Optimal. Leaf size=27 \[ \frac {a \tanh ^{-1}(\sin (c+d x))}{d}+\frac {i a \sec (c+d x)}{d} \]
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Rubi [A]
time = 0.01, antiderivative size = 27, 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 = {3567, 3855}
\begin {gather*} \frac {a \tanh ^{-1}(\sin (c+d x))}{d}+\frac {i a \sec (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3567
Rule 3855
Rubi steps
\begin {align*} \int \sec (c+d x) (a+i a \tan (c+d x)) \, dx &=\frac {i a \sec (c+d x)}{d}+a \int \sec (c+d x) \, dx\\ &=\frac {a \tanh ^{-1}(\sin (c+d x))}{d}+\frac {i a \sec (c+d x)}{d}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 27, normalized size = 1.00 \begin {gather*} \frac {a \tanh ^{-1}(\sin (c+d x))}{d}+\frac {i a \sec (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 34, normalized size = 1.26
method | result | size |
derivativedivides | \(\frac {\frac {i a}{\cos \left (d x +c \right )}+a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(34\) |
default | \(\frac {\frac {i a}{\cos \left (d x +c \right )}+a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(34\) |
risch | \(\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 32, normalized size = 1.19 \begin {gather*} \frac {a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + \frac {i \, a}{\cos \left (d x + c\right )}}{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. 82 vs. \(2 (25) = 50\).
time = 0.38, size = 82, normalized size = 3.04 \begin {gather*} \frac {2 i \, a e^{\left (i \, d x + i \, c\right )} + {\left (a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - {\left (a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{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 = 2.44, size = 41, normalized size = 1.52 \begin {gather*} \begin {cases} \frac {a \log {\left (\tan {\left (c + d x \right )} + \sec {\left (c + d x \right )} \right )} + i a \sec {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (i a \tan {\left (c \right )} + a\right ) \sec {\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 52 vs. \(2 (25) = 50\).
time = 0.49, size = 52, normalized size = 1.93 \begin {gather*} \frac {a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) - a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right ) - \frac {2 i \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.35, size = 39, normalized size = 1.44 \begin {gather*} \frac {2\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {a\,2{}\mathrm {i}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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