Integrand size = 10, antiderivative size = 77 \[ \int \frac {1}{\left (a \cot ^4(x)\right )^{3/2}} \, dx=\frac {\cot (x)}{a \sqrt {a \cot ^4(x)}}-\frac {x \cot ^2(x)}{a \sqrt {a \cot ^4(x)}}-\frac {\tan (x)}{3 a \sqrt {a \cot ^4(x)}}+\frac {\tan ^3(x)}{5 a \sqrt {a \cot ^4(x)}} \]
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Time = 0.03 (sec) , antiderivative size = 77, 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 = {3739, 3554, 8} \[ \int \frac {1}{\left (a \cot ^4(x)\right )^{3/2}} \, dx=\frac {\cot (x)}{a \sqrt {a \cot ^4(x)}}-\frac {x \cot ^2(x)}{a \sqrt {a \cot ^4(x)}}+\frac {\tan ^3(x)}{5 a \sqrt {a \cot ^4(x)}}-\frac {\tan (x)}{3 a \sqrt {a \cot ^4(x)}} \]
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Rule 8
Rule 3554
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \frac {\cot ^2(x) \int \tan ^6(x) \, dx}{a \sqrt {a \cot ^4(x)}} \\ & = \frac {\tan ^3(x)}{5 a \sqrt {a \cot ^4(x)}}-\frac {\cot ^2(x) \int \tan ^4(x) \, dx}{a \sqrt {a \cot ^4(x)}} \\ & = -\frac {\tan (x)}{3 a \sqrt {a \cot ^4(x)}}+\frac {\tan ^3(x)}{5 a \sqrt {a \cot ^4(x)}}+\frac {\cot ^2(x) \int \tan ^2(x) \, dx}{a \sqrt {a \cot ^4(x)}} \\ & = \frac {\cot (x)}{a \sqrt {a \cot ^4(x)}}-\frac {\tan (x)}{3 a \sqrt {a \cot ^4(x)}}+\frac {\tan ^3(x)}{5 a \sqrt {a \cot ^4(x)}}-\frac {\cot ^2(x) \int 1 \, dx}{a \sqrt {a \cot ^4(x)}} \\ & = \frac {\cot (x)}{a \sqrt {a \cot ^4(x)}}-\frac {x \cot ^2(x)}{a \sqrt {a \cot ^4(x)}}-\frac {\tan (x)}{3 a \sqrt {a \cot ^4(x)}}+\frac {\tan ^3(x)}{5 a \sqrt {a \cot ^4(x)}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.53 \[ \int \frac {1}{\left (a \cot ^4(x)\right )^{3/2}} \, dx=\frac {15 \cot (x)-15 \arctan (\tan (x)) \cot ^2(x)-5 \tan (x)+3 \tan ^3(x)}{15 a \sqrt {a \cot ^4(x)}} \]
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Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.55
method | result | size |
derivativedivides | \(\frac {\cot \left (x \right ) \left (15 \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (x \right )\right )\right ) \cot \left (x \right )^{5}+15 \cot \left (x \right )^{4}-5 \cot \left (x \right )^{2}+3\right )}{15 \left (a \cot \left (x \right )^{4}\right )^{\frac {3}{2}}}\) | \(42\) |
default | \(\frac {\cot \left (x \right ) \left (15 \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (x \right )\right )\right ) \cot \left (x \right )^{5}+15 \cot \left (x \right )^{4}-5 \cot \left (x \right )^{2}+3\right )}{15 \left (a \cot \left (x \right )^{4}\right )^{\frac {3}{2}}}\) | \(42\) |
risch | \(\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2} x}{a \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \left ({\mathrm e}^{2 i x}+1\right )^{4}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {2 i \left (45 \,{\mathrm e}^{8 i x}+90 \,{\mathrm e}^{6 i x}+140 \,{\mathrm e}^{4 i x}+70 \,{\mathrm e}^{2 i x}+23\right )}{15 a \left ({\mathrm e}^{2 i x}+1\right )^{3} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \left ({\mathrm e}^{2 i x}+1\right )^{4}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}\) | \(123\) |
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Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (65) = 130\).
Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.84 \[ \int \frac {1}{\left (a \cot ^4(x)\right )^{3/2}} \, dx=\frac {{\left (15 \, x \cos \left (2 \, x\right )^{4} + 30 \, x \cos \left (2 \, x\right )^{3} - 30 \, x \cos \left (2 \, x\right ) - {\left (23 \, \cos \left (2 \, x\right )^{3} + \cos \left (2 \, x\right )^{2} - 11 \, \cos \left (2 \, x\right ) - 13\right )} \sin \left (2 \, x\right ) - 15 \, x\right )} \sqrt {\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{15 \, {\left (a^{2} \cos \left (2 \, x\right )^{4} + 4 \, a^{2} \cos \left (2 \, x\right )^{3} + 6 \, a^{2} \cos \left (2 \, x\right )^{2} + 4 \, a^{2} \cos \left (2 \, x\right ) + a^{2}\right )}} \]
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\[ \int \frac {1}{\left (a \cot ^4(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a \cot ^{4}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]
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none
Time = 0.43 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.38 \[ \int \frac {1}{\left (a \cot ^4(x)\right )^{3/2}} \, dx=\frac {3 \, \tan \left (x\right )^{5} - 5 \, \tan \left (x\right )^{3} + 15 \, \tan \left (x\right )}{15 \, a^{\frac {3}{2}}} - \frac {x}{a^{\frac {3}{2}}} \]
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Exception generated. \[ \int \frac {1}{\left (a \cot ^4(x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{\left (a \cot ^4(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (a\,{\mathrm {cot}\left (x\right )}^4\right )}^{3/2}} \,d x \]
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