Optimal. Leaf size=23 \[ \frac {x}{a}+\frac {i \log (\cos (c+d x))}{a d} \]
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Rubi [A] time = 0.04, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3487, 31} \[ \frac {x}{a}+\frac {i \log (\cos (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3487
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{a+i a \tan (c+d x)} \, dx &=-\frac {i \operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=\frac {x}{a}+\frac {i \log (\cos (c+d x))}{a d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 31, normalized size = 1.35 \[ \frac {2 \tan ^{-1}(\tan (d x))+i \log \left (\cos ^2(c+d x)\right )}{2 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 26, normalized size = 1.13 \[ \frac {2 \, d x + i \, \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.27, size = 57, normalized size = 2.48 \[ -\frac {-\frac {i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a} + \frac {2 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}{a} - \frac {i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 23, normalized size = 1.00 \[ -\frac {i \ln \left (a +i a \tan \left (d x +c \right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 20, normalized size = 0.87 \[ -\frac {i \, \log \left (i \, a \tan \left (d x + c\right ) + a\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.35, size = 19, normalized size = 0.83 \[ -\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{a\,d} \]
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 \[ - \frac {i \int \frac {\sec ^{2}{\left (c + d x \right )}}{\tan {\left (c + d x \right )} - i}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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