} \left (\cot (x) \csc (x)+\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right ) \sin (x) \]
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Time = 0.46 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.65
method | result | size |
default | \(-\frac {a \sqrt {a \csc \left (x \right )^{2}}\, \left (-\sin \left (x \right ) \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+\cot \left (x \right )\right ) \sqrt {4}}{4}\) | \(30\) |
risch | \(-\frac {i a \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}+1\right )}{{\mathrm e}^{2 i x}-1}-a \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )+a \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )\) | \(104\) |
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none
Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.07 \[ \int \left (a \csc ^2(x)\right )^{3/2} \, dx=-\frac {{\left (2 \, a \cos \left (x\right ) + {\left (a \cos \left (x\right )^{2} - a\right )} \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right )\right )} \sqrt {-\frac {a}{\cos \left (x\right )^{2} - 1}}}{4 \, \sin \left (x\right )} \]
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\[ \int \left (a \csc ^2(x)\right )^{3/2} \, dx=\int \left (a \csc ^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (34) = 68\).
Time = 0.32 (sec) , antiderivative size = 318, normalized size of antiderivative = 6.91 \[ \int \left (a \csc ^2(x)\right )^{3/2} \, dx=-\frac {{\left (4 \, a \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \cos \left (x\right ) \sin \left (2 \, x\right ) - 4 \, a \cos \left (2 \, x\right ) \sin \left (x\right ) - {\left (a \cos \left (4 \, x\right )^{2} + 4 \, a \cos \left (2 \, x\right )^{2} + a \sin \left (4 \, x\right )^{2} - 4 \, a \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \sin \left (2 \, x\right )^{2} - 2 \, {\left (2 \, a \cos \left (2 \, x\right ) - a\right )} \cos \left (4 \, x\right ) - 4 \, a \cos \left (2 \, x\right ) + a\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) + {\left (a \cos \left (4 \, x\right )^{2} + 4 \, a \cos \left (2 \, x\right )^{2} + a \sin \left (4 \, x\right )^{2} - 4 \, a \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \sin \left (2 \, x\right )^{2} - 2 \, {\left (2 \, a \cos \left (2 \, x\right ) - a\right )} \cos \left (4 \, x\right ) - 4 \, a \cos \left (2 \, x\right ) + a\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right ) + 2 \, {\left (a \sin \left (3 \, x\right ) + a \sin \left (x\right )\right )} \cos \left (4 \, x\right ) - 2 \, {\left (a \cos \left (3 \, x\right ) + a \cos \left (x\right )\right )} \sin \left (4 \, x\right ) - 2 \, {\left (2 \, a \cos \left (2 \, x\right ) - a\right )} \sin \left (3 \, x\right ) + 2 \, a \sin \left (x\right )\right )} \sqrt {-a}}{2 \, {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (34) = 68\).
Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.57 \[ \int \left (a \csc ^2(x)\right )^{3/2} \, dx=\frac {1}{8} \, {\left (2 \, \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) - \frac {{\left (\frac {2 \, {\left (\cos \left (x\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1} - \mathrm {sgn}\left (\sin \left (x\right )\right )\right )} {\left (\cos \left (x\right ) + 1\right )}}{\cos \left (x\right ) - 1} - \frac {{\left (\cos \left (x\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1}\right )} a^{\frac {3}{2}} \]
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Timed out. \[ \int \left (a \csc ^2(x)\right )^{3/2} \, dx=\int {\left (\frac {a}{{\sin \left (x\right )}^2}\right )}^{3/2} \,d x \]
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