Integrand size = 24, antiderivative size = 233 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {9 d^4 \cos (2 a+2 b x)}{128 b^5}+\frac {9 d^2 (c+d x)^2 \cos (2 a+2 b x)}{64 b^3}-\frac {3 (c+d x)^4 \cos (2 a+2 b x)}{64 b}+\frac {d^4 \cos (6 a+6 b x)}{10368 b^5}-\frac {d^2 (c+d x)^2 \cos (6 a+6 b x)}{576 b^3}+\frac {(c+d x)^4 \cos (6 a+6 b x)}{192 b}-\frac {9 d^3 (c+d x) \sin (2 a+2 b x)}{64 b^4}+\frac {3 d (c+d x)^3 \sin (2 a+2 b x)}{32 b^2}+\frac {d^3 (c+d x) \sin (6 a+6 b x)}{1728 b^4}-\frac {d (c+d x)^3 \sin (6 a+6 b x)}{288 b^2} \]
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Time = 0.36 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4491, 3377, 2718} \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {9 d^4 \cos (2 a+2 b x)}{128 b^5}+\frac {d^4 \cos (6 a+6 b x)}{10368 b^5}-\frac {9 d^3 (c+d x) \sin (2 a+2 b x)}{64 b^4}+\frac {d^3 (c+d x) \sin (6 a+6 b x)}{1728 b^4}+\frac {9 d^2 (c+d x)^2 \cos (2 a+2 b x)}{64 b^3}-\frac {d^2 (c+d x)^2 \cos (6 a+6 b x)}{576 b^3}+\frac {3 d (c+d x)^3 \sin (2 a+2 b x)}{32 b^2}-\frac {d (c+d x)^3 \sin (6 a+6 b x)}{288 b^2}-\frac {3 (c+d x)^4 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^4 \cos (6 a+6 b x)}{192 b} \]
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Rule 2718
Rule 3377
Rule 4491
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{32} (c+d x)^4 \sin (2 a+2 b x)-\frac {1}{32} (c+d x)^4 \sin (6 a+6 b x)\right ) \, dx \\ & = -\left (\frac {1}{32} \int (c+d x)^4 \sin (6 a+6 b x) \, dx\right )+\frac {3}{32} \int (c+d x)^4 \sin (2 a+2 b x) \, dx \\ & = -\frac {3 (c+d x)^4 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^4 \cos (6 a+6 b x)}{192 b}-\frac {d \int (c+d x)^3 \cos (6 a+6 b x) \, dx}{48 b}+\frac {(3 d) \int (c+d x)^3 \cos (2 a+2 b x) \, dx}{16 b} \\ & = -\frac {3 (c+d x)^4 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^4 \cos (6 a+6 b x)}{192 b}+\frac {3 d (c+d x)^3 \sin (2 a+2 b x)}{32 b^2}-\frac {d (c+d x)^3 \sin (6 a+6 b x)}{288 b^2}+\frac {d^2 \int (c+d x)^2 \sin (6 a+6 b x) \, dx}{96 b^2}-\frac {\left (9 d^2\right ) \int (c+d x)^2 \sin (2 a+2 b x) \, dx}{32 b^2} \\ & = \frac {9 d^2 (c+d x)^2 \cos (2 a+2 b x)}{64 b^3}-\frac {3 (c+d x)^4 \cos (2 a+2 b x)}{64 b}-\frac {d^2 (c+d x)^2 \cos (6 a+6 b x)}{576 b^3}+\frac {(c+d x)^4 \cos (6 a+6 b x)}{192 b}+\frac {3 d (c+d x)^3 \sin (2 a+2 b x)}{32 b^2}-\frac {d (c+d x)^3 \sin (6 a+6 b x)}{288 b^2}+\frac {d^3 \int (c+d x) \cos (6 a+6 b x) \, dx}{288 b^3}-\frac {\left (9 d^3\right ) \int (c+d x) \cos (2 a+2 b x) \, dx}{32 b^3} \\ & = \frac {9 d^2 (c+d x)^2 \cos (2 a+2 b x)}{64 b^3}-\frac {3 (c+d x)^4 \cos (2 a+2 b x)}{64 b}-\frac {d^2 (c+d x)^2 \cos (6 a+6 b x)}{576 b^3}+\frac {(c+d x)^4 \cos (6 a+6 b x)}{192 b}-\frac {9 d^3 (c+d x) \sin (2 a+2 b x)}{64 b^4}+\frac {3 d (c+d x)^3 \sin (2 a+2 b x)}{32 b^2}+\frac {d^3 (c+d x) \sin (6 a+6 b x)}{1728 b^4}-\frac {d (c+d x)^3 \sin (6 a+6 b x)}{288 b^2}-\frac {d^4 \int \sin (6 a+6 b x) \, dx}{1728 b^4}+\frac {\left (9 d^4\right ) \int \sin (2 a+2 b x) \, dx}{64 b^4} \\ & = -\frac {9 d^4 \cos (2 a+2 b x)}{128 b^5}+\frac {9 d^2 (c+d x)^2 \cos (2 a+2 b x)}{64 b^3}-\frac {3 (c+d x)^4 \cos (2 a+2 b x)}{64 b}+\frac {d^4 \cos (6 a+6 b x)}{10368 b^5}-\frac {d^2 (c+d x)^2 \cos (6 a+6 b x)}{576 b^3}+\frac {(c+d x)^4 \cos (6 a+6 b x)}{192 b}-\frac {9 d^3 (c+d x) \sin (2 a+2 b x)}{64 b^4}+\frac {3 d (c+d x)^3 \sin (2 a+2 b x)}{32 b^2}+\frac {d^3 (c+d x) \sin (6 a+6 b x)}{1728 b^4}-\frac {d (c+d x)^3 \sin (6 a+6 b x)}{288 b^2} \\ \end{align*}
Time = 1.37 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.66 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {-243 \left (3 d^4-6 b^2 d^2 (c+d x)^2+2 b^4 (c+d x)^4\right ) \cos (2 (a+b x))+\left (d^4-18 b^2 d^2 (c+d x)^2+54 b^4 (c+d x)^4\right ) \cos (6 (a+b x))-12 b d (c+d x) \left (121 d^2-78 b^2 (c+d x)^2+\left (-d^2+6 b^2 (c+d x)^2\right ) \cos (4 (a+b x))\right ) \sin (2 (a+b x))}{10368 b^5} \]
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Time = 3.44 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.79
method | result | size |
parallelrisch | \(\frac {\left (-486 b^{4} \left (d x +c \right )^{4}+1458 d^{2} \left (d x +c \right )^{2} b^{2}-729 d^{4}\right ) \cos \left (2 x b +2 a \right )+\left (54 b^{4} \left (d x +c \right )^{4}-18 d^{2} \left (d x +c \right )^{2} b^{2}+d^{4}\right ) \cos \left (6 x b +6 a \right )+972 b \left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{2}\right ) d \left (d x +c \right ) \sin \left (2 x b +2 a \right )-36 \left (\left (d x +c \right )^{2} b^{2}-\frac {d^{2}}{6}\right ) b d \left (d x +c \right ) \sin \left (6 x b +6 a \right )+432 b^{4} c^{4}-1440 b^{2} c^{2} d^{2}+728 d^{4}}{10368 b^{5}}\) | \(185\) |
risch | \(\frac {\left (54 d^{4} x^{4} b^{4}+216 b^{4} c \,d^{3} x^{3}+324 b^{4} c^{2} d^{2} x^{2}+216 b^{4} c^{3} d x +54 b^{4} c^{4}-18 b^{2} d^{4} x^{2}-36 b^{2} c \,d^{3} x -18 b^{2} c^{2} d^{2}+d^{4}\right ) \cos \left (6 x b +6 a \right )}{10368 b^{5}}-\frac {d \left (6 b^{2} d^{3} x^{3}+18 b^{2} c \,d^{2} x^{2}+18 b^{2} c^{2} d x +6 b^{2} c^{3}-d^{3} x -c \,d^{2}\right ) \sin \left (6 x b +6 a \right )}{1728 b^{4}}-\frac {3 \left (2 d^{4} x^{4} b^{4}+8 b^{4} c \,d^{3} x^{3}+12 b^{4} c^{2} d^{2} x^{2}+8 b^{4} c^{3} d x +2 b^{4} c^{4}-6 b^{2} d^{4} x^{2}-12 b^{2} c \,d^{3} x -6 b^{2} c^{2} d^{2}+3 d^{4}\right ) \cos \left (2 x b +2 a \right )}{128 b^{5}}+\frac {3 d \left (2 b^{2} d^{3} x^{3}+6 b^{2} c \,d^{2} x^{2}+6 b^{2} c^{2} d x +2 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \sin \left (2 x b +2 a \right )}{64 b^{4}}\) | \(352\) |
derivativedivides | \(\text {Expression too large to display}\) | \(2135\) |
default | \(\text {Expression too large to display}\) | \(2135\) |
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Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (213) = 426\).
Time = 0.27 (sec) , antiderivative size = 546, normalized size of antiderivative = 2.34 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 2 \, {\left (54 \, b^{4} d^{4} x^{4} + 216 \, b^{4} c d^{3} x^{3} + 54 \, b^{4} c^{4} - 18 \, b^{2} c^{2} d^{2} + d^{4} + 18 \, {\left (18 \, b^{4} c^{2} d^{2} - b^{2} d^{4}\right )} x^{2} + 36 \, {\left (6 \, b^{4} c^{3} d - b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{6} - 3 \, {\left (54 \, b^{4} d^{4} x^{4} + 216 \, b^{4} c d^{3} x^{3} + 54 \, b^{4} c^{4} - 18 \, b^{2} c^{2} d^{2} + d^{4} + 18 \, {\left (18 \, b^{4} c^{2} d^{2} - b^{2} d^{4}\right )} x^{2} + 36 \, {\left (6 \, b^{4} c^{3} d - b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{4} + 18 \, {\left (9 \, b^{4} c^{2} d^{2} - 5 \, b^{2} d^{4}\right )} x^{2} + 18 \, {\left (9 \, b^{2} d^{4} x^{2} + 18 \, b^{2} c d^{3} x + 9 \, b^{2} c^{2} d^{2} - 5 \, d^{4}\right )} \cos \left (b x + a\right )^{2} + 36 \, {\left (3 \, b^{4} c^{3} d - 5 \, b^{2} c d^{3}\right )} x - 12 \, {\left ({\left (6 \, b^{3} d^{4} x^{3} + 18 \, b^{3} c d^{3} x^{2} + 6 \, b^{3} c^{3} d - b c d^{3} + {\left (18 \, b^{3} c^{2} d^{2} - b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{5} - {\left (6 \, b^{3} d^{4} x^{3} + 18 \, b^{3} c d^{3} x^{2} + 6 \, b^{3} c^{3} d - b c d^{3} + {\left (18 \, b^{3} c^{2} d^{2} - b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d - 5 \, b c d^{3} + {\left (9 \, b^{3} c^{2} d^{2} - 5 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{648 \, b^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1334 vs. \(2 (231) = 462\).
Time = 1.52 (sec) , antiderivative size = 1334, normalized size of antiderivative = 5.73 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1033 vs. \(2 (213) = 426\).
Time = 0.30 (sec) , antiderivative size = 1033, normalized size of antiderivative = 4.43 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]
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Time = 0.42 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.54 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {{\left (54 \, b^{4} d^{4} x^{4} + 216 \, b^{4} c d^{3} x^{3} + 324 \, b^{4} c^{2} d^{2} x^{2} + 216 \, b^{4} c^{3} d x + 54 \, b^{4} c^{4} - 18 \, b^{2} d^{4} x^{2} - 36 \, b^{2} c d^{3} x - 18 \, b^{2} c^{2} d^{2} + d^{4}\right )} \cos \left (6 \, b x + 6 \, a\right )}{10368 \, b^{5}} - \frac {3 \, {\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 12 \, b^{4} c^{2} d^{2} x^{2} + 8 \, b^{4} c^{3} d x + 2 \, b^{4} c^{4} - 6 \, b^{2} d^{4} x^{2} - 12 \, b^{2} c d^{3} x - 6 \, b^{2} c^{2} d^{2} + 3 \, d^{4}\right )} \cos \left (2 \, b x + 2 \, a\right )}{128 \, b^{5}} - \frac {{\left (6 \, b^{3} d^{4} x^{3} + 18 \, b^{3} c d^{3} x^{2} + 18 \, b^{3} c^{2} d^{2} x + 6 \, b^{3} c^{3} d - b d^{4} x - b c d^{3}\right )} \sin \left (6 \, b x + 6 \, a\right )}{1728 \, b^{5}} + \frac {3 \, {\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 6 \, b^{3} c^{2} d^{2} x + 2 \, b^{3} c^{3} d - 3 \, b d^{4} x - 3 \, b c d^{3}\right )} \sin \left (2 \, b x + 2 \, a\right )}{64 \, b^{5}} \]
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Time = 26.37 (sec) , antiderivative size = 576, normalized size of antiderivative = 2.47 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {729\,d^4\,\cos \left (2\,a+2\,b\,x\right )-d^4\,\cos \left (6\,a+6\,b\,x\right )+486\,b^4\,c^4\,\cos \left (2\,a+2\,b\,x\right )-54\,b^4\,c^4\,\cos \left (6\,a+6\,b\,x\right )-972\,b^3\,c^3\,d\,\sin \left (2\,a+2\,b\,x\right )+36\,b^3\,c^3\,d\,\sin \left (6\,a+6\,b\,x\right )-1458\,b^2\,c^2\,d^2\,\cos \left (2\,a+2\,b\,x\right )+18\,b^2\,c^2\,d^2\,\cos \left (6\,a+6\,b\,x\right )-1458\,b^2\,d^4\,x^2\,\cos \left (2\,a+2\,b\,x\right )+486\,b^4\,d^4\,x^4\,\cos \left (2\,a+2\,b\,x\right )+18\,b^2\,d^4\,x^2\,\cos \left (6\,a+6\,b\,x\right )-54\,b^4\,d^4\,x^4\,\cos \left (6\,a+6\,b\,x\right )-972\,b^3\,d^4\,x^3\,\sin \left (2\,a+2\,b\,x\right )+36\,b^3\,d^4\,x^3\,\sin \left (6\,a+6\,b\,x\right )+1458\,b\,c\,d^3\,\sin \left (2\,a+2\,b\,x\right )-6\,b\,c\,d^3\,\sin \left (6\,a+6\,b\,x\right )+1458\,b\,d^4\,x\,\sin \left (2\,a+2\,b\,x\right )-6\,b\,d^4\,x\,\sin \left (6\,a+6\,b\,x\right )+2916\,b^4\,c^2\,d^2\,x^2\,\cos \left (2\,a+2\,b\,x\right )-324\,b^4\,c^2\,d^2\,x^2\,\cos \left (6\,a+6\,b\,x\right )-2916\,b^2\,c\,d^3\,x\,\cos \left (2\,a+2\,b\,x\right )+1944\,b^4\,c^3\,d\,x\,\cos \left (2\,a+2\,b\,x\right )+36\,b^2\,c\,d^3\,x\,\cos \left (6\,a+6\,b\,x\right )-216\,b^4\,c^3\,d\,x\,\cos \left (6\,a+6\,b\,x\right )+1944\,b^4\,c\,d^3\,x^3\,\cos \left (2\,a+2\,b\,x\right )-216\,b^4\,c\,d^3\,x^3\,\cos \left (6\,a+6\,b\,x\right )-2916\,b^3\,c^2\,d^2\,x\,\sin \left (2\,a+2\,b\,x\right )-2916\,b^3\,c\,d^3\,x^2\,\sin \left (2\,a+2\,b\,x\right )+108\,b^3\,c^2\,d^2\,x\,\sin \left (6\,a+6\,b\,x\right )+108\,b^3\,c\,d^3\,x^2\,\sin \left (6\,a+6\,b\,x\right )}{10368\,b^5} \]
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