Optimal. Leaf size=15 \[ -\frac{\cot ^2(x)}{2}+i \cot (x) \]
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Rubi [A] time = 0.0327547, antiderivative size = 15, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0264791, size = 15, normalized size = 1. \[ -\frac{\csc ^2(x)}{2}+i \cot (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 15, normalized size = 1. \begin{align*} -{\frac{1}{2\, \left ( \tan \left ( x \right ) \right ) ^{2}}}+{\frac{i}{\tan \left ( x \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.23016, size = 16, normalized size = 1.07 \begin{align*} \frac{2 i \, \tan \left (x\right ) - 1}{2 \, \tan \left (x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.57706, size = 140, normalized size = 9.33 \begin{align*} \frac{2 \,{\left ({\left (3 \, e^{\left (2 i \, x\right )} - 1\right )} e^{\left (2 i \, x\right )} - 2 \, e^{\left (2 i \, x\right )}\right )} e^{\left (-2 i \, x\right )}}{3 \,{\left (e^{\left (6 i \, x\right )} - 3 \, e^{\left (4 i \, x\right )} + 3 \, e^{\left (2 i \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25502, size = 16, normalized size = 1.07 \begin{align*} -\frac{-2 i \, \tan \left (x\right ) + 1}{2 \, \tan \left (x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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