Integrand size = 24, antiderivative size = 105 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {(c+d x)^4}{32 d}+\frac {3 d^3 \cos (4 a+4 b x)}{1024 b^4}-\frac {3 d (c+d x)^2 \cos (4 a+4 b x)}{128 b^2}+\frac {3 d^2 (c+d x) \sin (4 a+4 b x)}{256 b^3}-\frac {(c+d x)^3 \sin (4 a+4 b x)}{32 b} \]
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Time = 0.14 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4491, 3377, 2718} \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {3 d^3 \cos (4 a+4 b x)}{1024 b^4}+\frac {3 d^2 (c+d x) \sin (4 a+4 b x)}{256 b^3}-\frac {3 d (c+d x)^2 \cos (4 a+4 b x)}{128 b^2}-\frac {(c+d x)^3 \sin (4 a+4 b x)}{32 b}+\frac {(c+d x)^4}{32 d} \]
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Rule 2718
Rule 3377
Rule 4491
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{8} (c+d x)^3-\frac {1}{8} (c+d x)^3 \cos (4 a+4 b x)\right ) \, dx \\ & = \frac {(c+d x)^4}{32 d}-\frac {1}{8} \int (c+d x)^3 \cos (4 a+4 b x) \, dx \\ & = \frac {(c+d x)^4}{32 d}-\frac {(c+d x)^3 \sin (4 a+4 b x)}{32 b}+\frac {(3 d) \int (c+d x)^2 \sin (4 a+4 b x) \, dx}{32 b} \\ & = \frac {(c+d x)^4}{32 d}-\frac {3 d (c+d x)^2 \cos (4 a+4 b x)}{128 b^2}-\frac {(c+d x)^3 \sin (4 a+4 b x)}{32 b}+\frac {\left (3 d^2\right ) \int (c+d x) \cos (4 a+4 b x) \, dx}{64 b^2} \\ & = \frac {(c+d x)^4}{32 d}-\frac {3 d (c+d x)^2 \cos (4 a+4 b x)}{128 b^2}+\frac {3 d^2 (c+d x) \sin (4 a+4 b x)}{256 b^3}-\frac {(c+d x)^3 \sin (4 a+4 b x)}{32 b}-\frac {\left (3 d^3\right ) \int \sin (4 a+4 b x) \, dx}{256 b^3} \\ & = \frac {(c+d x)^4}{32 d}+\frac {3 d^3 \cos (4 a+4 b x)}{1024 b^4}-\frac {3 d (c+d x)^2 \cos (4 a+4 b x)}{128 b^2}+\frac {3 d^2 (c+d x) \sin (4 a+4 b x)}{256 b^3}-\frac {(c+d x)^3 \sin (4 a+4 b x)}{32 b} \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.01 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {32 b^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )-3 d \left (-d^2+8 b^2 (c+d x)^2\right ) \cos (4 (a+b x))-4 b (c+d x) \left (-3 d^2+8 b^2 (c+d x)^2\right ) \sin (4 (a+b x))}{1024 b^4} \]
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Time = 1.81 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.15
method | result | size |
parallelrisch | \(\frac {-32 \left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{8}\right ) b \left (d x +c \right ) \sin \left (4 x b +4 a \right )-24 \left (\left (d x +c \right )^{2} b^{2}-\frac {d^{2}}{8}\right ) d \cos \left (4 x b +4 a \right )+32 \left (d^{3} x^{4}+4 d^{2} c \,x^{3}+6 d \,c^{2} x^{2}+4 c^{3} x \right ) b^{4}+24 b^{2} c^{2} d -3 d^{3}}{1024 b^{4}}\) | \(121\) |
risch | \(\frac {d^{3} x^{4}}{32}+\frac {d^{2} c \,x^{3}}{8}+\frac {3 d \,c^{2} x^{2}}{16}+\frac {c^{3} x}{8}+\frac {c^{4}}{32 d}-\frac {3 d \left (8 x^{2} d^{2} b^{2}+16 b^{2} c d x +8 b^{2} c^{2}-d^{2}\right ) \cos \left (4 x b +4 a \right )}{1024 b^{4}}-\frac {\left (8 b^{2} d^{3} x^{3}+24 b^{2} c \,d^{2} x^{2}+24 b^{2} c^{2} d x +8 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \sin \left (4 x b +4 a \right )}{256 b^{3}}\) | \(158\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1098\) |
default | \(\text {Expression too large to display}\) | \(1098\) |
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Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (95) = 190\).
Time = 0.25 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.93 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {4 \, b^{4} d^{3} x^{4} + 16 \, b^{4} c d^{2} x^{3} - 3 \, {\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{4} + 3 \, {\left (8 \, b^{4} c^{2} d - b^{2} d^{3}\right )} x^{2} + 3 \, {\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{2} + 2 \, {\left (8 \, b^{4} c^{3} - 3 \, b^{2} c d^{2}\right )} x - 2 \, {\left (2 \, {\left (8 \, b^{3} d^{3} x^{3} + 24 \, b^{3} c d^{2} x^{2} + 8 \, b^{3} c^{3} - 3 \, b c d^{2} + 3 \, {\left (8 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{3} - {\left (8 \, b^{3} d^{3} x^{3} + 24 \, b^{3} c d^{2} x^{2} + 8 \, b^{3} c^{3} - 3 \, b c d^{2} + 3 \, {\left (8 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{128 \, b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 835 vs. \(2 (100) = 200\).
Time = 0.63 (sec) , antiderivative size = 835, normalized size of antiderivative = 7.95 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\begin {cases} \frac {c^{3} x \sin ^{4}{\left (a + b x \right )}}{8} + \frac {c^{3} x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} + \frac {c^{3} x \cos ^{4}{\left (a + b x \right )}}{8} + \frac {3 c^{2} d x^{2} \sin ^{4}{\left (a + b x \right )}}{16} + \frac {3 c^{2} d x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{8} + \frac {3 c^{2} d x^{2} \cos ^{4}{\left (a + b x \right )}}{16} + \frac {c d^{2} x^{3} \sin ^{4}{\left (a + b x \right )}}{8} + \frac {c d^{2} x^{3} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} + \frac {c d^{2} x^{3} \cos ^{4}{\left (a + b x \right )}}{8} + \frac {d^{3} x^{4} \sin ^{4}{\left (a + b x \right )}}{32} + \frac {d^{3} x^{4} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16} + \frac {d^{3} x^{4} \cos ^{4}{\left (a + b x \right )}}{32} + \frac {c^{3} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} - \frac {c^{3} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} + \frac {3 c^{2} d x \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} - \frac {3 c^{2} d x \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} + \frac {3 c d^{2} x^{2} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} - \frac {3 c d^{2} x^{2} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} + \frac {d^{3} x^{3} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} - \frac {d^{3} x^{3} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} - \frac {3 c^{2} d \sin ^{4}{\left (a + b x \right )}}{32 b^{2}} - \frac {3 c^{2} d \cos ^{4}{\left (a + b x \right )}}{32 b^{2}} - \frac {3 c d^{2} x \sin ^{4}{\left (a + b x \right )}}{64 b^{2}} + \frac {9 c d^{2} x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32 b^{2}} - \frac {3 c d^{2} x \cos ^{4}{\left (a + b x \right )}}{64 b^{2}} - \frac {3 d^{3} x^{2} \sin ^{4}{\left (a + b x \right )}}{128 b^{2}} + \frac {9 d^{3} x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{64 b^{2}} - \frac {3 d^{3} x^{2} \cos ^{4}{\left (a + b x \right )}}{128 b^{2}} - \frac {3 c d^{2} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{64 b^{3}} + \frac {3 c d^{2} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{64 b^{3}} - \frac {3 d^{3} x \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{64 b^{3}} + \frac {3 d^{3} x \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{64 b^{3}} + \frac {3 d^{3} \sin ^{4}{\left (a + b x \right )}}{256 b^{4}} + \frac {3 d^{3} \cos ^{4}{\left (a + b x \right )}}{256 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 ^{2}{\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. 442 vs. \(2 (95) = 190\).
Time = 0.25 (sec) , antiderivative size = 442, normalized size of antiderivative = 4.21 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {32 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} c^{3} - \frac {96 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a c^{2} d}{b} + \frac {96 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a^{2} c d^{2}}{b^{2}} - \frac {32 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a^{3} d^{3}}{b^{3}} + \frac {24 \, {\left (8 \, {\left (b x + a\right )}^{2} - 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )\right )} c^{2} d}{b} - \frac {48 \, {\left (8 \, {\left (b x + a\right )}^{2} - 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )\right )} a c d^{2}}{b^{2}} + \frac {24 \, {\left (8 \, {\left (b x + a\right )}^{2} - 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )\right )} a^{2} d^{3}}{b^{3}} + \frac {4 \, {\left (32 \, {\left (b x + a\right )}^{3} - 12 \, {\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) - 3 \, {\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (4 \, b x + 4 \, a\right )\right )} c d^{2}}{b^{2}} - \frac {4 \, {\left (32 \, {\left (b x + a\right )}^{3} - 12 \, {\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) - 3 \, {\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (4 \, b x + 4 \, a\right )\right )} a d^{3}}{b^{3}} + \frac {{\left (32 \, {\left (b x + a\right )}^{4} - 3 \, {\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (4 \, b x + 4 \, a\right ) - 4 \, {\left (8 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \sin \left (4 \, b x + 4 \, a\right )\right )} d^{3}}{b^{3}}}{1024 \, b} \]
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Time = 0.31 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.46 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {1}{32} \, d^{3} x^{4} + \frac {1}{8} \, c d^{2} x^{3} + \frac {3}{16} \, c^{2} d x^{2} + \frac {1}{8} \, c^{3} x - \frac {3 \, {\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (4 \, b x + 4 \, a\right )}{1024 \, b^{4}} - \frac {{\left (8 \, b^{3} d^{3} x^{3} + 24 \, b^{3} c d^{2} x^{2} + 24 \, b^{3} c^{2} d x + 8 \, b^{3} c^{3} - 3 \, b d^{3} x - 3 \, b c d^{2}\right )} \sin \left (4 \, b x + 4 \, a\right )}{256 \, b^{4}} \]
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Time = 1.20 (sec) , antiderivative size = 329, normalized size of antiderivative = 3.13 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=x^2\,\left (\frac {3\,c^2\,d}{64}+\frac {9\,d^3}{512\,b^2}\right )+x^2\,\left (\frac {9\,c^2\,d}{64}-\frac {9\,d^3}{512\,b^2}\right )+x\,\left (\frac {c^3}{32}+\frac {9\,c\,d^2}{256\,b^2}\right )+x\,\left (\frac {3\,c^3}{32}-\frac {9\,c\,d^2}{256\,b^2}\right )+\frac {d^3\,x^4}{32}-\frac {x\,\cos \left (4\,a+4\,b\,x\right )\,\left (\frac {c^3}{4}+\frac {9\,c\,d^2}{32\,b^2}\right )}{8}+\frac {x\,\cos \left (4\,a+4\,b\,x\right )\,\left (\frac {c^3}{8}-\frac {3\,c\,d^2}{64\,b^2}\right )}{4}+\frac {c\,d^2\,x^3}{8}+\frac {\cos \left (4\,a+4\,b\,x\right )\,\left (\frac {3\,d^3}{128}-\frac {3\,b^2\,c^2\,d}{16}\right )}{8\,b^4}+\frac {\sin \left (4\,a+4\,b\,x\right )\,\left (3\,c\,d^2-8\,b^2\,c^3\right )}{256\,b^3}-\frac {x^2\,\cos \left (4\,a+4\,b\,x\right )\,\left (\frac {3\,c^2\,d}{8}+\frac {9\,d^3}{64\,b^2}\right )}{8}+\frac {x^2\,\cos \left (4\,a+4\,b\,x\right )\,\left (\frac {3\,c^2\,d}{16}-\frac {3\,d^3}{128\,b^2}\right )}{4}-\frac {d^3\,x^3\,\sin \left (4\,a+4\,b\,x\right )}{32\,b}+\frac {3\,x\,\sin \left (4\,a+4\,b\,x\right )\,\left (d^3-8\,b^2\,c^2\,d\right )}{256\,b^3}-\frac {3\,c\,d^2\,x^2\,\sin \left (4\,a+4\,b\,x\right )}{32\,b} \]
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