Optimal. Leaf size=20 \[ i x-i \tan (x)-\log (\sin (x))+\log (\tan (x)) \]
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Rubi [A] time = 0.04, antiderivative size = 20, 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 = {3516, 44} \[ i x-i \tan (x)-\log (\sin (x))+\log (\tan (x)) \]
Antiderivative was successfully verified.
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Rule 44
Rule 3516
Rubi steps
\begin {align*} \int \frac {\sec ^2(x)}{i+\cot (x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{x^2 (i+x)} \, dx,x,\cot (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {1}{-i-x}-\frac {i}{x^2}+\frac {1}{x}\right ) \, dx,x,\cot (x)\right )\\ &=i x-\log (\sin (x))+\log (\tan (x))-i \tan (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 17, normalized size = 0.85 \[ i (x-\tan (x)+i \log (\cos (x))) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 36, normalized size = 1.80 \[ \frac {2 i \, x e^{\left (2 i \, x\right )} - {\left (e^{\left (2 i \, x\right )} + 1\right )} \log \left (e^{\left (2 i \, x\right )} + 1\right ) + 2 i \, x + 2}{e^{\left (2 i \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.42, size = 53, normalized size = 2.65 \[ \frac {\tan \left (\frac {1}{2} \, x\right )^{2} + 2 i \, \tan \left (\frac {1}{2} \, x\right ) - 1}{\tan \left (\frac {1}{2} \, x\right )^{2} - 1} - \log \left (\tan \left (\frac {1}{2} \, x\right ) + 1\right ) + 2 \, \log \left (\tan \left (\frac {1}{2} \, x\right ) - i\right ) - \log \left (\tan \left (\frac {1}{2} \, x\right ) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 13, normalized size = 0.65 \[ -i \tan \relax (x )+\ln \left (\tan \relax (x )-i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 12, normalized size = 0.60 \[ \log \left (i \, \tan \relax (x) + 1\right ) - i \, \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 12, normalized size = 0.60 \[ \ln \left (\mathrm {tan}\relax (x)-\mathrm {i}\right )-\mathrm {tan}\relax (x)\,1{}\mathrm {i} \]
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 \[ \int \frac {\sec ^{2}{\relax (x )}}{\cot {\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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