Optimal. Leaf size=53 \[ -\frac {\tanh ^{-1}\left (\frac {\sin (3 x)}{\sqrt {2} \sqrt {1-\cos (3 x)}}\right )}{6 \sqrt {2}}-\frac {\sin (3 x)}{6 (1-\cos (3 x))^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2729, 2728,
212} \begin {gather*} -\frac {\sin (3 x)}{6 (1-\cos (3 x))^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sin (3 x)}{\sqrt {2} \sqrt {1-\cos (3 x)}}\right )}{6 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2728
Rule 2729
Rubi steps
\begin {align*} \int \frac {1}{(1-\cos (3 x))^{3/2}} \, dx &=-\frac {\sin (3 x)}{6 (1-\cos (3 x))^{3/2}}+\frac {1}{4} \int \frac {1}{\sqrt {1-\cos (3 x)}} \, dx\\ &=-\frac {\sin (3 x)}{6 (1-\cos (3 x))^{3/2}}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\frac {\sin (3 x)}{\sqrt {1-\cos (3 x)}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sin (3 x)}{\sqrt {2} \sqrt {1-\cos (3 x)}}\right )}{6 \sqrt {2}}-\frac {\sin (3 x)}{6 (1-\cos (3 x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 61, normalized size = 1.15 \begin {gather*} -\frac {\left (\csc ^2\left (\frac {3 x}{4}\right )+4 \log \left (\cos \left (\frac {3 x}{4}\right )\right )-4 \log \left (\sin \left (\frac {3 x}{4}\right )\right )-\sec ^2\left (\frac {3 x}{4}\right )\right ) \sin ^3\left (\frac {3 x}{2}\right )}{12 (1-\cos (3 x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 52, normalized size = 0.98
method | result | size |
default | \(-\frac {\left (\frac {\cos \left (\frac {3 x}{2}\right )}{2}+\frac {\left (\ln \left (\cos \left (\frac {3 x}{2}\right )+1\right )-\ln \left (\cos \left (\frac {3 x}{2}\right )-1\right )\right ) \left (\sin ^{2}\left (\frac {3 x}{2}\right )\right )}{4}\right ) \sqrt {2}}{3 \sin \left (\frac {3 x}{2}\right ) \sqrt {2-2 \cos \left (3 x \right )}}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 433 vs.
\(2 (42) = 84\).
time = 1.26, size = 433, normalized size = 8.17 \begin {gather*} \frac {4 \, {\left (\sin \left (6 \, x\right ) - 2 \, \sin \left (3 \, x\right )\right )} \cos \left (\frac {3}{2} \, \pi + \frac {3}{2} \, \arctan \left (\sin \left (3 \, x\right ), \cos \left (3 \, x\right )\right )\right ) - 4 \, {\left (\sin \left (6 \, x\right ) - 2 \, \sin \left (3 \, x\right )\right )} \cos \left (\frac {1}{2} \, \pi + \frac {1}{2} \, \arctan \left (\sin \left (3 \, x\right ), \cos \left (3 \, x\right )\right )\right ) + {\left (2 \, {\left (2 \, \cos \left (3 \, x\right ) - 1\right )} \cos \left (6 \, x\right ) - \cos \left (6 \, x\right )^{2} - 4 \, \cos \left (3 \, x\right )^{2} - \sin \left (6 \, x\right )^{2} + 4 \, \sin \left (6 \, x\right ) \sin \left (3 \, x\right ) - 4 \, \sin \left (3 \, x\right )^{2} + 4 \, \cos \left (3 \, x\right ) - 1\right )} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (3 \, x\right ), \cos \left (3 \, x\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (3 \, x\right ), \cos \left (3 \, x\right )\right )\right )^{2} + 2 \, \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (3 \, x\right ), \cos \left (3 \, x\right )\right )\right ) + 1\right ) - {\left (2 \, {\left (2 \, \cos \left (3 \, x\right ) - 1\right )} \cos \left (6 \, x\right ) - \cos \left (6 \, x\right )^{2} - 4 \, \cos \left (3 \, x\right )^{2} - \sin \left (6 \, x\right )^{2} + 4 \, \sin \left (6 \, x\right ) \sin \left (3 \, x\right ) - 4 \, \sin \left (3 \, x\right )^{2} + 4 \, \cos \left (3 \, x\right ) - 1\right )} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (3 \, x\right ), \cos \left (3 \, x\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (3 \, x\right ), \cos \left (3 \, x\right )\right )\right )^{2} - 2 \, \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (3 \, x\right ), \cos \left (3 \, x\right )\right )\right ) + 1\right ) - 4 \, {\left (\cos \left (6 \, x\right ) - 2 \, \cos \left (3 \, x\right ) + 1\right )} \sin \left (\frac {3}{2} \, \pi + \frac {3}{2} \, \arctan \left (\sin \left (3 \, x\right ), \cos \left (3 \, x\right )\right )\right ) + 4 \, {\left (\cos \left (6 \, x\right ) - 2 \, \cos \left (3 \, x\right ) + 1\right )} \sin \left (\frac {1}{2} \, \pi + \frac {1}{2} \, \arctan \left (\sin \left (3 \, x\right ), \cos \left (3 \, x\right )\right )\right )}{12 \, {\left (\sqrt {2} \cos \left (6 \, x\right )^{2} + 4 \, \sqrt {2} \cos \left (3 \, x\right )^{2} + \sqrt {2} \sin \left (6 \, x\right )^{2} - 4 \, \sqrt {2} \sin \left (6 \, x\right ) \sin \left (3 \, x\right ) + 4 \, \sqrt {2} \sin \left (3 \, x\right )^{2} - 2 \, {\left (2 \, \sqrt {2} \cos \left (3 \, x\right ) - \sqrt {2}\right )} \cos \left (6 \, x\right ) - 4 \, \sqrt {2} \cos \left (3 \, x\right ) + \sqrt {2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 107 vs.
\(2 (42) = 84\).
time = 1.34, size = 107, normalized size = 2.02 \begin {gather*} \frac {{\left (\sqrt {2} \cos \left (3 \, x\right ) - \sqrt {2}\right )} \log \left (-\frac {{\left (\cos \left (3 \, x\right ) + 3\right )} \sin \left (3 \, x\right ) - 2 \, {\left (\sqrt {2} \cos \left (3 \, x\right ) + \sqrt {2}\right )} \sqrt {-\cos \left (3 \, x\right ) + 1}}{{\left (\cos \left (3 \, x\right ) - 1\right )} \sin \left (3 \, x\right )}\right ) \sin \left (3 \, x\right ) + 4 \, {\left (\cos \left (3 \, x\right ) + 1\right )} \sqrt {-\cos \left (3 \, x\right ) + 1}}{24 \, {\left (\cos \left (3 \, x\right ) - 1\right )} \sin \left (3 \, x\right )} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (1 - \cos {\left (3 x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs.
\(2 (42) = 84\).
time = 1.26, size = 100, normalized size = 1.89 \begin {gather*} -\frac {\sqrt {2} {\left (\frac {2 \, {\left (\cos \left (\frac {3}{2} \, x\right ) - 1\right )}}{\cos \left (\frac {3}{2} \, x\right ) + 1} - 1\right )} {\left (\cos \left (\frac {3}{2} \, x\right ) + 1\right )}}{48 \, {\left (\cos \left (\frac {3}{2} \, x\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (\frac {3}{2} \, x\right )\right )} + \frac {\sqrt {2} \log \left (-\frac {\cos \left (\frac {3}{2} \, x\right ) - 1}{\cos \left (\frac {3}{2} \, x\right ) + 1}\right )}{24 \, \mathrm {sgn}\left (\sin \left (\frac {3}{2} \, x\right )\right )} - \frac {\sqrt {2} {\left (\cos \left (\frac {3}{2} \, x\right ) - 1\right )}}{48 \, {\left (\cos \left (\frac {3}{2} \, x\right ) + 1\right )} \mathrm {sgn}\left (\sin \left (\frac {3}{2} \, x\right )\right )} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (1-\cos \left (3\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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