Integrand size = 10, antiderivative size = 55 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{5/2}} \, dx=-\frac {\cot (x)}{5 \left (a \csc ^2(x)\right )^{5/2}}-\frac {4 \cot (x)}{15 a \left (a \csc ^2(x)\right )^{3/2}}-\frac {8 \cot (x)}{15 a^2 \sqrt {a \csc ^2(x)}} \]
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Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4207, 198, 197} \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{5/2}} \, dx=-\frac {8 \cot (x)}{15 a^2 \sqrt {a \csc ^2(x)}}-\frac {4 \cot (x)}{15 a \left (a \csc ^2(x)\right )^{3/2}}-\frac {\cot (x)}{5 \left (a \csc ^2(x)\right )^{5/2}} \]
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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 )^{7/2}} \, dx,x,\cot (x)\right )\right ) \\ & = -\frac {\cot (x)}{5 \left (a \csc ^2(x)\right )^{5/2}}-\frac {4}{5} \text {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{5/2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {\cot (x)}{5 \left (a \csc ^2(x)\right )^{5/2}}-\frac {4 \cot (x)}{15 a \left (a \csc ^2(x)\right )^{3/2}}-\frac {8 \text {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\cot (x)\right )}{15 a} \\ & = -\frac {\cot (x)}{5 \left (a \csc ^2(x)\right )^{5/2}}-\frac {4 \cot (x)}{15 a \left (a \csc ^2(x)\right )^{3/2}}-\frac {8 \cot (x)}{15 a^2 \sqrt {a \csc ^2(x)}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{5/2}} \, dx=-\frac {(150 \cos (x)-25 \cos (3 x)+3 \cos (5 x)) \sqrt {a \csc ^2(x)} \sin (x)}{240 a^3} \]
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Time = 0.48 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {\sin \left (x \right )^{3} \left (8+3 \cos \left (x \right )^{3}-6 \cos \left (x \right )^{2}-\cos \left (x \right )\right ) \sqrt {4}}{30 \left (\cos \left (x \right )-1\right )^{2} \sqrt {a \csc \left (x \right )^{2}}\, a^{2}}\) | \(45\) |
risch | \(-\frac {i {\mathrm e}^{6 i x}}{160 a^{2} \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {5 i {\mathrm e}^{2 i x}}{16 a^{2} \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {5 i}{16 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{2}}+\frac {5 i {\mathrm e}^{-2 i x}}{96 a^{2} \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}+\frac {11 i \cos \left (4 x \right )}{240 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{2}}-\frac {7 \sin \left (4 x \right )}{120 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{2}}\) | \(228\) |
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Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{5/2}} \, dx=-\frac {{\left (3 \, \cos \left (x\right )^{5} - 10 \, \cos \left (x\right )^{3} + 15 \, \cos \left (x\right )\right )} \sqrt {-\frac {a}{\cos \left (x\right )^{2} - 1}} \sin \left (x\right )}{15 \, a^{3}} \]
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Time = 1.17 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{5/2}} \, dx=- \frac {8 \cot ^{5}{\left (x \right )}}{15 \left (a \csc ^{2}{\left (x \right )}\right )^{\frac {5}{2}}} - \frac {4 \cot ^{3}{\left (x \right )}}{3 \left (a \csc ^{2}{\left (x \right )}\right )^{\frac {5}{2}}} - \frac {\cot {\left (x \right )}}{\left (a \csc ^{2}{\left (x \right )}\right )^{\frac {5}{2}}} \]
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\[ \int \frac {1}{\left (a \csc ^2(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \csc \left (x\right )^{2}\right )^{\frac {5}{2}}} \,d x } \]
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Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{5/2}} \, dx=\frac {16 \, \mathrm {sgn}\left (\sin \left (x\right )\right )}{15 \, a^{\frac {5}{2}}} - \frac {16 \, {\left (\frac {5 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac {10 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right )}}{15 \, a^{\frac {5}{2}} {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )}^{5} \mathrm {sgn}\left (\sin \left (x\right )\right )} \]
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Timed out. \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\sin \left (x\right )}^2}\right )}^{5/2}} \,d x \]
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