Optimal. Leaf size=19 \[ \frac {\tan ^2(x)}{2}-\frac {1}{3} i \tan ^3(x) \]
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Rubi [A] time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 43
Rule 848
Rule 3516
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*}
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Mathematica [A] time = 0.06, 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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fricas [B] time = 0.95, size = 30, normalized size = 1.58 \[ \frac {2 \, {\left (3 \, e^{\left (2 i \, x\right )} - 1\right )}}{3 \, {\left (e^{\left (6 i \, x\right )} + 3 \, e^{\left (4 i \, x\right )} + 3 \, e^{\left (2 i \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 13, normalized size = 0.68 \[ -\frac {1}{3} i \, \tan \relax (x)^{3} + \frac {1}{2} \, \tan \relax (x)^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 15, normalized size = 0.79 \[ \frac {\left (\tan ^{2}\relax (x )\right )}{2}-\frac {i \left (\tan ^{3}\relax (x )\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 13, normalized size = 0.68 \[ -\frac {1}{3} i \, \tan \relax (x)^{3} + \frac {1}{2} \, \tan \relax (x)^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 13, normalized size = 0.68 \[ -\frac {{\mathrm {tan}\relax (x)}^2\,\left (-3+\mathrm {tan}\relax (x)\,2{}\mathrm {i}\right )}{6} \]
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 \[ \int \frac {\sec ^{4}{\relax (x )}}{\cot {\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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