Integrand size = 21, antiderivative size = 37 \[ \int \frac {\sin ^3(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\cos ^3(c+d x)}{3 a d}+\frac {\sin ^2(c+d x)}{2 a d} \]
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Time = 0.16 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3957, 2914, 2644, 30, 2645} \[ \int \frac {\sin ^3(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\sin ^2(c+d x)}{2 a d}+\frac {\cos ^3(c+d x)}{3 a d} \]
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Rule 30
Rule 2644
Rule 2645
Rule 2914
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos (c+d x) \sin ^3(c+d x)}{-a-a \cos (c+d x)} \, dx \\ & = \frac {\int \cos (c+d x) \sin (c+d x) \, dx}{a}-\frac {\int \cos ^2(c+d x) \sin (c+d x) \, dx}{a} \\ & = \frac {\text {Subst}(\int x \, dx,x,\sin (c+d x))}{a d}+\frac {\text {Subst}\left (\int x^2 \, dx,x,\cos (c+d x)\right )}{a d} \\ & = \frac {\cos ^3(c+d x)}{3 a d}+\frac {\sin ^2(c+d x)}{2 a d} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.86 \[ \int \frac {\sin ^3(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {2 (1+2 \cos (c+d x)) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{3 a d} \]
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Time = 0.51 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\frac {\cos \left (d x +c \right )^{3}}{3}-\frac {\cos \left (d x +c \right )^{2}}{2}}{d a}\) | \(29\) |
default | \(\frac {\frac {\cos \left (d x +c \right )^{3}}{3}-\frac {\cos \left (d x +c \right )^{2}}{2}}{d a}\) | \(29\) |
parallelrisch | \(\frac {7+3 \cos \left (d x +c \right )+\cos \left (3 d x +3 c \right )-3 \cos \left (2 d x +2 c \right )}{12 d a}\) | \(39\) |
risch | \(\frac {\cos \left (d x +c \right )}{4 a d}+\frac {\cos \left (3 d x +3 c \right )}{12 a d}-\frac {\cos \left (2 d x +2 c \right )}{4 a d}\) | \(50\) |
norman | \(\frac {\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d a}+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d a}+\frac {2}{3 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}\) | \(64\) |
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Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int \frac {\sin ^3(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {2 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2}}{6 \, a d} \]
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Timed out. \[ \int \frac {\sin ^3(c+d x)}{a+a \sec (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int \frac {\sin ^3(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {2 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2}}{6 \, a d} \]
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Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.86 \[ \int \frac {\sin ^3(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\frac {2 \, \cos \left (d x + c\right )^{3}}{d} - \frac {3 \, \cos \left (d x + c\right )^{2}}{d}}{6 \, a} \]
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Time = 13.50 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.70 \[ \int \frac {\sin ^3(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {{\cos \left (c+d\,x\right )}^2\,\left (2\,\cos \left (c+d\,x\right )-3\right )}{6\,a\,d} \]
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