Integrand size = 10, antiderivative size = 164 \[ \int \left (a \csc ^4(x)\right )^{7/2} \, dx=-2 a^3 \cos ^2(x) \cot (x) \sqrt {a \csc ^4(x)}-3 a^3 \cos ^2(x) \cot ^3(x) \sqrt {a \csc ^4(x)}-\frac {20}{7} a^3 \cos ^2(x) \cot ^5(x) \sqrt {a \csc ^4(x)}-\frac {5}{3} a^3 \cos ^2(x) \cot ^7(x) \sqrt {a \csc ^4(x)}-\frac {6}{11} a^3 \cos ^2(x) \cot ^9(x) \sqrt {a \csc ^4(x)}-\frac {1}{13} a^3 \cos ^2(x) \cot ^{11}(x) \sqrt {a \csc ^4(x)}-a^3 \cos (x) \sqrt {a \csc ^4(x)} \sin (x) \]
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Time = 0.04 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4208, 3852} \[ \int \left (a \csc ^4(x)\right )^{7/2} \, dx=-\frac {1}{13} a^3 \cos ^2(x) \cot ^{11}(x) \sqrt {a \csc ^4(x)}-\frac {6}{11} a^3 \cos ^2(x) \cot ^9(x) \sqrt {a \csc ^4(x)}-\frac {5}{3} a^3 \cos ^2(x) \cot ^7(x) \sqrt {a \csc ^4(x)}-\frac {20}{7} a^3 \cos ^2(x) \cot ^5(x) \sqrt {a \csc ^4(x)}-3 a^3 \cos ^2(x) \cot ^3(x) \sqrt {a \csc ^4(x)}-2 a^3 \cos ^2(x) \cot (x) \sqrt {a \csc ^4(x)}-a^3 \sin (x) \cos (x) \sqrt {a \csc ^4(x)} \]
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Rule 3852
Rule 4208
Rubi steps \begin{align*} \text {integral}& = \left (a^3 \sqrt {a \csc ^4(x)} \sin ^2(x)\right ) \int \csc ^{14}(x) \, dx \\ & = -\left (\left (a^3 \sqrt {a \csc ^4(x)} \sin ^2(x)\right ) \text {Subst}\left (\int \left (1+6 x^2+15 x^4+20 x^6+15 x^8+6 x^{10}+x^{12}\right ) \, dx,x,\cot (x)\right )\right ) \\ & = -2 a^3 \cos ^2(x) \cot (x) \sqrt {a \csc ^4(x)}-3 a^3 \cos ^2(x) \cot ^3(x) \sqrt {a \csc ^4(x)}-\frac {20}{7} a^3 \cos ^2(x) \cot ^5(x) \sqrt {a \csc ^4(x)}-\frac {5}{3} a^3 \cos ^2(x) \cot ^7(x) \sqrt {a \csc ^4(x)}-\frac {6}{11} a^3 \cos ^2(x) \cot ^9(x) \sqrt {a \csc ^4(x)}-\frac {1}{13} a^3 \cos ^2(x) \cot ^{11}(x) \sqrt {a \csc ^4(x)}-a^3 \cos (x) \sqrt {a \csc ^4(x)} \sin (x) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.36 \[ \int \left (a \csc ^4(x)\right )^{7/2} \, dx=-\frac {a^3 \cos (x) \sqrt {a \csc ^4(x)} \left (1024+512 \csc ^2(x)+384 \csc ^4(x)+320 \csc ^6(x)+280 \csc ^8(x)+252 \csc ^{10}(x)+231 \csc ^{12}(x)\right ) \sin (x)}{3003} \]
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Time = 1.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.37
method | result | size |
default | \(-\frac {\cot \left (x \right ) \csc \left (x \right )^{10} \sqrt {a \csc \left (x \right )^{4}}\, a^{3} \left (1024 \cos \left (x \right )^{12}-6656 \cos \left (x \right )^{10}+18304 \cos \left (x \right )^{8}-27456 \cos \left (x \right )^{6}+24024 \cos \left (x \right )^{4}-12012 \cos \left (x \right )^{2}+3003\right ) \sqrt {16}}{12012}\) | \(61\) |
risch | \(\frac {2048 i a^{3} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}\, \left (1716 \,{\mathrm e}^{10 i x}-1287 \,{\mathrm e}^{8 i x}+715 \,{\mathrm e}^{6 i x}-286 \,{\mathrm e}^{4 i x}-13+79 \cos \left (2 x \right )+77 i \sin \left (2 x \right )\right )}{3003 \left ({\mathrm e}^{2 i x}-1\right )^{11}}\) | \(77\) |
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Time = 0.25 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.72 \[ \int \left (a \csc ^4(x)\right )^{7/2} \, dx=\frac {{\left (1024 \, a^{3} \cos \left (x\right )^{13} - 6656 \, a^{3} \cos \left (x\right )^{11} + 18304 \, a^{3} \cos \left (x\right )^{9} - 27456 \, a^{3} \cos \left (x\right )^{7} + 24024 \, a^{3} \cos \left (x\right )^{5} - 12012 \, a^{3} \cos \left (x\right )^{3} + 3003 \, a^{3} \cos \left (x\right )\right )} \sqrt {\frac {a}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}}}{3003 \, {\left (\cos \left (x\right )^{10} - 5 \, \cos \left (x\right )^{8} + 10 \, \cos \left (x\right )^{6} - 10 \, \cos \left (x\right )^{4} + 5 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \]
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Timed out. \[ \int \left (a \csc ^4(x)\right )^{7/2} \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.40 \[ \int \left (a \csc ^4(x)\right )^{7/2} \, dx=-\frac {3003 \, a^{\frac {7}{2}} \tan \left (x\right )^{12} + 6006 \, a^{\frac {7}{2}} \tan \left (x\right )^{10} + 9009 \, a^{\frac {7}{2}} \tan \left (x\right )^{8} + 8580 \, a^{\frac {7}{2}} \tan \left (x\right )^{6} + 5005 \, a^{\frac {7}{2}} \tan \left (x\right )^{4} + 1638 \, a^{\frac {7}{2}} \tan \left (x\right )^{2} + 231 \, a^{\frac {7}{2}}}{3003 \, \tan \left (x\right )^{13}} \]
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Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.42 \[ \int \left (a \csc ^4(x)\right )^{7/2} \, dx=-\frac {{\left (3003 \, a^{3} \tan \left (x\right )^{12} + 6006 \, a^{3} \tan \left (x\right )^{10} + 9009 \, a^{3} \tan \left (x\right )^{8} + 8580 \, a^{3} \tan \left (x\right )^{6} + 5005 \, a^{3} \tan \left (x\right )^{4} + 1638 \, a^{3} \tan \left (x\right )^{2} + 231 \, a^{3}\right )} \sqrt {a}}{3003 \, \tan \left (x\right )^{13}} \]
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Time = 23.06 (sec) , antiderivative size = 603, normalized size of antiderivative = 3.68 \[ \int \left (a \csc ^4(x)\right )^{7/2} \, dx=\text {Too large to display} \]
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