Optimal. Leaf size=43 \[ \frac {\sin (6 x)}{60 \left (11-5 \sin ^2(6 x)\right )}-\frac {\tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sin (6 x)\right )}{60 \sqrt {55}} \]
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Rubi [A] time = 0.06, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4338, 288, 206} \[ \frac {\sin (6 x)}{60 \left (11-5 \sin ^2(6 x)\right )}-\frac {\tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sin (6 x)\right )}{60 \sqrt {55}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 288
Rule 4338
Rubi steps
\begin {align*} \int \frac {\cot (6 x) \csc (6 x)}{\left (5-11 \csc ^2(6 x)\right )^2} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {x^2}{\left (11-5 x^2\right )^2} \, dx,x,\sin (6 x)\right )\\ &=\frac {\sin (6 x)}{60 \left (11-5 \sin ^2(6 x)\right )}-\frac {1}{60} \operatorname {Subst}\left (\int \frac {1}{11-5 x^2} \, dx,x,\sin (6 x)\right )\\ &=-\frac {\tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sin (6 x)\right )}{60 \sqrt {55}}+\frac {\sin (6 x)}{60 \left (11-5 \sin ^2(6 x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.66, size = 41, normalized size = 0.95 \[ \frac {\sin (6 x)}{30 (5 \cos (12 x)+17)}-\frac {\tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sin (6 x)\right )}{60 \sqrt {55}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 73, normalized size = 1.70 \[ \frac {{\left (5 \, \sqrt {55} \cos \left (6 \, x\right )^{2} + 6 \, \sqrt {55}\right )} \log \left (-\frac {5 \, \cos \left (6 \, x\right )^{2} + 2 \, \sqrt {55} \sin \left (6 \, x\right ) - 16}{5 \, \cos \left (6 \, x\right )^{2} + 6}\right ) + 110 \, \sin \left (6 \, x\right )}{6600 \, {\left (5 \, \cos \left (6 \, x\right )^{2} + 6\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 48, normalized size = 1.12 \[ \frac {1}{6600} \, \sqrt {55} \log \left (\frac {\sqrt {55} - 5 \, \sin \left (6 \, x\right )}{\sqrt {55} + 5 \, \sin \left (6 \, x\right )}\right ) - \frac {\sin \left (6 \, x\right )}{60 \, {\left (5 \, \sin \left (6 \, x\right )^{2} - 11\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 35, normalized size = 0.81 \[ \frac {\csc \left (6 x \right )}{660 \left (\csc ^{2}\left (6 x \right )\right )-300}-\frac {\sqrt {55}\, \arctanh \left (\frac {\csc \left (6 x \right ) \sqrt {55}}{5}\right )}{3300} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 49, normalized size = 1.14 \[ \frac {1}{6600} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sin \left (6 \, x\right )}{\sqrt {55} + 5 \, \sin \left (6 \, x\right )}\right ) - \frac {\sin \left (6 \, x\right )}{60 \, {\left (5 \, \sin \left (6 \, x\right )^{2} - 11\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.15, size = 57, normalized size = 1.33 \[ -\frac {55\,\sin \left (6\,x\right )-11\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sin \left (6\,x\right )}{11}\right )+5\,\sqrt {55}\,{\sin \left (6\,x\right )}^2\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sin \left (6\,x\right )}{11}\right )}{16500\,{\sin \left (6\,x\right )}^2-36300} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.72, size = 151, normalized size = 3.51 \[ \frac {11 \sqrt {55} \log {\left (\csc {\left (6 x \right )} - \frac {\sqrt {55}}{11} \right )} \csc ^{2}{\left (6 x \right )}}{72600 \csc ^{2}{\left (6 x \right )} - 33000} - \frac {5 \sqrt {55} \log {\left (\csc {\left (6 x \right )} - \frac {\sqrt {55}}{11} \right )}}{72600 \csc ^{2}{\left (6 x \right )} - 33000} - \frac {11 \sqrt {55} \log {\left (\csc {\left (6 x \right )} + \frac {\sqrt {55}}{11} \right )} \csc ^{2}{\left (6 x \right )}}{72600 \csc ^{2}{\left (6 x \right )} - 33000} + \frac {5 \sqrt {55} \log {\left (\csc {\left (6 x \right )} + \frac {\sqrt {55}}{11} \right )}}{72600 \csc ^{2}{\left (6 x \right )} - 33000} + \frac {110 \csc {\left (6 x \right )}}{72600 \csc ^{2}{\left (6 x \right )} - 33000} \]
Verification of antiderivative is not currently implemented for this CAS.
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