Optimal. Leaf size=33 \[ -\frac {1}{4} \cot ^4(x)+\frac {1}{3} i \cot ^3(x)-\frac {\cot ^2(x)}{2}+i \cot (x) \]
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Rubi [A] time = 0.04, 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 = {3487, 43} \[ -\frac {1}{4} \cot ^4(x)+\frac {1}{3} i \cot ^3(x)-\frac {\cot ^2(x)}{2}+i \cot (x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rubi steps
\begin {align*} \int \frac {\csc ^6(x)}{i+\cot (x)} \, dx &=-\operatorname {Subst}\left (\int (i-x)^2 (i+x) \, dx,x,\cot (x)\right )\\ &=-\operatorname {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 = 0.05, size = 23, normalized size = 0.70 \[ -\frac {\csc ^4(x)}{4}+\frac {1}{3} i \cot (x) \left (\csc ^2(x)+2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 36, normalized size = 1.09 \[ -\frac {4 \, {\left (4 \, e^{\left (2 i \, x\right )} - 1\right )}}{3 \, {\left (e^{\left (8 i \, x\right )} - 4 \, e^{\left (6 i \, x\right )} + 6 \, e^{\left (4 i \, x\right )} - 4 \, e^{\left (2 i \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.88, size = 24, normalized size = 0.73 \[ -\frac {-12 i \, \tan \relax (x)^{3} + 6 \, \tan \relax (x)^{2} - 4 i \, \tan \relax (x) + 3}{12 \, \tan \relax (x)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 28, normalized size = 0.85 \[ -\frac {1}{2 \tan \relax (x )^{2}}+\frac {i}{3 \tan \relax (x )^{3}}-\frac {1}{4 \tan \relax (x )^{4}}+\frac {i}{\tan \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 24, normalized size = 0.73 \[ \frac {12 i \, \tan \relax (x)^{3} - 6 \, \tan \relax (x)^{2} + 4 i \, \tan \relax (x) - 3}{12 \, \tan \relax (x)^{4}} \]
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 \[ -\frac {{\mathrm {cot}\relax (x)}^4}{4}+\frac {{\mathrm {cot}\relax (x)}^3\,1{}\mathrm {i}}{3}-\frac {{\mathrm {cot}\relax (x)}^2}{2}+\mathrm {cot}\relax (x)\,1{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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