Integrand size = 10, antiderivative size = 74 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=-\frac {\cot (x)}{7 \left (a \csc ^2(x)\right )^{7/2}}-\frac {6 \cot (x)}{35 a \left (a \csc ^2(x)\right )^{5/2}}-\frac {8 \cot (x)}{35 a^2 \left (a \csc ^2(x)\right )^{3/2}}-\frac {16 \cot (x)}{35 a^3 \sqrt {a \csc ^2(x)}} \]
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Time = 0.05 (sec) , antiderivative size = 74, 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 = {4207, 198, 197} \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=-\frac {16 \cot (x)}{35 a^3 \sqrt {a \csc ^2(x)}}-\frac {8 \cot (x)}{35 a^2 \left (a \csc ^2(x)\right )^{3/2}}-\frac {6 \cot (x)}{35 a \left (a \csc ^2(x)\right )^{5/2}}-\frac {\cot (x)}{7 \left (a \csc ^2(x)\right )^{7/2}} \]
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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 )^{9/2}} \, dx,x,\cot (x)\right )\right ) \\ & = -\frac {\cot (x)}{7 \left (a \csc ^2(x)\right )^{7/2}}-\frac {6}{7} \text {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{7/2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {\cot (x)}{7 \left (a \csc ^2(x)\right )^{7/2}}-\frac {6 \cot (x)}{35 a \left (a \csc ^2(x)\right )^{5/2}}-\frac {24 \text {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{5/2}} \, dx,x,\cot (x)\right )}{35 a} \\ & = -\frac {\cot (x)}{7 \left (a \csc ^2(x)\right )^{7/2}}-\frac {6 \cot (x)}{35 a \left (a \csc ^2(x)\right )^{5/2}}-\frac {8 \cot (x)}{35 a^2 \left (a \csc ^2(x)\right )^{3/2}}-\frac {16 \text {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\cot (x)\right )}{35 a^2} \\ & = -\frac {\cot (x)}{7 \left (a \csc ^2(x)\right )^{7/2}}-\frac {6 \cot (x)}{35 a \left (a \csc ^2(x)\right )^{5/2}}-\frac {8 \cot (x)}{35 a^2 \left (a \csc ^2(x)\right )^{3/2}}-\frac {16 \cot (x)}{35 a^3 \sqrt {a \csc ^2(x)}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=\frac {(-1225 \cos (x)+245 \cos (3 x)-49 \cos (5 x)+5 \cos (7 x)) \sqrt {a \csc ^2(x)} \sin (x)}{2240 a^4} \]
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Time = 0.48 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.69
method | result | size |
default | \(-\frac {\sin \left (x \right )^{5} \left (-16+5 \cos \left (x \right )^{4}-15 \cos \left (x \right )^{3}+9 \cos \left (x \right )^{2}+13 \cos \left (x \right )\right ) \sqrt {4}}{70 \left (\cos \left (x \right )-1\right )^{3} \sqrt {a \csc \left (x \right )^{2}}\, a^{3}}\) | \(51\) |
risch | \(\frac {i {\mathrm e}^{8 i x}}{896 a^{3} \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {35 i {\mathrm e}^{2 i x}}{128 a^{3} \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {35 i}{128 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{3}}+\frac {7 i {\mathrm e}^{-2 i x}}{128 a^{3} \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {11 i \cos \left (6 x \right )}{1120 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{3}}+\frac {27 \sin \left (6 x \right )}{2240 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{3}}+\frac {7 i \cos \left (4 x \right )}{160 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{3}}-\frac {21 \sin \left (4 x \right )}{320 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{3}}\) | \(303\) |
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Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=\frac {{\left (5 \, \cos \left (x\right )^{7} - 21 \, \cos \left (x\right )^{5} + 35 \, \cos \left (x\right )^{3} - 35 \, \cos \left (x\right )\right )} \sqrt {-\frac {a}{\cos \left (x\right )^{2} - 1}} \sin \left (x\right )}{35 \, a^{4}} \]
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Time = 10.63 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=- \frac {16 \cot ^{7}{\left (x \right )}}{35 \left (a \csc ^{2}{\left (x \right )}\right )^{\frac {7}{2}}} - \frac {8 \cot ^{5}{\left (x \right )}}{5 \left (a \csc ^{2}{\left (x \right )}\right )^{\frac {7}{2}}} - \frac {2 \cot ^{3}{\left (x \right )}}{\left (a \csc ^{2}{\left (x \right )}\right )^{\frac {7}{2}}} - \frac {\cot {\left (x \right )}}{\left (a \csc ^{2}{\left (x \right )}\right )^{\frac {7}{2}}} \]
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\[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=\int { \frac {1}{\left (a \csc \left (x\right )^{2}\right )^{\frac {7}{2}}} \,d x } \]
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Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=\frac {32 \, \mathrm {sgn}\left (\sin \left (x\right )\right )}{35 \, a^{\frac {7}{2}}} - \frac {32 \, {\left (\frac {7 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac {21 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {35 \, {\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - 1\right )}}{35 \, a^{\frac {7}{2}} {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )}^{7} \mathrm {sgn}\left (\sin \left (x\right )\right )} \]
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Timed out. \[ \int \frac {1}{\left (a \csc ^2(x)\right )^{7/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\sin \left (x\right )}^2}\right )}^{7/2}} \,d x \]
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