Integrand size = 23, antiderivative size = 27 \[ \int \sec (c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=-a x+\frac {a \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d} \]
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Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2917, 2686, 8, 3554} \[ \int \sec (c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=\frac {a \tan (c+d x)}{d}+\frac {a \sec (c+d x)}{d}-a x \]
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Rule 8
Rule 2686
Rule 2917
Rule 3554
Rubi steps \begin{align*} \text {integral}& = a \int \sec (c+d x) \tan (c+d x) \, dx+a \int \tan ^2(c+d x) \, dx \\ & = \frac {a \tan (c+d x)}{d}-a \int 1 \, dx+\frac {a \text {Subst}(\int 1 \, dx,x,\sec (c+d x))}{d} \\ & = -a x+\frac {a \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \sec (c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=-\frac {a \arctan (\tan (c+d x))}{d}+\frac {a \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d} \]
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Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96
method | result | size |
risch | \(-a x +\frac {2 a}{d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}\) | \(26\) |
derivativedivides | \(\frac {a \left (\tan \left (d x +c \right )-d x -c \right )+\frac {a}{\cos \left (d x +c \right )}}{d}\) | \(32\) |
default | \(\frac {a \left (\tan \left (d x +c \right )-d x -c \right )+\frac {a}{\cos \left (d x +c \right )}}{d}\) | \(32\) |
parallelrisch | \(-\frac {a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) d x -d x +2\right )}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(38\) |
norman | \(\frac {a x -\frac {2 a}{d}-\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-a x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(106\) |
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \sec (c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=-\frac {a d x + {\left (a d x - a\right )} \cos \left (d x + c\right ) - {\left (a d x + a\right )} \sin \left (d x + c\right ) - a}{d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d} \]
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\[ \int \sec (c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=a \left (\int \sin {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
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none
Time = 0.32 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \sec (c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=-\frac {{\left (d x + c - \tan \left (d x + c\right )\right )} a - \frac {a}{\cos \left (d x + c\right )}}{d} \]
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none
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \sec (c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=-\frac {{\left (d x + c\right )} a + \frac {2 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}}{d} \]
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Time = 9.59 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \sec (c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx=-a\,x-\frac {2\,a}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )} \]
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