Integrand size = 23, antiderivative size = 27 \[ \int \sec (c+d x) (a+b \sin (c+d x)) \tan (c+d x) \, dx=-b x+\frac {a \sec (c+d x)}{d}+\frac {b \tan (c+d x)}{d} \]
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Time = 0.03 (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} \[ \int \sec (c+d x) (a+b \sin (c+d x)) \tan (c+d x) \, dx=\frac {a \sec (c+d x)}{d}+\frac {b \tan (c+d x)}{d}-b x \]
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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+b \int \tan ^2(c+d x) \, dx \\ & = \frac {b \tan (c+d x)}{d}-b \int 1 \, dx+\frac {a \text {Subst}(\int 1 \, dx,x,\sec (c+d x))}{d} \\ & = -b x+\frac {a \sec (c+d x)}{d}+\frac {b \tan (c+d x)}{d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \sec (c+d x) (a+b \sin (c+d x)) \tan (c+d x) \, dx=-\frac {b \arctan (\tan (c+d x))}{d}+\frac {a \sec (c+d x)}{d}+\frac {b \tan (c+d x)}{d} \]
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Time = 0.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19
method | result | size |
derivativedivides | \(\frac {\frac {a}{\cos \left (d x +c \right )}+b \left (\tan \left (d x +c \right )-d x -c \right )}{d}\) | \(32\) |
default | \(\frac {\frac {a}{\cos \left (d x +c \right )}+b \left (\tan \left (d x +c \right )-d x -c \right )}{d}\) | \(32\) |
risch | \(-b x +\frac {2 i \left (-i a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\) | \(40\) |
parallelrisch | \(\frac {-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) x d +b x d -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 a}{d \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(56\) |
norman | \(\frac {b x -\frac {2 a}{d}-\frac {2 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-b x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(106\) |
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Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \sec (c+d x) (a+b \sin (c+d x)) \tan (c+d x) \, dx=-\frac {b d x \cos \left (d x + c\right ) - b \sin \left (d x + c\right ) - a}{d \cos \left (d x + c\right )} \]
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\[ \int \sec (c+d x) (a+b \sin (c+d x)) \tan (c+d x) \, dx=\int \left (a + b \sin {\left (c + d x \right )}\right ) \sin {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \sec (c+d x) (a+b \sin (c+d x)) \tan (c+d x) \, dx=-\frac {{\left (d x + c - \tan \left (d x + c\right )\right )} b - \frac {a}{\cos \left (d x + c\right )}}{d} \]
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Time = 0.39 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59 \[ \int \sec (c+d x) (a+b \sin (c+d x)) \tan (c+d x) \, dx=-\frac {{\left (d x + c\right )} b + \frac {2 \, {\left (b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}}{d} \]
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Time = 10.89 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \sec (c+d x) (a+b \sin (c+d x)) \tan (c+d x) \, dx=-b\,x-\frac {2\,a+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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