Optimal. Leaf size=11 \[ \csc (x)-\frac {\csc ^3(x)}{3} \]
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Rubi [A] time = 0.02, 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 = {2606} \[ \csc (x)-\frac {\csc ^3(x)}{3} \]
Antiderivative was successfully verified.
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Rule 2606
Rubi steps
\begin {align*} \int \cot ^3(x) \csc (x) \, dx &=-\operatorname {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 \[ \csc (x)-\frac {\csc ^3(x)}{3} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cot ^3(x) \csc (x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.84, size = 22, normalized size = 2.00 \[ \frac {3 \, \cos \relax (x)^{2} - 2}{3 \, {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 14, normalized size = 1.27 \[ \frac {3 \, \sin \relax (x)^{2} - 1}{3 \, \sin \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.31, size = 32, normalized size = 2.91
method | result | size |
default | \(-\frac {\cos ^{4}\relax (x )}{3 \sin \relax (x )^{3}}+\frac {\cos ^{4}\relax (x )}{3 \sin \relax (x )}+\frac {\left (2+\cos ^{2}\relax (x )\right ) \sin \relax (x )}{3}\) | \(32\) |
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 = 0.56, size = 14, normalized size = 1.27 \[ \frac {3 \, \sin \relax (x)^{2} - 1}{3 \, \sin \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 11, normalized size = 1.00 \[ \frac {{\sin \relax (x)}^2-\frac {1}{3}}{{\sin \relax (x)}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 15, normalized size = 1.36 \[ - \frac {1 - 3 \sin ^{2}{\relax (x )}}{3 \sin ^{3}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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