Optimal. Leaf size=27 \[ -\frac {i (a+i a \tan (c+d x))^2}{2 a d} \]
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Rubi [A] time = 0.03, 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 8
Rule 3486
Rule 3767
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*}
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Mathematica [A] time = 0.01, 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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fricas [B] time = 0.40, size = 44, normalized size = 1.63 \[ \frac {4 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i \, a}{d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.76, size = 26, normalized size = 0.96 \[ -\frac {-i \, a \tan \left (d x + c\right )^{2} - 2 \, a \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 26, normalized size = 0.96 \[ \frac {\frac {i a}{2 \cos \left (d x +c \right )^{2}}+a \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 21, normalized size = 0.78 \[ -\frac {i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}{2 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.21, size = 23, normalized size = 0.85 \[ \frac {a\,\mathrm {tan}\left (c+d\,x\right )\,\left (2+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.36, size = 37, normalized size = 1.37 \[ \begin {cases} \frac {\frac {i a \tan ^{2}{\left (c + d x \right )}}{2} + a \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (i a \tan {\relax (c )} + a\right ) \sec ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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