Optimal. Leaf size=13 \[ -\cot (x)-\frac {\cot ^3(x)}{3} \]
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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3852}
\begin {gather*} -\frac {1}{3} \cot ^3(x)-\cot (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 3852
Rubi steps
\begin {align*} \int \csc ^4(x) \, dx &=-\text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (x)\right )\\ &=-\cot (x)-\frac {\cot ^3(x)}{3}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.31 \begin {gather*} -\frac {2 \cot (x)}{3}-\frac {1}{3} \cot (x) \csc ^2(x) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.96, size = 15, normalized size = 1.15 \begin {gather*} \frac {-2}{3 \text {Tan}\left [x\right ]}-\frac {\text {Cos}\left [x\right ]}{3 \text {Sin}\left [x\right ]^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 12, normalized size = 0.92
method | result | size |
default | \(\left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (x \right )\right )}{3}\right ) \cot \left (x \right )\) | \(12\) |
risch | \(\frac {4 i \left (3 \,{\mathrm e}^{2 i x}-1\right )}{3 \left ({\mathrm e}^{2 i x}-1\right )^{3}}\) | \(22\) |
norman | \(\frac {-\frac {1}{24}-\frac {3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {3 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{8}+\frac {\left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{24}}{\tan \left (\frac {x}{2}\right )^{3}}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 14, normalized size = 1.08 \begin {gather*} -\frac {3 \, \tan \left (x\right )^{2} + 1}{3 \, \tan \left (x\right )^{3}} \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. 25 vs.
\(2 (11) = 22\).
time = 0.33, size = 25, normalized size = 1.92 \begin {gather*} -\frac {2 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{3 \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 20, normalized size = 1.54 \begin {gather*} - \frac {2 \cos {\left (x \right )}}{3 \sin {\left (x \right )}} - \frac {\cos {\left (x \right )}}{3 \sin ^{3}{\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 18, normalized size = 1.38 \begin {gather*} \frac {2 \left (-3 \tan ^{2}x-1\right )}{6 \tan ^{3}x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 17, normalized size = 1.31 \begin {gather*} -\frac {2\,\cos \left (x\right )\,{\sin \left (x\right )}^2+\cos \left (x\right )}{3\,{\sin \left (x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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