Optimal. Leaf size=25 \[ \frac {\sec ^9(x)}{9}-\frac {2 \sec ^7(x)}{7}+\frac {\sec ^5(x)}{5} \]
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Rubi [A] time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4397, 2606, 270} \[ \frac {\sec ^9(x)}{9}-\frac {2 \sec ^7(x)}{7}+\frac {\sec ^5(x)}{5} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2606
Rule 4397
Rubi steps
\begin {align*} \int \frac {1}{(\csc (x)-\sin (x))^5} \, dx &=\int \sec ^5(x) \tan ^5(x) \, dx\\ &=\operatorname {Subst}\left (\int x^4 \left (-1+x^2\right )^2 \, dx,x,\sec (x)\right )\\ &=\operatorname {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sec (x)\right )\\ &=\frac {\sec ^5(x)}{5}-\frac {2 \sec ^7(x)}{7}+\frac {\sec ^9(x)}{9}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 1.00 \[ \frac {\sec ^9(x)}{9}-\frac {2 \sec ^7(x)}{7}+\frac {\sec ^5(x)}{5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 20, normalized size = 0.80 \[ \frac {63 \, \cos \relax (x)^{4} - 90 \, \cos \relax (x)^{2} + 35}{315 \, \cos \relax (x)^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 101, normalized size = 4.04 \[ \frac {16 \, {\left (\frac {9 \, {\left (\cos \relax (x) - 1\right )}}{\cos \relax (x) + 1} + \frac {36 \, {\left (\cos \relax (x) - 1\right )}^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {126 \, {\left (\cos \relax (x) - 1\right )}^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {441 \, {\left (\cos \relax (x) - 1\right )}^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} - \frac {315 \, {\left (\cos \relax (x) - 1\right )}^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} + \frac {210 \, {\left (\cos \relax (x) - 1\right )}^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} + 1\right )}}{315 \, {\left (\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1} + 1\right )}^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 20, normalized size = 0.80 \[ -\frac {2}{7 \cos \relax (x )^{7}}+\frac {1}{9 \cos \relax (x )^{9}}+\frac {1}{5 \cos \relax (x )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 187, normalized size = 7.48 \[ \frac {16 \, {\left (\frac {9 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {36 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} - \frac {126 \, \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} - \frac {441 \, \sin \relax (x)^{8}}{{\left (\cos \relax (x) + 1\right )}^{8}} - \frac {315 \, \sin \relax (x)^{10}}{{\left (\cos \relax (x) + 1\right )}^{10}} - \frac {210 \, \sin \relax (x)^{12}}{{\left (\cos \relax (x) + 1\right )}^{12}} - 1\right )}}{315 \, {\left (\frac {9 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {36 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {84 \, \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} - \frac {126 \, \sin \relax (x)^{8}}{{\left (\cos \relax (x) + 1\right )}^{8}} + \frac {126 \, \sin \relax (x)^{10}}{{\left (\cos \relax (x) + 1\right )}^{10}} - \frac {84 \, \sin \relax (x)^{12}}{{\left (\cos \relax (x) + 1\right )}^{12}} + \frac {36 \, \sin \relax (x)^{14}}{{\left (\cos \relax (x) + 1\right )}^{14}} - \frac {9 \, \sin \relax (x)^{16}}{{\left (\cos \relax (x) + 1\right )}^{16}} + \frac {\sin \relax (x)^{18}}{{\left (\cos \relax (x) + 1\right )}^{18}} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.94, size = 19, normalized size = 0.76 \[ \frac {1}{5\,{\cos \relax (x)}^5}-\frac {2}{7\,{\cos \relax (x)}^7}+\frac {1}{9\,{\cos \relax (x)}^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- \sin {\relax (x )} + \csc {\relax (x )}\right )^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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