Integrand size = 10, antiderivative size = 36 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{3/2}} \, dx=-\frac {\cot (x)}{3 \left (a \csc ^2(x)\right )^{3/2}}-\frac {2 \cot (x)}{3 a \sqrt {a \csc ^2(x)}} \]
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Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4207, 198, 197} \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{3/2}} \, dx=-\frac {2 \cot (x)}{3 a \sqrt {a \csc ^2(x)}}-\frac {\cot (x)}{3 \left (a \csc ^2(x)\right )^{3/2}} \]
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Rule 197
Rule 198
Rule 4207
Rubi steps \begin{align*} \text {integral}& = -\left (a \text {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{5/2}} \, dx,x,\cot (x)\right )\right ) \\ & = -\frac {\cot (x)}{3 \left (a \csc ^2(x)\right )^{3/2}}-\frac {2}{3} \text {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {\cot (x)}{3 \left (a \csc ^2(x)\right )^{3/2}}-\frac {2 \cot (x)}{3 a \sqrt {a \csc ^2(x)}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{3/2}} \, dx=\frac {(-9 \cos (x)+\cos (3 x)) \csc ^3(x)}{12 \left (a \csc ^2(x)\right )^{3/2}} \]
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Time = 0.44 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {\sin \left (x \right ) \left (-2+\cos \left (x \right )^{2}-\cos \left (x \right )\right ) \sqrt {4}}{6 \left (\cos \left (x \right )-1\right ) \sqrt {a \csc \left (x \right )^{2}}\, a}\) | \(35\) |
risch | \(\frac {i {\mathrm e}^{4 i x}}{24 a \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {3 i {\mathrm e}^{2 i x}}{8 a \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {3 i}{8 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a}+\frac {i {\mathrm e}^{-2 i x}}{24 a \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}\) | \(153\) |
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Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{3/2}} \, dx=\frac {{\left (\cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )} \sqrt {-\frac {a}{\cos \left (x\right )^{2} - 1}} \sin \left (x\right )}{3 \, a^{2}} \]
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Time = 0.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{3/2}} \, dx=- \frac {2 \cot ^{3}{\left (x \right )}}{3 \left (a \csc ^{2}{\left (x \right )}\right )^{\frac {3}{2}}} - \frac {\cot {\left (x \right )}}{\left (a \csc ^{2}{\left (x \right )}\right )^{\frac {3}{2}}} \]
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\[ \int \frac {1}{\left (a \csc ^2(x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (a \csc \left (x\right )^{2}\right )^{\frac {3}{2}}} \,d x } \]
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Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{3/2}} \, dx=\frac {4 \, {\left (\frac {\mathrm {sgn}\left (\sin \left (x\right )\right )}{\sqrt {a}} - \frac {\frac {3 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 1}{\sqrt {a} {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )}^{3} \mathrm {sgn}\left (\sin \left (x\right )\right )}\right )}}{3 \, a} \]
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Timed out. \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\sin \left (x\right )}^2}\right )}^{3/2}} \,d x \]
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