Integrand size = 24, antiderivative size = 259 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\frac {3 d^2 (c+d x) \cos (a+b x)}{4 b^3}-\frac {(c+d x)^3 \cos (a+b x)}{8 b}+\frac {d^2 (c+d x) \cos (3 a+3 b x)}{72 b^3}-\frac {(c+d x)^3 \cos (3 a+3 b x)}{48 b}-\frac {3 d^2 (c+d x) \cos (5 a+5 b x)}{1000 b^3}+\frac {(c+d x)^3 \cos (5 a+5 b x)}{80 b}-\frac {3 d^3 \sin (a+b x)}{4 b^4}+\frac {3 d (c+d x)^2 \sin (a+b x)}{8 b^2}-\frac {d^3 \sin (3 a+3 b x)}{216 b^4}+\frac {d (c+d x)^2 \sin (3 a+3 b x)}{48 b^2}+\frac {3 d^3 \sin (5 a+5 b x)}{5000 b^4}-\frac {3 d (c+d x)^2 \sin (5 a+5 b x)}{400 b^2} \]
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Time = 0.31 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4491, 3377, 2717} \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=-\frac {3 d^3 \sin (a+b x)}{4 b^4}-\frac {d^3 \sin (3 a+3 b x)}{216 b^4}+\frac {3 d^3 \sin (5 a+5 b x)}{5000 b^4}+\frac {3 d^2 (c+d x) \cos (a+b x)}{4 b^3}+\frac {d^2 (c+d x) \cos (3 a+3 b x)}{72 b^3}-\frac {3 d^2 (c+d x) \cos (5 a+5 b x)}{1000 b^3}+\frac {3 d (c+d x)^2 \sin (a+b x)}{8 b^2}+\frac {d (c+d x)^2 \sin (3 a+3 b x)}{48 b^2}-\frac {3 d (c+d x)^2 \sin (5 a+5 b x)}{400 b^2}-\frac {(c+d x)^3 \cos (a+b x)}{8 b}-\frac {(c+d x)^3 \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^3 \cos (5 a+5 b x)}{80 b} \]
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Rule 2717
Rule 3377
Rule 4491
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{8} (c+d x)^3 \sin (a+b x)+\frac {1}{16} (c+d x)^3 \sin (3 a+3 b x)-\frac {1}{16} (c+d x)^3 \sin (5 a+5 b x)\right ) \, dx \\ & = \frac {1}{16} \int (c+d x)^3 \sin (3 a+3 b x) \, dx-\frac {1}{16} \int (c+d x)^3 \sin (5 a+5 b x) \, dx+\frac {1}{8} \int (c+d x)^3 \sin (a+b x) \, dx \\ & = -\frac {(c+d x)^3 \cos (a+b x)}{8 b}-\frac {(c+d x)^3 \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^3 \cos (5 a+5 b x)}{80 b}-\frac {(3 d) \int (c+d x)^2 \cos (5 a+5 b x) \, dx}{80 b}+\frac {d \int (c+d x)^2 \cos (3 a+3 b x) \, dx}{16 b}+\frac {(3 d) \int (c+d x)^2 \cos (a+b x) \, dx}{8 b} \\ & = -\frac {(c+d x)^3 \cos (a+b x)}{8 b}-\frac {(c+d x)^3 \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^3 \cos (5 a+5 b x)}{80 b}+\frac {3 d (c+d x)^2 \sin (a+b x)}{8 b^2}+\frac {d (c+d x)^2 \sin (3 a+3 b x)}{48 b^2}-\frac {3 d (c+d x)^2 \sin (5 a+5 b x)}{400 b^2}+\frac {\left (3 d^2\right ) \int (c+d x) \sin (5 a+5 b x) \, dx}{200 b^2}-\frac {d^2 \int (c+d x) \sin (3 a+3 b x) \, dx}{24 b^2}-\frac {\left (3 d^2\right ) \int (c+d x) \sin (a+b x) \, dx}{4 b^2} \\ & = \frac {3 d^2 (c+d x) \cos (a+b x)}{4 b^3}-\frac {(c+d x)^3 \cos (a+b x)}{8 b}+\frac {d^2 (c+d x) \cos (3 a+3 b x)}{72 b^3}-\frac {(c+d x)^3 \cos (3 a+3 b x)}{48 b}-\frac {3 d^2 (c+d x) \cos (5 a+5 b x)}{1000 b^3}+\frac {(c+d x)^3 \cos (5 a+5 b x)}{80 b}+\frac {3 d (c+d x)^2 \sin (a+b x)}{8 b^2}+\frac {d (c+d x)^2 \sin (3 a+3 b x)}{48 b^2}-\frac {3 d (c+d x)^2 \sin (5 a+5 b x)}{400 b^2}+\frac {\left (3 d^3\right ) \int \cos (5 a+5 b x) \, dx}{1000 b^3}-\frac {d^3 \int \cos (3 a+3 b x) \, dx}{72 b^3}-\frac {\left (3 d^3\right ) \int \cos (a+b x) \, dx}{4 b^3} \\ & = \frac {3 d^2 (c+d x) \cos (a+b x)}{4 b^3}-\frac {(c+d x)^3 \cos (a+b x)}{8 b}+\frac {d^2 (c+d x) \cos (3 a+3 b x)}{72 b^3}-\frac {(c+d x)^3 \cos (3 a+3 b x)}{48 b}-\frac {3 d^2 (c+d x) \cos (5 a+5 b x)}{1000 b^3}+\frac {(c+d x)^3 \cos (5 a+5 b x)}{80 b}-\frac {3 d^3 \sin (a+b x)}{4 b^4}+\frac {3 d (c+d x)^2 \sin (a+b x)}{8 b^2}-\frac {d^3 \sin (3 a+3 b x)}{216 b^4}+\frac {d (c+d x)^2 \sin (3 a+3 b x)}{48 b^2}+\frac {3 d^3 \sin (5 a+5 b x)}{5000 b^4}-\frac {3 d (c+d x)^2 \sin (5 a+5 b x)}{400 b^2} \\ \end{align*}
Time = 1.51 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.42 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\frac {-33750 b (c+d x) \left (-6 d^2+b^2 (c+d x)^2\right ) \cos (a+b x)-1875 b (c+d x) \left (-2 d^2+3 b^2 (c+d x)^2\right ) \cos (3 (a+b x))+3375 b^3 c^3 \cos (5 (a+b x))-810 b c d^2 \cos (5 (a+b x))+10125 b^3 c^2 d x \cos (5 (a+b x))-810 b d^3 x \cos (5 (a+b x))+10125 b^3 c d^2 x^2 \cos (5 (a+b x))+3375 b^3 d^3 x^3 \cos (5 (a+b x))+101250 b^2 c^2 d \sin (a+b x)-202500 d^3 \sin (a+b x)+202500 b^2 c d^2 x \sin (a+b x)+101250 b^2 d^3 x^2 \sin (a+b x)+5625 b^2 c^2 d \sin (3 (a+b x))-1250 d^3 \sin (3 (a+b x))+11250 b^2 c d^2 x \sin (3 (a+b x))+5625 b^2 d^3 x^2 \sin (3 (a+b x))-2025 b^2 c^2 d \sin (5 (a+b x))+162 d^3 \sin (5 (a+b x))-4050 b^2 c d^2 x \sin (5 (a+b x))-2025 b^2 d^3 x^2 \sin (5 (a+b x))}{270000 b^4} \]
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Time = 2.25 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.79
method | result | size |
parallelrisch | \(\frac {-375 b \left (\left (d x +c \right )^{2} b^{2}-\frac {2 d^{2}}{3}\right ) \left (d x +c \right ) \cos \left (3 x b +3 a \right )+225 b \left (\left (d x +c \right )^{2} b^{2}-\frac {6 d^{2}}{25}\right ) \left (d x +c \right ) \cos \left (5 x b +5 a \right )+375 d \left (\left (d x +c \right )^{2} b^{2}-\frac {2 d^{2}}{9}\right ) \sin \left (3 x b +3 a \right )-135 d \left (\left (d x +c \right )^{2} b^{2}-\frac {2 d^{2}}{25}\right ) \sin \left (5 x b +5 a \right )-2250 b \left (\left (d x +c \right )^{2} b^{2}-6 d^{2}\right ) \left (d x +c \right ) \cos \left (x b +a \right )+6750 d \left (\left (d x +c \right )^{2} b^{2}-2 d^{2}\right ) \sin \left (x b +a \right )-2400 b^{3} c^{3}+13696 c \,d^{2} b}{18000 b^{4}}\) | \(205\) |
risch | \(-\frac {\left (b^{2} d^{3} x^{3}+3 b^{2} c \,d^{2} x^{2}+3 b^{2} c^{2} d x +b^{2} c^{3}-6 d^{3} x -6 c \,d^{2}\right ) \cos \left (x b +a \right )}{8 b^{3}}+\frac {3 d \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}-2 d^{2}\right ) \sin \left (x b +a \right )}{8 b^{4}}+\frac {\left (25 b^{2} d^{3} x^{3}+75 b^{2} c \,d^{2} x^{2}+75 b^{2} c^{2} d x +25 b^{2} c^{3}-6 d^{3} x -6 c \,d^{2}\right ) \cos \left (5 x b +5 a \right )}{2000 b^{3}}-\frac {3 d \left (25 x^{2} d^{2} b^{2}+50 b^{2} c d x +25 b^{2} c^{2}-2 d^{2}\right ) \sin \left (5 x b +5 a \right )}{10000 b^{4}}-\frac {\left (3 b^{2} d^{3} x^{3}+9 b^{2} c \,d^{2} x^{2}+9 b^{2} c^{2} d x +3 b^{2} c^{3}-2 d^{3} x -2 c \,d^{2}\right ) \cos \left (3 x b +3 a \right )}{144 b^{3}}+\frac {d \left (9 x^{2} d^{2} b^{2}+18 b^{2} c d x +9 b^{2} c^{2}-2 d^{2}\right ) \sin \left (3 x b +3 a \right )}{432 b^{4}}\) | \(340\) |
derivativedivides | \(\text {Expression too large to display}\) | \(992\) |
default | \(\text {Expression too large to display}\) | \(992\) |
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Time = 0.26 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.14 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\frac {135 \, {\left (25 \, b^{3} d^{3} x^{3} + 75 \, b^{3} c d^{2} x^{2} + 25 \, b^{3} c^{3} - 6 \, b c d^{2} + 3 \, {\left (25 \, b^{3} c^{2} d - 2 \, b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{5} - 75 \, {\left (75 \, b^{3} d^{3} x^{3} + 225 \, b^{3} c d^{2} x^{2} + 75 \, b^{3} c^{3} - 26 \, b c d^{2} + {\left (225 \, b^{3} c^{2} d - 26 \, b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{3} + 11700 \, {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right ) + {\left (5850 \, b^{2} d^{3} x^{2} + 11700 \, b^{2} c d^{2} x + 5850 \, b^{2} c^{2} d - 81 \, {\left (25 \, b^{2} d^{3} x^{2} + 50 \, b^{2} c d^{2} x + 25 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (b x + a\right )^{4} - 12568 \, d^{3} + {\left (2925 \, b^{2} d^{3} x^{2} + 5850 \, b^{2} c d^{2} x + 2925 \, b^{2} c^{2} d - 434 \, d^{3}\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )}{16875 \, b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 690 vs. \(2 (255) = 510\).
Time = 0.84 (sec) , antiderivative size = 690, normalized size of antiderivative = 2.66 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\begin {cases} - \frac {c^{3} \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {2 c^{3} \cos ^{5}{\left (a + b x \right )}}{15 b} - \frac {c^{2} d x \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{b} - \frac {2 c^{2} d x \cos ^{5}{\left (a + b x \right )}}{5 b} - \frac {c d^{2} x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{b} - \frac {2 c d^{2} x^{2} \cos ^{5}{\left (a + b x \right )}}{5 b} - \frac {d^{3} x^{3} \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {2 d^{3} x^{3} \cos ^{5}{\left (a + b x \right )}}{15 b} + \frac {26 c^{2} d \sin ^{5}{\left (a + b x \right )}}{75 b^{2}} + \frac {13 c^{2} d \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{15 b^{2}} + \frac {2 c^{2} d \sin {\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{5 b^{2}} + \frac {52 c d^{2} x \sin ^{5}{\left (a + b x \right )}}{75 b^{2}} + \frac {26 c d^{2} x \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{15 b^{2}} + \frac {4 c d^{2} x \sin {\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{5 b^{2}} + \frac {26 d^{3} x^{2} \sin ^{5}{\left (a + b x \right )}}{75 b^{2}} + \frac {13 d^{3} x^{2} \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{15 b^{2}} + \frac {2 d^{3} x^{2} \sin {\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{5 b^{2}} + \frac {52 c d^{2} \sin ^{4}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{75 b^{3}} + \frac {338 c d^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{225 b^{3}} + \frac {856 c d^{2} \cos ^{5}{\left (a + b x \right )}}{1125 b^{3}} + \frac {52 d^{3} x \sin ^{4}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{75 b^{3}} + \frac {338 d^{3} x \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{225 b^{3}} + \frac {856 d^{3} x \cos ^{5}{\left (a + b x \right )}}{1125 b^{3}} - \frac {12568 d^{3} \sin ^{5}{\left (a + b x \right )}}{16875 b^{4}} - \frac {5114 d^{3} \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3375 b^{4}} - \frac {856 d^{3} \sin {\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{1125 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 ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 766 vs. \(2 (235) = 470\).
Time = 0.27 (sec) , antiderivative size = 766, normalized size of antiderivative = 2.96 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]
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Time = 0.34 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.36 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\frac {{\left (25 \, b^{3} d^{3} x^{3} + 75 \, b^{3} c d^{2} x^{2} + 75 \, b^{3} c^{2} d x + 25 \, b^{3} c^{3} - 6 \, b d^{3} x - 6 \, b c d^{2}\right )} \cos \left (5 \, b x + 5 \, a\right )}{2000 \, b^{4}} - \frac {{\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 9 \, b^{3} c^{2} d x + 3 \, b^{3} c^{3} - 2 \, b d^{3} x - 2 \, b c d^{2}\right )} \cos \left (3 \, b x + 3 \, a\right )}{144 \, b^{4}} - \frac {{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3} - 6 \, b d^{3} x - 6 \, b c d^{2}\right )} \cos \left (b x + a\right )}{8 \, b^{4}} - \frac {3 \, {\left (25 \, b^{2} d^{3} x^{2} + 50 \, b^{2} c d^{2} x + 25 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \sin \left (5 \, b x + 5 \, a\right )}{10000 \, b^{4}} + \frac {{\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \sin \left (3 \, b x + 3 \, a\right )}{432 \, b^{4}} + \frac {3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d - 2 \, d^{3}\right )} \sin \left (b x + a\right )}{8 \, b^{4}} \]
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Time = 3.00 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.99 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=-\frac {\frac {3\,d^3\,\sin \left (a+b\,x\right )}{4}+\frac {d^3\,\sin \left (3\,a+3\,b\,x\right )}{216}-\frac {3\,d^3\,\sin \left (5\,a+5\,b\,x\right )}{5000}+\frac {b^3\,c^3\,\cos \left (a+b\,x\right )}{8}+\frac {b^3\,c^3\,\cos \left (3\,a+3\,b\,x\right )}{48}-\frac {b^3\,c^3\,\cos \left (5\,a+5\,b\,x\right )}{80}-\frac {b^2\,c^2\,d\,\sin \left (3\,a+3\,b\,x\right )}{48}+\frac {3\,b^2\,c^2\,d\,\sin \left (5\,a+5\,b\,x\right )}{400}+\frac {b^3\,d^3\,x^3\,\cos \left (a+b\,x\right )}{8}-\frac {3\,b^2\,d^3\,x^2\,\sin \left (a+b\,x\right )}{8}-\frac {3\,b\,c\,d^2\,\cos \left (a+b\,x\right )}{4}-\frac {3\,b\,d^3\,x\,\cos \left (a+b\,x\right )}{4}+\frac {b^3\,d^3\,x^3\,\cos \left (3\,a+3\,b\,x\right )}{48}-\frac {b^3\,d^3\,x^3\,\cos \left (5\,a+5\,b\,x\right )}{80}-\frac {b^2\,d^3\,x^2\,\sin \left (3\,a+3\,b\,x\right )}{48}+\frac {3\,b^2\,d^3\,x^2\,\sin \left (5\,a+5\,b\,x\right )}{400}-\frac {b\,c\,d^2\,\cos \left (3\,a+3\,b\,x\right )}{72}+\frac {3\,b\,c\,d^2\,\cos \left (5\,a+5\,b\,x\right )}{1000}-\frac {3\,b^2\,c^2\,d\,\sin \left (a+b\,x\right )}{8}-\frac {b\,d^3\,x\,\cos \left (3\,a+3\,b\,x\right )}{72}+\frac {3\,b\,d^3\,x\,\cos \left (5\,a+5\,b\,x\right )}{1000}+\frac {3\,b^3\,c^2\,d\,x\,\cos \left (a+b\,x\right )}{8}-\frac {3\,b^2\,c\,d^2\,x\,\sin \left (a+b\,x\right )}{4}+\frac {b^3\,c^2\,d\,x\,\cos \left (3\,a+3\,b\,x\right )}{16}-\frac {3\,b^3\,c^2\,d\,x\,\cos \left (5\,a+5\,b\,x\right )}{80}+\frac {3\,b^3\,c\,d^2\,x^2\,\cos \left (a+b\,x\right )}{8}-\frac {b^2\,c\,d^2\,x\,\sin \left (3\,a+3\,b\,x\right )}{24}+\frac {3\,b^2\,c\,d^2\,x\,\sin \left (5\,a+5\,b\,x\right )}{200}+\frac {b^3\,c\,d^2\,x^2\,\cos \left (3\,a+3\,b\,x\right )}{16}-\frac {3\,b^3\,c\,d^2\,x^2\,\cos \left (5\,a+5\,b\,x\right )}{80}}{b^4} \]
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