Optimal. Leaf size=19 \[ \frac{\tan ^2(x)}{2}-\frac{1}{3} i \tan ^3(x) \]
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Rubi [A] time = 0.0447802, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3516, 848, 43} \[ \frac{\tan ^2(x)}{2}-\frac{1}{3} i \tan ^3(x) \]
Antiderivative was successfully verified.
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Rule 3516
Rule 848
Rule 43
Rubi steps
\begin{align*} \int \frac{\sec ^4(x)}{i+\cot (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1+x^2}{x^4 (i+x)} \, dx,x,\cot (x)\right )\\ &=-\operatorname{Subst}\left (\int \frac{-i+x}{x^4} \, dx,x,\cot (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{i}{x^4}+\frac{1}{x^3}\right ) \, dx,x,\cot (x)\right )\\ &=\frac{\tan ^2(x)}{2}-\frac{1}{3} i \tan ^3(x)\\ \end{align*}
Mathematica [A] time = 0.0542781, size = 24, normalized size = 1.26 \[ \frac{1}{6} \left (2 i \tan (x)+(3-2 i \tan (x)) \sec ^2(x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 15, normalized size = 0.8 \begin{align*}{\frac{ \left ( \tan \left ( x \right ) \right ) ^{2}}{2}}-{\frac{i}{3}} \left ( \tan \left ( x \right ) \right ) ^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20208, size = 18, normalized size = 0.95 \begin{align*} -\frac{1}{3} i \, \tan \left (x\right )^{3} + \frac{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 = 1.85279, size = 140, normalized size = 7.37 \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27745, size = 18, normalized size = 0.95 \begin{align*} -\frac{1}{3} i \, \tan \left (x\right )^{3} + \frac{1}{2} \, \tan \left (x\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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