Optimal. Leaf size=23 \[ \frac {x}{a}+\frac {i \log (\cos (c+d x))}{a d} \]
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Rubi [A] time = 0.07, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3092, 3090, 3475} \[ \frac {x}{a}+\frac {i \log (\cos (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 3090
Rule 3092
Rule 3475
Rubi steps
\begin {align*} \int \frac {\sec (c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx &=-\frac {i \int \sec (c+d x) (i a \cos (c+d x)+a \sin (c+d x)) \, dx}{a^2}\\ &=-\frac {i \int (i a+a \tan (c+d x)) \, dx}{a^2}\\ &=\frac {x}{a}-\frac {i \int \tan (c+d x) \, dx}{a}\\ &=\frac {x}{a}+\frac {i \log (\cos (c+d x))}{a d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 23, normalized size = 1.00 \[ \frac {i \log (\cos (c+d x))+c+d x}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, 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 = 0.20, 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 = 0.18, size = 22, normalized size = 0.96 \[ -\frac {i \ln \left (i \tan \left (d x +c \right )+1\right )}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 101, normalized size = 4.39 \[ -\frac {-\frac {i \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} - \frac {i \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} + \frac {i \, \log \left (-\frac {2 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.74, size = 41, normalized size = 1.78 \[ -\frac {\left (2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\mathrm {i}\right )-\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )\right )\,1{}\mathrm {i}}{a\,d} \]
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 \[ \frac {\int \frac {\sec {\left (c + d x \right )}}{i \sin {\left (c + d x \right )} + \cos {\left (c + d x \right )}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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