Optimal. Leaf size=15 \[ -\frac {\cot ^2(x)}{2}+i \cot (x) \]
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Rubi [A] time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3487} \[ -\frac {\cot ^2(x)}{2}+i \cot (x) \]
Antiderivative was successfully verified.
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Rule 3487
Rubi steps
\begin {align*} \int \frac {\csc ^4(x)}{i+\cot (x)} \, dx &=\operatorname {Subst}(\int (i-x) \, dx,x,\cot (x))\\ &=i \cot (x)-\frac {\cot ^2(x)}{2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 15, normalized size = 1.00 \[ -\frac {\csc ^2(x)}{2}+i \cot (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 16, normalized size = 1.07 \[ \frac {2}{e^{\left (4 i \, x\right )} - 2 \, e^{\left (2 i \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.97, size = 12, normalized size = 0.80 \[ -\frac {-2 i \, \tan \relax (x) + 1}{2 \, \tan \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 15, normalized size = 1.00 \[ -\frac {1}{2 \tan \relax (x )^{2}}+\frac {i}{\tan \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 12, normalized size = 0.80 \[ \frac {2 i \, \tan \relax (x) - 1}{2 \, \tan \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 9, normalized size = 0.60 \[ -\frac {\mathrm {cot}\relax (x)\,\left (\mathrm {cot}\relax (x)-2{}\mathrm {i}\right )}{2} \]
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 \[ \int \frac {\csc ^{4}{\relax (x )}}{\cot {\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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