Integrand size = 17, antiderivative size = 15 \[ \int \cos (c+d x) (a+a \sec (c+d x)) \, dx=a x+\frac {a \sin (c+d x)}{d} \]
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Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3872, 2717, 8} \[ \int \cos (c+d x) (a+a \sec (c+d x)) \, dx=\frac {a \sin (c+d x)}{d}+a x \]
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Rule 8
Rule 2717
Rule 3872
Rubi steps \begin{align*} \text {integral}& = a \int 1 \, dx+a \int \cos (c+d x) \, dx \\ & = a x+\frac {a \sin (c+d x)}{d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.73 \[ \int \cos (c+d x) (a+a \sec (c+d x)) \, dx=a x+\frac {a \cos (d x) \sin (c)}{d}+\frac {a \cos (c) \sin (d x)}{d} \]
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Time = 0.44 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07
method | result | size |
risch | \(a x +\frac {a \sin \left (d x +c \right )}{d}\) | \(16\) |
parallelrisch | \(\frac {a \left (d x +\sin \left (d x +c \right )\right )}{d}\) | \(16\) |
derivativedivides | \(\frac {a \sin \left (d x +c \right )+a \left (d x +c \right )}{d}\) | \(21\) |
default | \(\frac {a \sin \left (d x +c \right )+a \left (d x +c \right )}{d}\) | \(21\) |
norman | \(\frac {a x +a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\) | \(50\) |
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none
Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \cos (c+d x) (a+a \sec (c+d x)) \, dx=\frac {a d x + a \sin \left (d x + c\right )}{d} \]
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Time = 0.78 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \cos (c+d x) (a+a \sec (c+d x)) \, dx=a x + a \left (\begin {cases} x \cos {\left (c \right )} & \text {for}\: d = 0 \\\frac {\sin {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right ) \]
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none
Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33 \[ \int \cos (c+d x) (a+a \sec (c+d x)) \, dx=\frac {{\left (d x + c\right )} a + a \sin \left (d x + c\right )}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (15) = 30\).
Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.60 \[ \int \cos (c+d x) (a+a \sec (c+d x)) \, dx=\frac {{\left (d x + c\right )} a + \frac {2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1}}{d} \]
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Time = 13.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \cos (c+d x) (a+a \sec (c+d x)) \, dx=a\,x+\frac {a\,\sin \left (c+d\,x\right )}{d} \]
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