Optimal. Leaf size=12 \[ -\frac {\log (\cos (c+d x))}{d} \]
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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3556}
\begin {gather*} -\frac {\log (\cos (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rubi steps
\begin {align*} \int \tan (c+d x) \, dx &=-\frac {\log (\cos (c+d x))}{d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 12, normalized size = 1.00 \begin {gather*} -\frac {\log (\cos (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 17, normalized size = 1.42
method | result | size |
derivativedivides | \(\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(17\) |
default | \(\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(17\) |
norman | \(\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(17\) |
risch | \(i x +\frac {2 i c}{d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 11, normalized size = 0.92 \begin {gather*} \frac {\log \left (\sec \left (d x + c\right )\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 18, normalized size = 1.50 \begin {gather*} -\frac {\log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 19, normalized size = 1.58 \begin {gather*} \begin {cases} \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text {for}\: d \neq 0 \\x \tan {\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 = 0.42, size = 13, normalized size = 1.08 \begin {gather*} -\frac {\log \left ({\left | \cos \left (d x + c\right ) \right |}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.57, size = 16, normalized size = 1.33 \begin {gather*} \frac {\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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