Optimal. Leaf size=11 \[ \csc (x)-\frac {\csc ^3(x)}{3} \]
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Rubi [A]
time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2686}
\begin {gather*} \csc (x)-\frac {\csc ^3(x)}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2686
Rubi steps
\begin {align*} \int \cot ^3(x) \csc (x) \, dx &=-\text {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\csc (x)\right )\\ &=\csc (x)-\frac {\csc ^3(x)}{3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 11, normalized size = 1.00 \begin {gather*} \csc (x)-\frac {\csc ^3(x)}{3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(31\) vs.
\(2(11)=22\).
time = 0.03, size = 32, normalized size = 2.91
method | result | size |
default | \(-\frac {\cos ^{4}\left (x \right )}{3 \sin \left (x \right )^{3}}+\frac {\cos ^{4}\left (x \right )}{3 \sin \left (x \right )}+\frac {\left (2+\cos ^{2}\left (x \right )\right ) \sin \left (x \right )}{3}\) | \(32\) |
norman | \(\frac {-\frac {1}{24}+\frac {3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {3 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{8}-\frac {\left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{24}}{\tan \left (\frac {x}{2}\right )^{3}}\) | \(34\) |
risch | \(\frac {2 i \left (3 \,{\mathrm e}^{5 i x}-2 \,{\mathrm e}^{3 i x}+3 \,{\mathrm e}^{i x}\right )}{3 \left ({\mathrm e}^{2 i x}-1\right )^{3}}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.75, size = 14, normalized size = 1.27 \begin {gather*} \frac {3 \, \sin \left (x\right )^{2} - 1}{3 \, \sin \left (x\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.55, size = 22, normalized size = 2.00 \begin {gather*} \frac {3 \, \cos \left (x\right )^{2} - 2}{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 = 0.02, size = 15, normalized size = 1.36 \begin {gather*} - \frac {1 - 3 \sin ^{2}{\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 = 0.68, size = 14, normalized size = 1.27 \begin {gather*} \frac {3 \, \sin \left (x\right )^{2} - 1}{3 \, \sin \left (x\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 11, normalized size = 1.00 \begin {gather*} \frac {{\sin \left (x\right )}^2-\frac {1}{3}}{{\sin \left (x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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