Optimal. Leaf size=34 \[ \frac {i a \tan (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d}-i a x \]
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Rubi [A] time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3525, 3475} \[ \frac {i a \tan (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d}-i a x \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3525
Rubi steps
\begin {align*} \int \tan (c+d x) (a+i a \tan (c+d x)) \, dx &=-i a x+\frac {i a \tan (c+d x)}{d}+a \int \tan (c+d x) \, dx\\ &=-i a x-\frac {a \log (\cos (c+d x))}{d}+\frac {i a \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 1.26 \[ -\frac {i a \tan ^{-1}(\tan (c+d x))}{d}+\frac {i a \tan (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 47, normalized size = 1.38 \[ -\frac {{\left (a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2 \, a}{d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 58, normalized size = 1.71 \[ -\frac {a e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + a \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2 \, a}{d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 46, normalized size = 1.35 \[ \frac {i a \tan \left (d x +c \right )}{d}+\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {i a \arctan \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.81, size = 37, normalized size = 1.09 \[ -\frac {2 i \, {\left (d x + c\right )} a - a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 i \, a \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.78, size = 25, normalized size = 0.74 \[ \frac {a\,\left (\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 44, normalized size = 1.29 \[ \frac {2 a}{- d e^{2 i c} e^{2 i d x} - d} - \frac {a \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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