Optimal. Leaf size=21 \[ \frac {1}{6} \csc ^3(2 x)-\frac {1}{10} \csc ^5(2 x) \]
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Rubi [A] time = 0.03, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2606, 14} \[ \frac {1}{6} \csc ^3(2 x)-\frac {1}{10} \csc ^5(2 x) \]
Antiderivative was successfully verified.
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Rule 14
Rule 2606
Rubi steps
\begin {align*} \int \cot ^3(2 x) \csc ^3(2 x) \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\csc (2 x)\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\csc (2 x)\right )\right )\\ &=\frac {1}{6} \csc ^3(2 x)-\frac {1}{10} \csc ^5(2 x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 21, normalized size = 1.00 \[ \frac {1}{6} \csc ^3(2 x)-\frac {1}{10} \csc ^5(2 x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 36, normalized size = 1.71 \[ -\frac {5 \, \cos \left (2 \, x\right )^{2} - 2}{30 \, {\left (\cos \left (2 \, x\right )^{4} - 2 \, \cos \left (2 \, x\right )^{2} + 1\right )} \sin \left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 18, normalized size = 0.86 \[ \frac {5 \, \sin \left (2 \, x\right )^{2} - 3}{30 \, \sin \left (2 \, x\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 58, normalized size = 2.76 \[ \frac {\cos ^{4}\left (2 x \right )}{30 \sin \left (2 x \right )}-\frac {\cos ^{4}\left (2 x \right )}{30 \sin \left (2 x \right )^{3}}+\frac {\left (\cos ^{2}\left (2 x \right )+2\right ) \sin \left (2 x \right )}{30}-\frac {\cos ^{4}\left (2 x \right )}{10 \sin \left (2 x \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 18, normalized size = 0.86 \[ \frac {5 \, \sin \left (2 \, x\right )^{2} - 3}{30 \, \sin \left (2 \, x\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 18, normalized size = 0.86 \[ \frac {5\,{\sin \left (2\,x\right )}^2-3}{30\,{\sin \left (2\,x\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 19, normalized size = 0.90 \[ - \frac {3 - 5 \sin ^{2}{\left (2 x \right )}}{30 \sin ^{5}{\left (2 x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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