Integrand size = 21, antiderivative size = 30 \[ \int \cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {(A+C) \sin (c+d x)}{d}-\frac {A \sin ^3(c+d x)}{3 d} \]
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Time = 0.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4129, 3092} \[ \int \cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {(A+C) \sin (c+d x)}{d}-\frac {A \sin ^3(c+d x)}{3 d} \]
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Rule 3092
Rule 4129
Rubi steps \begin{align*} \text {integral}& = \int \cos (c+d x) \left (C+A \cos ^2(c+d x)\right ) \, dx \\ & = -\frac {\text {Subst}\left (\int \left (A+C-A x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d} \\ & = \frac {(A+C) \sin (c+d x)}{d}-\frac {A \sin ^3(c+d x)}{3 d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.67 \[ \int \cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {C \cos (d x) \sin (c)}{d}+\frac {C \cos (c) \sin (d x)}{d}+\frac {A \sin (c+d x)}{d}-\frac {A \sin ^3(c+d x)}{3 d} \]
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Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03
method | result | size |
parallelrisch | \(\frac {A \sin \left (3 d x +3 c \right )+9 \left (A +\frac {4 C}{3}\right ) \sin \left (d x +c \right )}{12 d}\) | \(31\) |
derivativedivides | \(\frac {\frac {A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+C \sin \left (d x +c \right )}{d}\) | \(33\) |
default | \(\frac {\frac {A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+C \sin \left (d x +c \right )}{d}\) | \(33\) |
risch | \(\frac {3 A \sin \left (d x +c \right )}{4 d}+\frac {C \sin \left (d x +c \right )}{d}+\frac {A \sin \left (3 d x +3 c \right )}{12 d}\) | \(40\) |
norman | \(\frac {\frac {2 \left (A -3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}-\frac {2 \left (A -3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d}-\frac {2 \left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 \left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\) | \(111\) |
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Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (A \cos \left (d x + c\right )^{2} + 2 \, A + 3 \, C\right )} \sin \left (d x + c\right )}{3 \, d} \]
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\[ \int \cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{3}{\left (c + d x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {A \sin \left (d x + c\right )^{3} - 3 \, {\left (A + C\right )} \sin \left (d x + c\right )}{3 \, d} \]
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Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {A \sin \left (d x + c\right )^{3} - 3 \, A \sin \left (d x + c\right ) - 3 \, C \sin \left (d x + c\right )}{3 \, d} \]
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Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {\frac {A\,{\sin \left (c+d\,x\right )}^3}{3}-\sin \left (c+d\,x\right )\,\left (A+C\right )}{d} \]
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