Integrand size = 10, antiderivative size = 86 \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{3/2}} \, dx=-\frac {5 \cot (x)}{16 a \sqrt {a \csc ^4(x)}}+\frac {5 x \csc ^2(x)}{16 a \sqrt {a \csc ^4(x)}}-\frac {5 \cos (x) \sin (x)}{24 a \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^3(x)}{6 a \sqrt {a \csc ^4(x)}} \]
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Time = 0.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4208, 2715, 8} \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{3/2}} \, dx=\frac {5 x \csc ^2(x)}{16 a \sqrt {a \csc ^4(x)}}-\frac {5 \cot (x)}{16 a \sqrt {a \csc ^4(x)}}-\frac {\sin ^3(x) \cos (x)}{6 a \sqrt {a \csc ^4(x)}}-\frac {5 \sin (x) \cos (x)}{24 a \sqrt {a \csc ^4(x)}} \]
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Rule 8
Rule 2715
Rule 4208
Rubi steps \begin{align*} \text {integral}& = \frac {\csc ^2(x) \int \sin ^6(x) \, dx}{a \sqrt {a \csc ^4(x)}} \\ & = -\frac {\cos (x) \sin ^3(x)}{6 a \sqrt {a \csc ^4(x)}}+\frac {\left (5 \csc ^2(x)\right ) \int \sin ^4(x) \, dx}{6 a \sqrt {a \csc ^4(x)}} \\ & = -\frac {5 \cos (x) \sin (x)}{24 a \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^3(x)}{6 a \sqrt {a \csc ^4(x)}}+\frac {\left (5 \csc ^2(x)\right ) \int \sin ^2(x) \, dx}{8 a \sqrt {a \csc ^4(x)}} \\ & = -\frac {5 \cot (x)}{16 a \sqrt {a \csc ^4(x)}}-\frac {5 \cos (x) \sin (x)}{24 a \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^3(x)}{6 a \sqrt {a \csc ^4(x)}}+\frac {\left (5 \csc ^2(x)\right ) \int 1 \, dx}{16 a \sqrt {a \csc ^4(x)}} \\ & = -\frac {5 \cot (x)}{16 a \sqrt {a \csc ^4(x)}}+\frac {5 x \csc ^2(x)}{16 a \sqrt {a \csc ^4(x)}}-\frac {5 \cos (x) \sin (x)}{24 a \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^3(x)}{6 a \sqrt {a \csc ^4(x)}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{3/2}} \, dx=-\frac {\csc ^6(x) (-60 x+45 \sin (2 x)-9 \sin (4 x)+\sin (6 x))}{192 \left (a \csc ^4(x)\right )^{3/2}} \]
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Time = 0.55 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.52
method | result | size |
default | \(-\frac {\left (8 \cos \left (x \right )^{4} \cot \left (x \right )-26 \cos \left (x \right )^{2} \cot \left (x \right )+33 \cot \left (x \right )-15 \csc \left (x \right )^{2} x \right ) \sqrt {16}}{192 \sqrt {a \csc \left (x \right )^{4}}\, a}\) | \(45\) |
risch | \(-\frac {5 \,{\mathrm e}^{2 i x} x}{16 a \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {i {\mathrm e}^{8 i x}}{384 a \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}+\frac {3 i {\mathrm e}^{6 i x}}{128 a \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {15 i {\mathrm e}^{4 i x}}{128 a \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}+\frac {15 i}{128 a \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {3 i {\mathrm e}^{-2 i x}}{128 a \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}+\frac {i {\mathrm e}^{-4 i x}}{384 a \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}\) | \(263\) |
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Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{3/2}} \, dx=-\frac {{\left (15 \, x \cos \left (x\right )^{2} - {\left (8 \, \cos \left (x\right )^{7} - 34 \, \cos \left (x\right )^{5} + 59 \, \cos \left (x\right )^{3} - 33 \, \cos \left (x\right )\right )} \sin \left (x\right ) - 15 \, x\right )} \sqrt {\frac {a}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}}}{48 \, a^{2}} \]
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\[ \int \frac {1}{\left (a \csc ^4(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a \csc ^{4}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{3/2}} \, dx=-\frac {33 \, \tan \left (x\right )^{5} + 40 \, \tan \left (x\right )^{3} + 15 \, \tan \left (x\right )}{48 \, {\left (a^{\frac {3}{2}} \tan \left (x\right )^{6} + 3 \, a^{\frac {3}{2}} \tan \left (x\right )^{4} + 3 \, a^{\frac {3}{2}} \tan \left (x\right )^{2} + a^{\frac {3}{2}}\right )}} + \frac {5 \, x}{16 \, a^{\frac {3}{2}}} \]
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Exception generated. \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\sin \left (x\right )}^4}\right )}^{3/2}} \,d x \]
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