Integrand size = 29, antiderivative size = 42 \[ \int \cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{2} (A+2 C) x+\frac {B \sin (c+d x)}{d}+\frac {A \cos (c+d x) \sin (c+d x)}{2 d} \]
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Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {4132, 2717, 4130, 8} \[ \int \cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {A \sin (c+d x) \cos (c+d x)}{2 d}+\frac {1}{2} x (A+2 C)+\frac {B \sin (c+d x)}{d} \]
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Rule 8
Rule 2717
Rule 4130
Rule 4132
Rubi steps \begin{align*} \text {integral}& = B \int \cos (c+d x) \, dx+\int \cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx \\ & = \frac {B \sin (c+d x)}{d}+\frac {A \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} (A+2 C) \int 1 \, dx \\ & = \frac {1}{2} (A+2 C) x+\frac {B \sin (c+d x)}{d}+\frac {A \cos (c+d x) \sin (c+d x)}{2 d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.31 \[ \int \cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=C x+\frac {A (c+d x)}{2 d}+\frac {B \cos (d x) \sin (c)}{d}+\frac {B \cos (c) \sin (d x)}{d}+\frac {A \sin (2 (c+d x))}{4 d} \]
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Time = 0.15 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.83
method | result | size |
risch | \(\frac {A x}{2}+C x +\frac {B \sin \left (d x +c \right )}{d}+\frac {A \sin \left (2 d x +2 c \right )}{4 d}\) | \(35\) |
parallelrisch | \(\frac {A \sin \left (2 d x +2 c \right )+4 B \sin \left (d x +c \right )+2 \left (A +2 C \right ) x d}{4 d}\) | \(36\) |
derivativedivides | \(\frac {A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \sin \left (d x +c \right )+C \left (d x +c \right )}{d}\) | \(45\) |
default | \(\frac {A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \sin \left (d x +c \right )+C \left (d x +c \right )}{d}\) | \(45\) |
norman | \(\frac {\left (-\frac {A}{2}-C \right ) x +\left (-\frac {A}{2}-C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {A}{2}+C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {A}{2}+C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {2 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}-\frac {\left (A -2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-\frac {\left (A +2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\) | \(155\) |
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.79 \[ \int \cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (A + 2 \, C\right )} d x + {\left (A \cos \left (d x + c\right ) + 2 \, B\right )} \sin \left (d x + c\right )}{2 \, d} \]
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\[ \int \cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A + 4 \, {\left (d x + c\right )} C + 4 \, B \sin \left (d x + c\right )}{4 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (38) = 76\).
Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.05 \[ \int \cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (d x + c\right )} {\left (A + 2 \, C\right )} - \frac {2 \, {\left (A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
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Time = 14.85 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.81 \[ \int \cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {A\,x}{2}+C\,x+\frac {A\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,\sin \left (c+d\,x\right )}{d} \]
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