Integrand size = 24, antiderivative size = 181 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {9 d^2 (c+d x) \cos (2 a+2 b x)}{128 b^3}-\frac {3 (c+d x)^3 \cos (2 a+2 b x)}{64 b}-\frac {d^2 (c+d x) \cos (6 a+6 b x)}{1152 b^3}+\frac {(c+d x)^3 \cos (6 a+6 b x)}{192 b}-\frac {9 d^3 \sin (2 a+2 b x)}{256 b^4}+\frac {9 d (c+d x)^2 \sin (2 a+2 b x)}{128 b^2}+\frac {d^3 \sin (6 a+6 b x)}{6912 b^4}-\frac {d (c+d x)^2 \sin (6 a+6 b x)}{384 b^2} \]
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Time = 0.27 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4491, 3377, 2717} \[ \int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {9 d^3 \sin (2 a+2 b x)}{256 b^4}+\frac {d^3 \sin (6 a+6 b x)}{6912 b^4}+\frac {9 d^2 (c+d x) \cos (2 a+2 b x)}{128 b^3}-\frac {d^2 (c+d x) \cos (6 a+6 b x)}{1152 b^3}+\frac {9 d (c+d x)^2 \sin (2 a+2 b x)}{128 b^2}-\frac {d (c+d x)^2 \sin (6 a+6 b x)}{384 b^2}-\frac {3 (c+d x)^3 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^3 \cos (6 a+6 b x)}{192 b} \]
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Rule 2717
Rule 3377
Rule 4491
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{32} (c+d x)^3 \sin (2 a+2 b x)-\frac {1}{32} (c+d x)^3 \sin (6 a+6 b x)\right ) \, dx \\ & = -\left (\frac {1}{32} \int (c+d x)^3 \sin (6 a+6 b x) \, dx\right )+\frac {3}{32} \int (c+d x)^3 \sin (2 a+2 b x) \, dx \\ & = -\frac {3 (c+d x)^3 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^3 \cos (6 a+6 b x)}{192 b}-\frac {d \int (c+d x)^2 \cos (6 a+6 b x) \, dx}{64 b}+\frac {(9 d) \int (c+d x)^2 \cos (2 a+2 b x) \, dx}{64 b} \\ & = -\frac {3 (c+d x)^3 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^3 \cos (6 a+6 b x)}{192 b}+\frac {9 d (c+d x)^2 \sin (2 a+2 b x)}{128 b^2}-\frac {d (c+d x)^2 \sin (6 a+6 b x)}{384 b^2}+\frac {d^2 \int (c+d x) \sin (6 a+6 b x) \, dx}{192 b^2}-\frac {\left (9 d^2\right ) \int (c+d x) \sin (2 a+2 b x) \, dx}{64 b^2} \\ & = \frac {9 d^2 (c+d x) \cos (2 a+2 b x)}{128 b^3}-\frac {3 (c+d x)^3 \cos (2 a+2 b x)}{64 b}-\frac {d^2 (c+d x) \cos (6 a+6 b x)}{1152 b^3}+\frac {(c+d x)^3 \cos (6 a+6 b x)}{192 b}+\frac {9 d (c+d x)^2 \sin (2 a+2 b x)}{128 b^2}-\frac {d (c+d x)^2 \sin (6 a+6 b x)}{384 b^2}+\frac {d^3 \int \cos (6 a+6 b x) \, dx}{1152 b^3}-\frac {\left (9 d^3\right ) \int \cos (2 a+2 b x) \, dx}{128 b^3} \\ & = \frac {9 d^2 (c+d x) \cos (2 a+2 b x)}{128 b^3}-\frac {3 (c+d x)^3 \cos (2 a+2 b x)}{64 b}-\frac {d^2 (c+d x) \cos (6 a+6 b x)}{1152 b^3}+\frac {(c+d x)^3 \cos (6 a+6 b x)}{192 b}-\frac {9 d^3 \sin (2 a+2 b x)}{256 b^4}+\frac {9 d (c+d x)^2 \sin (2 a+2 b x)}{128 b^2}+\frac {d^3 \sin (6 a+6 b x)}{6912 b^4}-\frac {d (c+d x)^2 \sin (6 a+6 b x)}{384 b^2} \\ \end{align*}
Time = 2.45 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.73 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {-324 b (c+d x) \left (-3 d^2+2 b^2 (c+d x)^2\right ) \cos (2 (a+b x))+12 b (c+d x) \left (-d^2+6 b^2 (c+d x)^2\right ) \cos (6 (a+b x))-4 d \left (121 d^2-234 b^2 (c+d x)^2+\left (-d^2+18 b^2 (c+d x)^2\right ) \cos (4 (a+b x))\right ) \sin (2 (a+b x))}{13824 b^4} \]
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Time = 2.54 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.82
method | result | size |
parallelrisch | \(\frac {-54 b \left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{2}\right ) \left (d x +c \right ) \cos \left (2 x b +2 a \right )+6 b \left (\left (d x +c \right )^{2} b^{2}-\frac {d^{2}}{6}\right ) \left (d x +c \right ) \cos \left (6 x b +6 a \right )+81 d \left (\left (d x +c \right )^{2} b^{2}-\frac {d^{2}}{2}\right ) \sin \left (2 x b +2 a \right )-3 \left (\left (d x +c \right )^{2} b^{2}-\frac {d^{2}}{18}\right ) d \sin \left (6 x b +6 a \right )+48 b^{3} c^{3}-80 c \,d^{2} b}{1152 b^{4}}\) | \(148\) |
risch | \(\frac {\left (6 b^{2} d^{3} x^{3}+18 b^{2} c \,d^{2} x^{2}+18 b^{2} c^{2} d x +6 b^{2} c^{3}-d^{3} x -c \,d^{2}\right ) \cos \left (6 x b +6 a \right )}{1152 b^{3}}-\frac {d \left (18 x^{2} d^{2} b^{2}+36 b^{2} c d x +18 b^{2} c^{2}-d^{2}\right ) \sin \left (6 x b +6 a \right )}{6912 b^{4}}-\frac {3 \left (2 b^{2} d^{3} x^{3}+6 b^{2} c \,d^{2} x^{2}+6 b^{2} c^{2} d x +2 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \cos \left (2 x b +2 a \right )}{128 b^{3}}+\frac {9 d \left (2 x^{2} d^{2} b^{2}+4 b^{2} c d x +2 b^{2} c^{2}-d^{2}\right ) \sin \left (2 x b +2 a \right )}{256 b^{4}}\) | \(234\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1132\) |
default | \(\text {Expression too large to display}\) | \(1132\) |
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Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (165) = 330\).
Time = 0.25 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.93 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {9 \, b^{3} d^{3} x^{3} + 27 \, b^{3} c d^{2} x^{2} + 6 \, {\left (6 \, b^{3} d^{3} x^{3} + 18 \, b^{3} c d^{2} x^{2} + 6 \, b^{3} c^{3} - b c d^{2} + {\left (18 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{6} - 9 \, {\left (6 \, b^{3} d^{3} x^{3} + 18 \, b^{3} c d^{2} x^{2} + 6 \, b^{3} c^{3} - b c d^{2} + {\left (18 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{4} + 27 \, {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right )^{2} + 3 \, {\left (9 \, b^{3} c^{2} d - 5 \, b d^{3}\right )} x - {\left ({\left (18 \, b^{2} d^{3} x^{2} + 36 \, b^{2} c d^{2} x + 18 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{5} - {\left (18 \, b^{2} d^{3} x^{2} + 36 \, b^{2} c d^{2} x + 18 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d - 5 \, d^{3}\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{216 \, b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 857 vs. \(2 (178) = 356\).
Time = 1.13 (sec) , antiderivative size = 857, normalized size of antiderivative = 4.73 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\begin {cases} - \frac {c^{3} \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{4 b} - \frac {c^{3} \cos ^{6}{\left (a + b x \right )}}{12 b} + \frac {c^{2} d x \sin ^{6}{\left (a + b x \right )}}{8 b} + \frac {3 c^{2} d x \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{8 b} - \frac {3 c^{2} d x \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{8 b} - \frac {c^{2} d x \cos ^{6}{\left (a + b x \right )}}{8 b} + \frac {c d^{2} x^{2} \sin ^{6}{\left (a + b x \right )}}{8 b} + \frac {3 c d^{2} x^{2} \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{8 b} - \frac {3 c d^{2} x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{8 b} - \frac {c d^{2} x^{2} \cos ^{6}{\left (a + b x \right )}}{8 b} + \frac {d^{3} x^{3} \sin ^{6}{\left (a + b x \right )}}{24 b} + \frac {d^{3} x^{3} \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{8 b} - \frac {d^{3} x^{3} \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{8 b} - \frac {d^{3} x^{3} \cos ^{6}{\left (a + b x \right )}}{24 b} + \frac {c^{2} d \sin ^{5}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b^{2}} + \frac {c^{2} d \sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {c^{2} d \sin {\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{8 b^{2}} + \frac {c d^{2} x \sin ^{5}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{4 b^{2}} + \frac {2 c d^{2} x \sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {c d^{2} x \sin {\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{4 b^{2}} + \frac {d^{3} x^{2} \sin ^{5}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b^{2}} + \frac {d^{3} x^{2} \sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {d^{3} x^{2} \sin {\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{8 b^{2}} - \frac {c d^{2} \sin ^{6}{\left (a + b x \right )}}{24 b^{3}} + \frac {c d^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{6 b^{3}} + \frac {7 c d^{2} \cos ^{6}{\left (a + b x \right )}}{72 b^{3}} - \frac {5 d^{3} x \sin ^{6}{\left (a + b x \right )}}{72 b^{3}} - \frac {d^{3} x \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{12 b^{3}} + \frac {d^{3} x \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{12 b^{3}} + \frac {5 d^{3} x \cos ^{6}{\left (a + b x \right )}}{72 b^{3}} - \frac {5 d^{3} \sin ^{5}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{72 b^{4}} - \frac {31 d^{3} \sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{216 b^{4}} - \frac {5 d^{3} \sin {\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{72 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \sin ^{3}{\left (a \right )} \cos ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (165) = 330\).
Time = 0.26 (sec) , antiderivative size = 602, normalized size of antiderivative = 3.33 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {576 \, {\left (2 \, \sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4}\right )} c^{3} - \frac {1728 \, {\left (2 \, \sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4}\right )} a c^{2} d}{b} + \frac {1728 \, {\left (2 \, \sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4}\right )} a^{2} c d^{2}}{b^{2}} - \frac {576 \, {\left (2 \, \sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4}\right )} a^{3} d^{3}}{b^{3}} - \frac {18 \, {\left (6 \, {\left (b x + a\right )} \cos \left (6 \, b x + 6 \, a\right ) - 54 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (6 \, b x + 6 \, a\right ) + 27 \, \sin \left (2 \, b x + 2 \, a\right )\right )} c^{2} d}{b} + \frac {36 \, {\left (6 \, {\left (b x + a\right )} \cos \left (6 \, b x + 6 \, a\right ) - 54 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (6 \, b x + 6 \, a\right ) + 27 \, \sin \left (2 \, b x + 2 \, a\right )\right )} a c d^{2}}{b^{2}} - \frac {18 \, {\left (6 \, {\left (b x + a\right )} \cos \left (6 \, b x + 6 \, a\right ) - 54 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (6 \, b x + 6 \, a\right ) + 27 \, \sin \left (2 \, b x + 2 \, a\right )\right )} a^{2} d^{3}}{b^{3}} - \frac {6 \, {\left ({\left (18 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (6 \, b x + 6 \, a\right ) - 81 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 6 \, {\left (b x + a\right )} \sin \left (6 \, b x + 6 \, a\right ) + 162 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} c d^{2}}{b^{2}} + \frac {6 \, {\left ({\left (18 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (6 \, b x + 6 \, a\right ) - 81 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 6 \, {\left (b x + a\right )} \sin \left (6 \, b x + 6 \, a\right ) + 162 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} a d^{3}}{b^{3}} - \frac {{\left (6 \, {\left (6 \, {\left (b x + a\right )}^{3} - b x - a\right )} \cos \left (6 \, b x + 6 \, a\right ) - 162 \, {\left (2 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \cos \left (2 \, b x + 2 \, a\right ) - {\left (18 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (6 \, b x + 6 \, a\right ) + 243 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} d^{3}}{b^{3}}}{6912 \, b} \]
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Time = 0.38 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.33 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {{\left (6 \, b^{3} d^{3} x^{3} + 18 \, b^{3} c d^{2} x^{2} + 18 \, b^{3} c^{2} d x + 6 \, b^{3} c^{3} - b d^{3} x - b c d^{2}\right )} \cos \left (6 \, b x + 6 \, a\right )}{1152 \, b^{4}} - \frac {3 \, {\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 6 \, b^{3} c^{2} d x + 2 \, b^{3} c^{3} - 3 \, b d^{3} x - 3 \, b c d^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )}{128 \, b^{4}} - \frac {{\left (18 \, b^{2} d^{3} x^{2} + 36 \, b^{2} c d^{2} x + 18 \, b^{2} c^{2} d - d^{3}\right )} \sin \left (6 \, b x + 6 \, a\right )}{6912 \, b^{4}} + \frac {9 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \sin \left (2 \, b x + 2 \, a\right )}{256 \, b^{4}} \]
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Time = 1.55 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.02 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {243\,d^3\,\sin \left (2\,a+2\,b\,x\right )-d^3\,\sin \left (6\,a+6\,b\,x\right )+324\,b^3\,c^3\,\cos \left (2\,a+2\,b\,x\right )-36\,b^3\,c^3\,\cos \left (6\,a+6\,b\,x\right )-486\,b^2\,c^2\,d\,\sin \left (2\,a+2\,b\,x\right )+18\,b^2\,c^2\,d\,\sin \left (6\,a+6\,b\,x\right )+324\,b^3\,d^3\,x^3\,\cos \left (2\,a+2\,b\,x\right )-36\,b^3\,d^3\,x^3\,\cos \left (6\,a+6\,b\,x\right )-486\,b^2\,d^3\,x^2\,\sin \left (2\,a+2\,b\,x\right )+18\,b^2\,d^3\,x^2\,\sin \left (6\,a+6\,b\,x\right )-486\,b\,c\,d^2\,\cos \left (2\,a+2\,b\,x\right )+6\,b\,c\,d^2\,\cos \left (6\,a+6\,b\,x\right )-486\,b\,d^3\,x\,\cos \left (2\,a+2\,b\,x\right )+6\,b\,d^3\,x\,\cos \left (6\,a+6\,b\,x\right )+972\,b^3\,c^2\,d\,x\,\cos \left (2\,a+2\,b\,x\right )-108\,b^3\,c^2\,d\,x\,\cos \left (6\,a+6\,b\,x\right )-972\,b^2\,c\,d^2\,x\,\sin \left (2\,a+2\,b\,x\right )+36\,b^2\,c\,d^2\,x\,\sin \left (6\,a+6\,b\,x\right )+972\,b^3\,c\,d^2\,x^2\,\cos \left (2\,a+2\,b\,x\right )-108\,b^3\,c\,d^2\,x^2\,\cos \left (6\,a+6\,b\,x\right )}{6912\,b^4} \]
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