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 = {3567, 3852, 8}
\begin {gather*} \frac {a \tan (c+d x)}{d}+\frac {i a \sec ^2(c+d x)}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3567
Rule 3852
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 \text {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 = 0.02, size = 30, normalized size = 1.11 \begin {gather*} \frac {i a \sec ^2(c+d x)}{2 d}+\frac {a \tan (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 26, normalized size = 0.96
method | result | size |
derivativedivides | \(\frac {\frac {i a}{2 \cos \left (d x +c \right )^{2}}+a \tan \left (d x +c \right )}{d}\) | \(26\) |
default | \(\frac {\frac {i a}{2 \cos \left (d x +c \right )^{2}}+a \tan \left (d x +c \right )}{d}\) | \(26\) |
risch | \(\frac {2 i a \left (2 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 21, normalized size = 0.78 \begin {gather*} -\frac {i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}{2 \, a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 45 vs. \(2 (21) = 42\).
time = 0.36, size = 45, normalized size = 1.67 \begin {gather*} -\frac {2 \, {\left (-2 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a\right )}}{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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.28, size = 37, normalized size = 1.37 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 26, normalized size = 0.96 \begin {gather*} -\frac {-i \, a \tan \left (d x + c\right )^{2} - 2 \, a \tan \left (d x + c\right )}{2 \, d} \end {gather*}
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 \begin {gather*} \frac {a\,\mathrm {tan}\left (c+d\,x\right )\,\left (2+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}{2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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