Optimal. Leaf size=43 \[ -\frac {8 \cot (x)}{15 \sqrt {\csc ^2(x)}}-\frac {4 \cot (x)}{15 \csc ^2(x)^{3/2}}-\frac {\cot (x)}{5 \csc ^2(x)^{5/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4122, 192, 191} \[ -\frac {8 \cot (x)}{15 \sqrt {\csc ^2(x)}}-\frac {4 \cot (x)}{15 \csc ^2(x)^{3/2}}-\frac {\cot (x)}{5 \csc ^2(x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\csc ^2(x)^{5/2}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{7/2}} \, dx,x,\cot (x)\right )\\ &=-\frac {\cot (x)}{5 \csc ^2(x)^{5/2}}-\frac {4}{5} \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{5/2}} \, dx,x,\cot (x)\right )\\ &=-\frac {\cot (x)}{5 \csc ^2(x)^{5/2}}-\frac {4 \cot (x)}{15 \csc ^2(x)^{3/2}}-\frac {8}{15} \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{3/2}} \, dx,x,\cot (x)\right )\\ &=-\frac {\cot (x)}{5 \csc ^2(x)^{5/2}}-\frac {4 \cot (x)}{15 \csc ^2(x)^{3/2}}-\frac {8 \cot (x)}{15 \sqrt {\csc ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 31, normalized size = 0.72 \[ -\frac {(150 \cos (x)-25 \cos (3 x)+3 \cos (5 x)) \csc (x)}{240 \sqrt {\csc ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 17, normalized size = 0.40 \[ -\frac {1}{5} \, \cos \relax (x)^{5} + \frac {2}{3} \, \cos \relax (x)^{3} - \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 61, normalized size = 1.42 \[ -\frac {16 \, {\left (\frac {5 \, {\left (\cos \relax (x) - 1\right )} \mathrm {sgn}\left (\sin \relax (x)\right )}{\cos \relax (x) + 1} - \frac {10 \, {\left (\cos \relax (x) - 1\right )}^{2} \mathrm {sgn}\left (\sin \relax (x)\right )}{{\left (\cos \relax (x) + 1\right )}^{2}} - \mathrm {sgn}\left (\sin \relax (x)\right )\right )}}{15 \, {\left (\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1} - 1\right )}^{5}} + \frac {16}{15} \, \mathrm {sgn}\left (\sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 38, normalized size = 0.88 \[ \frac {\sin \relax (x ) \left (3 \left (\cos ^{2}\relax (x )\right )-9 \cos \relax (x )+8\right ) \sqrt {4}}{30 \left (-1+\cos \relax (x )\right )^{3} \left (-\frac {1}{-1+\cos ^{2}\relax (x )}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 17, normalized size = 0.40 \[ -\frac {1}{80} \, \cos \left (5 \, x\right ) + \frac {5}{48} \, \cos \left (3 \, x\right ) - \frac {5}{8} \, \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (\frac {1}{{\sin \relax (x)}^2}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.32, size = 46, normalized size = 1.07 \[ - \frac {8 \cot ^{5}{\relax (x )}}{15 \left (\csc ^{2}{\relax (x )}\right )^{\frac {5}{2}}} - \frac {4 \cot ^{3}{\relax (x )}}{3 \left (\csc ^{2}{\relax (x )}\right )^{\frac {5}{2}}} - \frac {\cot {\relax (x )}}{\left (\csc ^{2}{\relax (x )}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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