Optimal. Leaf size=27 \[ -\frac{i (a+i a \tan (c+d x))^2}{2 a d} \]
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Rubi [A] time = 0.0306043, antiderivative size = 30, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {3486, 3767, 8} \[ \frac{a \tan (c+d x)}{d}+\frac{i a \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3486
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+i a \tan (c+d x)) \, dx &=\frac{i a \sec ^2(c+d x)}{2 d}+a \int \sec ^2(c+d x) \, dx\\ &=\frac{i a \sec ^2(c+d x)}{2 d}-\frac{a \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac{i a \sec ^2(c+d x)}{2 d}+\frac{a \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0132175, size = 30, normalized size = 1.11 \[ \frac{a \tan (c+d x)}{d}+\frac{i a \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 26, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({\frac{{\frac{i}{2}}a}{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+a\tan \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1059, size = 28, normalized size = 1.04 \begin{align*} -\frac{i \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}{2 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.1028, size = 123, normalized size = 4.56 \begin{align*} \frac{4 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i \, a}{d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.56283, size = 37, normalized size = 1.37 \begin{align*} \begin{cases} \frac{\frac{i a \tan ^{2}{\left (c + d x \right )}}{2} + a \tan{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (i a \tan{\left (c \right )} + a\right ) \sec ^{2}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12703, size = 35, normalized size = 1.3 \begin{align*} -\frac{-i \, a \tan \left (d x + c\right )^{2} - 2 \, a \tan \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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