Integrand size = 19, antiderivative size = 17 \[ \int \frac {\tan (c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\log (1+\cos (c+d x))}{a d} \]
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Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3964, 31} \[ \int \frac {\tan (c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\log (\cos (c+d x)+1)}{a d} \]
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Rule 31
Rule 3964
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{a+a x} \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {\log (1+\cos (c+d x))}{a d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {\tan (c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{a d} \]
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Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.59
method | result | size |
derivativedivides | \(\frac {-\ln \left (1+\sec \left (d x +c \right )\right )+\ln \left (\sec \left (d x +c \right )\right )}{d a}\) | \(27\) |
default | \(\frac {-\ln \left (1+\sec \left (d x +c \right )\right )+\ln \left (\sec \left (d x +c \right )\right )}{d a}\) | \(27\) |
risch | \(\frac {i x}{a}+\frac {2 i c}{d a}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d a}\) | \(39\) |
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Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {\tan (c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{a d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (14) = 28\).
Time = 2.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.41 \[ \int \frac {\tan (c+d x)}{a+a \sec (c+d x)} \, dx=\begin {cases} \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a d} - \frac {\log {\left (\sec {\left (c + d x \right )} + 1 \right )}}{a d} & \text {for}\: d \neq 0 \\\frac {x \tan {\left (c \right )}}{a \sec {\left (c \right )} + a} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {\tan (c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\log \left (\cos \left (d x + c\right ) + 1\right )}{a d} \]
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Time = 0.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int \frac {\tan (c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a d} \]
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Time = 14.49 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int \frac {\tan (c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d} \]
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