Optimal. Leaf size=29 \[ -\frac {2 \cot (x)}{3 \sqrt {\csc ^2(x)}}-\frac {\cot (x)}{3 \csc ^2(x)^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4122, 192, 191} \[ -\frac {2 \cot (x)}{3 \sqrt {\csc ^2(x)}}-\frac {\cot (x)}{3 \csc ^2(x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\csc ^2(x)^{3/2}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{5/2}} \, dx,x,\cot (x)\right )\\ &=-\frac {\cot (x)}{3 \csc ^2(x)^{3/2}}-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{3/2}} \, dx,x,\cot (x)\right )\\ &=-\frac {\cot (x)}{3 \csc ^2(x)^{3/2}}-\frac {2 \cot (x)}{3 \sqrt {\csc ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 23, normalized size = 0.79 \[ \frac {(\cos (3 x)-9 \cos (x)) \csc (x)}{12 \sqrt {\csc ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 11, normalized size = 0.38 \[ \frac {1}{3} \, \cos \relax (x)^{3} - \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.42, size = 44, normalized size = 1.52 \[ -\frac {4 \, {\left (\frac {3 \, {\left (\cos \relax (x) - 1\right )} \mathrm {sgn}\left (\sin \relax (x)\right )}{\cos \relax (x) + 1} - \mathrm {sgn}\left (\sin \relax (x)\right )\right )}}{3 \, {\left (\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1} - 1\right )}^{3}} + \frac {4}{3} \, \mathrm {sgn}\left (\sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 30, normalized size = 1.03 \[ \frac {\sin \relax (x ) \left (\cos \relax (x )-2\right ) \sqrt {4}}{6 \left (-1+\cos \relax (x )\right )^{2} \left (-\frac {1}{-1+\cos ^{2}\relax (x )}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 11, normalized size = 0.38 \[ \frac {1}{12} \, \cos \left (3 \, x\right ) - \frac {3}{4} \, \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\left (\frac {1}{{\sin \relax (x)}^2}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.94, size = 29, normalized size = 1.00 \[ - \frac {2 \cot ^{3}{\relax (x )}}{3 \left (\csc ^{2}{\relax (x )}\right )^{\frac {3}{2}}} - \frac {\cot {\relax (x )}}{\left (\csc ^{2}{\relax (x )}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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