Integrand size = 17, antiderivative size = 138 \[ \int \cos ^2(c+d x) \sin ^3(a+b x) \, dx=-\frac {3 \cos (a+b x)}{8 b}+\frac {\cos (3 a+3 b x)}{24 b}-\frac {3 \cos (a-2 c+(b-2 d) x)}{16 (b-2 d)}+\frac {\cos (3 a-2 c+(3 b-2 d) x)}{16 (3 b-2 d)}-\frac {3 \cos (a+2 c+(b+2 d) x)}{16 (b+2 d)}+\frac {\cos (3 a+2 c+(3 b+2 d) x)}{16 (3 b+2 d)} \]
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Time = 0.11 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4670, 2718} \[ \int \cos ^2(c+d x) \sin ^3(a+b x) \, dx=-\frac {3 \cos (a+x (b-2 d)-2 c)}{16 (b-2 d)}+\frac {\cos (3 a+x (3 b-2 d)-2 c)}{16 (3 b-2 d)}-\frac {3 \cos (a+x (b+2 d)+2 c)}{16 (b+2 d)}+\frac {\cos (3 a+x (3 b+2 d)+2 c)}{16 (3 b+2 d)}-\frac {3 \cos (a+b x)}{8 b}+\frac {\cos (3 a+3 b x)}{24 b} \]
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Rule 2718
Rule 4670
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{8} \sin (a+b x)-\frac {1}{8} \sin (3 a+3 b x)+\frac {3}{16} \sin (a-2 c+(b-2 d) x)-\frac {1}{16} \sin (3 a-2 c+(3 b-2 d) x)+\frac {3}{16} \sin (a+2 c+(b+2 d) x)-\frac {1}{16} \sin (3 a+2 c+(3 b+2 d) x)\right ) \, dx \\ & = -\left (\frac {1}{16} \int \sin (3 a-2 c+(3 b-2 d) x) \, dx\right )-\frac {1}{16} \int \sin (3 a+2 c+(3 b+2 d) x) \, dx-\frac {1}{8} \int \sin (3 a+3 b x) \, dx+\frac {3}{16} \int \sin (a-2 c+(b-2 d) x) \, dx+\frac {3}{16} \int \sin (a+2 c+(b+2 d) x) \, dx+\frac {3}{8} \int \sin (a+b x) \, dx \\ & = -\frac {3 \cos (a+b x)}{8 b}+\frac {\cos (3 a+3 b x)}{24 b}-\frac {3 \cos (a-2 c+(b-2 d) x)}{16 (b-2 d)}+\frac {\cos (3 a-2 c+(3 b-2 d) x)}{16 (3 b-2 d)}-\frac {3 \cos (a+2 c+(b+2 d) x)}{16 (b+2 d)}+\frac {\cos (3 a+2 c+(3 b+2 d) x)}{16 (3 b+2 d)} \\ \end{align*}
Time = 1.95 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.11 \[ \int \cos ^2(c+d x) \sin ^3(a+b x) \, dx=\frac {1}{48} \left (-\frac {18 \cos (a) \cos (b x)}{b}+\frac {2 \cos (3 a) \cos (3 b x)}{b}-\frac {9 \cos (a-2 c+b x-2 d x)}{b-2 d}+\frac {3 \cos (3 a-2 c+3 b x-2 d x)}{3 b-2 d}-\frac {9 \cos (a+2 c+b x+2 d x)}{b+2 d}+\frac {3 \cos (3 a+2 c+3 b x+2 d x)}{3 b+2 d}+\frac {18 \sin (a) \sin (b x)}{b}-\frac {2 \sin (3 a) \sin (3 b x)}{b}\right ) \]
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Time = 1.64 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {3 \cos \left (x b +a \right )}{8 b}+\frac {\cos \left (3 x b +3 a \right )}{24 b}-\frac {3 \cos \left (a -2 c +\left (b -2 d \right ) x \right )}{16 \left (b -2 d \right )}+\frac {\cos \left (3 a -2 c +\left (3 b -2 d \right ) x \right )}{48 b -32 d}-\frac {3 \cos \left (a +2 c +\left (b +2 d \right ) x \right )}{16 \left (b +2 d \right )}+\frac {\cos \left (3 a +2 c +\left (3 b +2 d \right ) x \right )}{48 b +32 d}\) | \(127\) |
parallelrisch | \(\frac {9 \left (b +\frac {2 d}{3}\right ) b \left (b -2 d \right ) \left (b +2 d \right ) \cos \left (3 a -2 c +\left (3 b -2 d \right ) x \right )+9 b \left (b -2 d \right ) \left (b +2 d \right ) \left (b -\frac {2 d}{3}\right ) \cos \left (3 a +2 c +\left (3 b +2 d \right ) x \right )-81 \left (b +\frac {2 d}{3}\right ) b \left (b +2 d \right ) \left (b -\frac {2 d}{3}\right ) \cos \left (a -2 c +\left (b -2 d \right ) x \right )-81 \left (b +\frac {2 d}{3}\right ) b \left (b -2 d \right ) \left (b -\frac {2 d}{3}\right ) \cos \left (a +2 c +\left (b +2 d \right ) x \right )+\left (18 b^{4}-80 b^{2} d^{2}+32 d^{4}\right ) \cos \left (3 x b +3 a \right )+\left (-162 b^{4}+720 b^{2} d^{2}-288 d^{4}\right ) \cos \left (x b +a \right )-288 b^{4}+640 b^{2} d^{2}-256 d^{4}}{432 b^{5}-1920 b^{3} d^{2}+768 b \,d^{4}}\) | \(230\) |
risch | \(-\frac {3 \cos \left (x b +a \right )}{8 b}-\frac {27 \cos \left (x b -2 d x +a -2 c \right ) b^{3}}{16 \left (b +2 d \right ) \left (3 b +2 d \right ) \left (3 b -2 d \right ) \left (b -2 d \right )}-\frac {27 \cos \left (x b -2 d x +a -2 c \right ) b^{2} d}{8 \left (b +2 d \right ) \left (3 b +2 d \right ) \left (3 b -2 d \right ) \left (b -2 d \right )}+\frac {3 \cos \left (x b -2 d x +a -2 c \right ) b \,d^{2}}{4 \left (b +2 d \right ) \left (3 b +2 d \right ) \left (3 b -2 d \right ) \left (b -2 d \right )}+\frac {3 \cos \left (x b -2 d x +a -2 c \right ) d^{3}}{2 \left (b +2 d \right ) \left (3 b +2 d \right ) \left (3 b -2 d \right ) \left (b -2 d \right )}-\frac {27 \cos \left (x b +2 d x +a +2 c \right ) b^{3}}{16 \left (b +2 d \right ) \left (3 b +2 d \right ) \left (3 b -2 d \right ) \left (b -2 d \right )}+\frac {27 \cos \left (x b +2 d x +a +2 c \right ) b^{2} d}{8 \left (b +2 d \right ) \left (3 b +2 d \right ) \left (3 b -2 d \right ) \left (b -2 d \right )}+\frac {3 \cos \left (x b +2 d x +a +2 c \right ) b \,d^{2}}{4 \left (b +2 d \right ) \left (3 b +2 d \right ) \left (3 b -2 d \right ) \left (b -2 d \right )}-\frac {3 \cos \left (x b +2 d x +a +2 c \right ) d^{3}}{2 \left (b +2 d \right ) \left (3 b +2 d \right ) \left (3 b -2 d \right ) \left (b -2 d \right )}+\frac {3 \cos \left (3 x b -2 d x +3 a -2 c \right ) b^{3}}{16 \left (b +2 d \right ) \left (3 b +2 d \right ) \left (3 b -2 d \right ) \left (b -2 d \right )}+\frac {\cos \left (3 x b -2 d x +3 a -2 c \right ) b^{2} d}{8 \left (b +2 d \right ) \left (3 b +2 d \right ) \left (3 b -2 d \right ) \left (b -2 d \right )}-\frac {3 \cos \left (3 x b -2 d x +3 a -2 c \right ) b \,d^{2}}{4 \left (b +2 d \right ) \left (3 b +2 d \right ) \left (3 b -2 d \right ) \left (b -2 d \right )}-\frac {\cos \left (3 x b -2 d x +3 a -2 c \right ) d^{3}}{2 \left (b +2 d \right ) \left (3 b +2 d \right ) \left (3 b -2 d \right ) \left (b -2 d \right )}+\frac {3 \cos \left (3 x b +2 d x +3 a +2 c \right ) b^{3}}{16 \left (b +2 d \right ) \left (3 b +2 d \right ) \left (3 b -2 d \right ) \left (b -2 d \right )}-\frac {\cos \left (3 x b +2 d x +3 a +2 c \right ) b^{2} d}{8 \left (b +2 d \right ) \left (3 b +2 d \right ) \left (3 b -2 d \right ) \left (b -2 d \right )}-\frac {3 \cos \left (3 x b +2 d x +3 a +2 c \right ) b \,d^{2}}{4 \left (b +2 d \right ) \left (3 b +2 d \right ) \left (3 b -2 d \right ) \left (b -2 d \right )}+\frac {\cos \left (3 x b +2 d x +3 a +2 c \right ) d^{3}}{2 \left (b +2 d \right ) \left (3 b +2 d \right ) \left (3 b -2 d \right ) \left (b -2 d \right )}+\frac {\cos \left (3 x b +3 a \right )}{24 b}\) | \(859\) |
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Time = 0.27 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.30 \[ \int \cos ^2(c+d x) \sin ^3(a+b x) \, dx=-\frac {2 \, {\left (b^{2} d^{2} - 4 \, d^{4}\right )} \cos \left (b x + a\right )^{3} + 6 \, {\left (7 \, b^{3} d - 4 \, b d^{3} - {\left (b^{3} d - 4 \, b d^{3}\right )} \cos \left (b x + a\right )^{2}\right )} \cos \left (d x + c\right ) \sin \left (b x + a\right ) \sin \left (d x + c\right ) - 9 \, {\left ({\left (b^{4} - 4 \, b^{2} d^{2}\right )} \cos \left (b x + a\right )^{3} - {\left (3 \, b^{4} - 4 \, b^{2} d^{2}\right )} \cos \left (b x + a\right )\right )} \cos \left (d x + c\right )^{2} - 6 \, {\left (7 \, b^{2} d^{2} - 4 \, d^{4}\right )} \cos \left (b x + a\right )}{3 \, {\left (9 \, b^{5} - 40 \, b^{3} d^{2} + 16 \, b d^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2020 vs. \(2 (116) = 232\).
Time = 5.64 (sec) , antiderivative size = 2020, normalized size of antiderivative = 14.64 \[ \int \cos ^2(c+d x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1360 vs. \(2 (126) = 252\).
Time = 0.31 (sec) , antiderivative size = 1360, normalized size of antiderivative = 9.86 \[ \int \cos ^2(c+d x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]
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Time = 0.29 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.90 \[ \int \cos ^2(c+d x) \sin ^3(a+b x) \, dx=\frac {\cos \left (3 \, b x + 2 \, d x + 3 \, a + 2 \, c\right )}{16 \, {\left (3 \, b + 2 \, d\right )}} + \frac {\cos \left (3 \, b x - 2 \, d x + 3 \, a - 2 \, c\right )}{16 \, {\left (3 \, b - 2 \, d\right )}} + \frac {\cos \left (3 \, b x + 3 \, a\right )}{24 \, b} - \frac {3 \, \cos \left (b x + 2 \, d x + a + 2 \, c\right )}{16 \, {\left (b + 2 \, d\right )}} - \frac {3 \, \cos \left (b x - 2 \, d x + a - 2 \, c\right )}{16 \, {\left (b - 2 \, d\right )}} - \frac {3 \, \cos \left (b x + a\right )}{8 \, b} \]
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Time = 22.24 (sec) , antiderivative size = 438, normalized size of antiderivative = 3.17 \[ \int \cos ^2(c+d x) \sin ^3(a+b x) \, dx=-\frac {81\,b^4\,\cos \left (a-2\,c+b\,x-2\,d\,x\right )+81\,b^4\,\cos \left (a+2\,c+b\,x+2\,d\,x\right )+162\,b^4\,\cos \left (a+b\,x\right )+288\,d^4\,\cos \left (a+b\,x\right )-9\,b^4\,\cos \left (3\,a-2\,c+3\,b\,x-2\,d\,x\right )-9\,b^4\,\cos \left (3\,a+2\,c+3\,b\,x+2\,d\,x\right )-18\,b^4\,\cos \left (3\,a+3\,b\,x\right )-32\,d^4\,\cos \left (3\,a+3\,b\,x\right )+24\,b\,d^3\,\cos \left (3\,a-2\,c+3\,b\,x-2\,d\,x\right )-24\,b\,d^3\,\cos \left (3\,a+2\,c+3\,b\,x+2\,d\,x\right )-6\,b^3\,d\,\cos \left (3\,a-2\,c+3\,b\,x-2\,d\,x\right )+6\,b^3\,d\,\cos \left (3\,a+2\,c+3\,b\,x+2\,d\,x\right )-36\,b^2\,d^2\,\cos \left (a-2\,c+b\,x-2\,d\,x\right )-36\,b^2\,d^2\,\cos \left (a+2\,c+b\,x+2\,d\,x\right )-720\,b^2\,d^2\,\cos \left (a+b\,x\right )+36\,b^2\,d^2\,\cos \left (3\,a-2\,c+3\,b\,x-2\,d\,x\right )+36\,b^2\,d^2\,\cos \left (3\,a+2\,c+3\,b\,x+2\,d\,x\right )+80\,b^2\,d^2\,\cos \left (3\,a+3\,b\,x\right )-72\,b\,d^3\,\cos \left (a-2\,c+b\,x-2\,d\,x\right )+72\,b\,d^3\,\cos \left (a+2\,c+b\,x+2\,d\,x\right )+162\,b^3\,d\,\cos \left (a-2\,c+b\,x-2\,d\,x\right )-162\,b^3\,d\,\cos \left (a+2\,c+b\,x+2\,d\,x\right )}{48\,\left (9\,b^5-40\,b^3\,d^2+16\,b\,d^4\right )} \]
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