Integrand size = 21, antiderivative size = 31 \[ \int \cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{2} (A+2 C) x+\frac {A \cos (c+d x) \sin (c+d x)}{2 d} \]
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Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4130, 8} \[ \int \cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {A \sin (c+d x) \cos (c+d x)}{2 d}+\frac {1}{2} x (A+2 C) \]
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Rule 8
Rule 4130
Rubi steps \begin{align*} \text {integral}& = \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 {A \cos (c+d x) \sin (c+d x)}{2 d} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=C x+\frac {A (c+d x)}{2 d}+\frac {A \sin (2 (c+d x))}{4 d} \]
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Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77
method | result | size |
risch | \(\frac {A x}{2}+C x +\frac {A \sin \left (2 d x +2 c \right )}{4 d}\) | \(24\) |
parallelrisch | \(\frac {A \sin \left (2 d x +2 c \right )+2 \left (A +2 C \right ) x d}{4 d}\) | \(27\) |
derivativedivides | \(\frac {A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \left (d x +c \right )}{d}\) | \(37\) |
default | \(\frac {A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \left (d x +c \right )}{d}\) | \(37\) |
norman | \(\frac {\left (-\frac {A}{2}-C \right ) x +\left (-\frac {A}{2}-C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {A}{2}+C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {A}{2}+C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\frac {A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}-\frac {A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\) | \(147\) |
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Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (A + 2 \, C\right )} d x + A \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, d} \]
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Time = 2.52 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=A \left (\begin {cases} \frac {x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {\sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases}\right ) + C x \]
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Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (d x + c\right )} {\left (A + 2 \, C\right )} + \frac {A \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \]
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (d x + c\right )} {\left (A + 2 \, C\right )} + \frac {A \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \]
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Time = 15.91 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\frac {A\,\sin \left (2\,c+2\,d\,x\right )}{4}+d\,x\,\left (\frac {A}{2}+C\right )}{d} \]
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