Integrand size = 10, antiderivative size = 36 \[ \int \frac {1}{\sqrt {a \csc ^4(x)}} \, dx=-\frac {\cot (x)}{2 \sqrt {a \csc ^4(x)}}+\frac {x \csc ^2(x)}{2 \sqrt {a \csc ^4(x)}} \]
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Time = 0.02 (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 = {4208, 2715, 8} \[ \int \frac {1}{\sqrt {a \csc ^4(x)}} \, dx=\frac {x \csc ^2(x)}{2 \sqrt {a \csc ^4(x)}}-\frac {\cot (x)}{2 \sqrt {a \csc ^4(x)}} \]
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Rule 8
Rule 2715
Rule 4208
Rubi steps \begin{align*} \text {integral}& = \frac {\csc ^2(x) \int \sin ^2(x) \, dx}{\sqrt {a \csc ^4(x)}} \\ & = -\frac {\cot (x)}{2 \sqrt {a \csc ^4(x)}}+\frac {\csc ^2(x) \int 1 \, dx}{2 \sqrt {a \csc ^4(x)}} \\ & = -\frac {\cot (x)}{2 \sqrt {a \csc ^4(x)}}+\frac {x \csc ^2(x)}{2 \sqrt {a \csc ^4(x)}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\sqrt {a \csc ^4(x)}} \, dx=\frac {-\cot (x)+x \csc ^2(x)}{2 \sqrt {a \csc ^4(x)}} \]
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Time = 0.52 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.67
method | result | size |
default | \(-\frac {\left (\cot \left (x \right )-\csc \left (x \right )^{2} x \right ) \sqrt {16}}{8 \sqrt {a \csc \left (x \right )^{4}}}\) | \(24\) |
risch | \(-\frac {{\mathrm e}^{2 i x} x}{2 \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}\, \left ({\mathrm e}^{2 i x}-1\right )^{2}}-\frac {i {\mathrm e}^{4 i x}}{8 \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}\, \left ({\mathrm e}^{2 i x}-1\right )^{2}}+\frac {i}{8 \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}\, \left ({\mathrm e}^{2 i x}-1\right )^{2}}\) | \(102\) |
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Time = 0.24 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.28 \[ \int \frac {1}{\sqrt {a \csc ^4(x)}} \, dx=-\frac {{\left (x \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{3} - \cos \left (x\right )\right )} \sin \left (x\right ) - x\right )} \sqrt {\frac {a}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}}}{2 \, a} \]
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\[ \int \frac {1}{\sqrt {a \csc ^4(x)}} \, dx=\int \frac {1}{\sqrt {a \csc ^{4}{\left (x \right )}}}\, dx \]
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none
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\sqrt {a \csc ^4(x)}} \, dx=\frac {x}{2 \, \sqrt {a}} - \frac {\tan \left (x\right )}{2 \, {\left (\sqrt {a} \tan \left (x\right )^{2} + \sqrt {a}\right )}} \]
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Exception generated. \[ \int \frac {1}{\sqrt {a \csc ^4(x)}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{\sqrt {a \csc ^4(x)}} \, dx=\int \frac {1}{\sqrt {\frac {a}{{\sin \left (x\right )}^4}}} \,d x \]
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