Optimal. Leaf size=33 \[ i \cot (x)-\frac {\cot ^2(x)}{2}+\frac {1}{3} i \cot ^3(x)-\frac {\cot ^4(x)}{4} \]
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Rubi [A]
time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3568, 45}
\begin {gather*} -\frac {1}{4} \cot ^4(x)+\frac {1}{3} i \cot ^3(x)-\frac {\cot ^2(x)}{2}+i \cot (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3568
Rubi steps
\begin {align*} \int \frac {\csc ^6(x)}{i+\cot (x)} \, dx &=-\text {Subst}\left (\int (i-x)^2 (i+x) \, dx,x,\cot (x)\right )\\ &=-\text {Subst}\left (\int \left (-i+x-i x^2+x^3\right ) \, dx,x,\cot (x)\right )\\ &=i \cot (x)-\frac {\cot ^2(x)}{2}+\frac {1}{3} i \cot ^3(x)-\frac {\cot ^4(x)}{4}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 23, normalized size = 0.70 \begin {gather*} -\frac {1}{4} \csc ^4(x)+\frac {1}{3} i \cot (x) \left (2+\csc ^2(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 28, normalized size = 0.85
method | result | size |
risch | \(-\frac {4 \left (4 \,{\mathrm e}^{2 i x}-1\right )}{3 \left ({\mathrm e}^{2 i x}-1\right )^{4}}\) | \(21\) |
default | \(-\frac {1}{4 \tan \left (x \right )^{4}}-\frac {1}{2 \tan \left (x \right )^{2}}+\frac {i}{\tan \left (x \right )}+\frac {i}{3 \tan \left (x \right )^{3}}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 24, normalized size = 0.73 \begin {gather*} \frac {12 i \, \tan \left (x\right )^{3} - 6 \, \tan \left (x\right )^{2} + 4 i \, \tan \left (x\right ) - 3}{12 \, \tan \left (x\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.43, size = 36, normalized size = 1.09 \begin {gather*} -\frac {4 \, {\left (4 \, e^{\left (2 i \, x\right )} - 1\right )}}{3 \, {\left (e^{\left (8 i \, x\right )} - 4 \, e^{\left (6 i \, x\right )} + 6 \, e^{\left (4 i \, x\right )} - 4 \, e^{\left (2 i \, x\right )} + 1\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 {\csc ^{6}{\left (x \right )}}{\cot {\left (x \right )} + i}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 24, normalized size = 0.73 \begin {gather*} -\frac {-12 i \, \tan \left (x\right )^{3} + 6 \, \tan \left (x\right )^{2} - 4 i \, \tan \left (x\right ) + 3}{12 \, \tan \left (x\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 25, normalized size = 0.76 \begin {gather*} -\frac {{\mathrm {cot}\left (x\right )}^4}{4}+\frac {{\mathrm {cot}\left (x\right )}^3\,1{}\mathrm {i}}{3}-\frac {{\mathrm {cot}\left (x\right )}^2}{2}+\mathrm {cot}\left (x\right )\,1{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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