Integrand size = 24, antiderivative size = 131 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {(c+d x)^5}{40 d}+\frac {3 d^3 (c+d x) \cos (4 a+4 b x)}{256 b^4}-\frac {d (c+d x)^3 \cos (4 a+4 b x)}{32 b^2}-\frac {3 d^4 \sin (4 a+4 b x)}{1024 b^5}+\frac {3 d^2 (c+d x)^2 \sin (4 a+4 b x)}{128 b^3}-\frac {(c+d x)^4 \sin (4 a+4 b x)}{32 b} \]
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Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4491, 3377, 2717} \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=-\frac {3 d^4 \sin (4 a+4 b x)}{1024 b^5}+\frac {3 d^3 (c+d x) \cos (4 a+4 b x)}{256 b^4}+\frac {3 d^2 (c+d x)^2 \sin (4 a+4 b x)}{128 b^3}-\frac {d (c+d x)^3 \cos (4 a+4 b x)}{32 b^2}-\frac {(c+d x)^4 \sin (4 a+4 b x)}{32 b}+\frac {(c+d x)^5}{40 d} \]
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Rule 2717
Rule 3377
Rule 4491
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{8} (c+d x)^4-\frac {1}{8} (c+d x)^4 \cos (4 a+4 b x)\right ) \, dx \\ & = \frac {(c+d x)^5}{40 d}-\frac {1}{8} \int (c+d x)^4 \cos (4 a+4 b x) \, dx \\ & = \frac {(c+d x)^5}{40 d}-\frac {(c+d x)^4 \sin (4 a+4 b x)}{32 b}+\frac {d \int (c+d x)^3 \sin (4 a+4 b x) \, dx}{8 b} \\ & = \frac {(c+d x)^5}{40 d}-\frac {d (c+d x)^3 \cos (4 a+4 b x)}{32 b^2}-\frac {(c+d x)^4 \sin (4 a+4 b x)}{32 b}+\frac {\left (3 d^2\right ) \int (c+d x)^2 \cos (4 a+4 b x) \, dx}{32 b^2} \\ & = \frac {(c+d x)^5}{40 d}-\frac {d (c+d x)^3 \cos (4 a+4 b x)}{32 b^2}+\frac {3 d^2 (c+d x)^2 \sin (4 a+4 b x)}{128 b^3}-\frac {(c+d x)^4 \sin (4 a+4 b x)}{32 b}-\frac {\left (3 d^3\right ) \int (c+d x) \sin (4 a+4 b x) \, dx}{64 b^3} \\ & = \frac {(c+d x)^5}{40 d}+\frac {3 d^3 (c+d x) \cos (4 a+4 b x)}{256 b^4}-\frac {d (c+d x)^3 \cos (4 a+4 b x)}{32 b^2}+\frac {3 d^2 (c+d x)^2 \sin (4 a+4 b x)}{128 b^3}-\frac {(c+d x)^4 \sin (4 a+4 b x)}{32 b}-\frac {\left (3 d^4\right ) \int \cos (4 a+4 b x) \, dx}{256 b^4} \\ & = \frac {(c+d x)^5}{40 d}+\frac {3 d^3 (c+d x) \cos (4 a+4 b x)}{256 b^4}-\frac {d (c+d x)^3 \cos (4 a+4 b x)}{32 b^2}-\frac {3 d^4 \sin (4 a+4 b x)}{1024 b^5}+\frac {3 d^2 (c+d x)^2 \sin (4 a+4 b x)}{128 b^3}-\frac {(c+d x)^4 \sin (4 a+4 b x)}{32 b} \\ \end{align*}
Time = 1.25 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.01 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {128 b^5 x \left (5 c^4+10 c^3 d x+10 c^2 d^2 x^2+5 c d^3 x^3+d^4 x^4\right )+20 b d (c+d x) \left (3 d^2-8 b^2 (c+d x)^2\right ) \cos (4 (a+b x))-5 \left (3 d^4-24 b^2 d^2 (c+d x)^2+32 b^4 (c+d x)^4\right ) \sin (4 (a+b x))}{5120 b^5} \]
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Time = 2.63 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.11
method | result | size |
parallelrisch | \(\frac {\left (-32 b^{4} \left (d x +c \right )^{4}+24 d^{2} \left (d x +c \right )^{2} b^{2}-3 d^{4}\right ) \sin \left (4 x b +4 a \right )+128 b \left (-\frac {\left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{8}\right ) d \left (d x +c \right ) \cos \left (4 x b +4 a \right )}{4}+x \left (\frac {1}{5} d^{4} x^{4}+c \,d^{3} x^{3}+2 c^{2} d^{2} x^{2}+2 c^{3} d x +c^{4}\right ) b^{4}+\frac {b^{2} c^{3} d}{4}-\frac {3 d^{3} c}{32}\right )}{1024 b^{5}}\) | \(146\) |
risch | \(\frac {d^{4} x^{5}}{40}+\frac {d^{3} c \,x^{4}}{8}+\frac {d^{2} c^{2} x^{3}}{4}+\frac {d \,c^{3} x^{2}}{4}+\frac {c^{4} x}{8}+\frac {c^{5}}{40 d}-\frac {d \left (8 b^{2} d^{3} x^{3}+24 b^{2} c \,d^{2} x^{2}+24 b^{2} c^{2} d x +8 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \cos \left (4 x b +4 a \right )}{256 b^{4}}-\frac {\left (32 d^{4} x^{4} b^{4}+128 b^{4} c \,d^{3} x^{3}+192 b^{4} c^{2} d^{2} x^{2}+128 b^{4} c^{3} d x +32 b^{4} c^{4}-24 b^{2} d^{4} x^{2}-48 b^{2} c \,d^{3} x -24 b^{2} c^{2} d^{2}+3 d^{4}\right ) \sin \left (4 x b +4 a \right )}{1024 b^{5}}\) | \(229\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1951\) |
default | \(\text {Expression too large to display}\) | \(1951\) |
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Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (119) = 238\).
Time = 0.26 (sec) , antiderivative size = 466, normalized size of antiderivative = 3.56 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {32 \, b^{5} d^{4} x^{5} + 160 \, b^{5} c d^{3} x^{4} - 40 \, {\left (8 \, b^{3} d^{4} x^{3} + 24 \, b^{3} c d^{3} x^{2} + 8 \, b^{3} c^{3} d - 3 \, b c d^{3} + 3 \, {\left (8 \, b^{3} c^{2} d^{2} - b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{4} + 40 \, {\left (8 \, b^{5} c^{2} d^{2} - b^{3} d^{4}\right )} x^{3} + 40 \, {\left (8 \, b^{5} c^{3} d - 3 \, b^{3} c d^{3}\right )} x^{2} + 40 \, {\left (8 \, b^{3} d^{4} x^{3} + 24 \, b^{3} c d^{3} x^{2} + 8 \, b^{3} c^{3} d - 3 \, b c d^{3} + 3 \, {\left (8 \, b^{3} c^{2} d^{2} - b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{2} + 5 \, {\left (32 \, b^{5} c^{4} - 24 \, b^{3} c^{2} d^{2} + 3 \, b d^{4}\right )} x - 5 \, {\left (2 \, {\left (32 \, b^{4} d^{4} x^{4} + 128 \, b^{4} c d^{3} x^{3} + 32 \, b^{4} c^{4} - 24 \, b^{2} c^{2} d^{2} + 3 \, d^{4} + 24 \, {\left (8 \, b^{4} c^{2} d^{2} - b^{2} d^{4}\right )} x^{2} + 16 \, {\left (8 \, b^{4} c^{3} d - 3 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{3} - {\left (32 \, b^{4} d^{4} x^{4} + 128 \, b^{4} c d^{3} x^{3} + 32 \, b^{4} c^{4} - 24 \, b^{2} c^{2} d^{2} + 3 \, d^{4} + 24 \, {\left (8 \, b^{4} c^{2} d^{2} - b^{2} d^{4}\right )} x^{2} + 16 \, {\left (8 \, b^{4} c^{3} d - 3 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{1280 \, b^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1231 vs. \(2 (126) = 252\).
Time = 0.91 (sec) , antiderivative size = 1231, normalized size of antiderivative = 9.40 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 735 vs. \(2 (119) = 238\).
Time = 0.27 (sec) , antiderivative size = 735, normalized size of antiderivative = 5.61 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {160 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} c^{4} - \frac {640 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a c^{3} d}{b} + \frac {960 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a^{2} c^{2} d^{2}}{b^{2}} - \frac {640 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a^{3} c d^{3}}{b^{3}} + \frac {160 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a^{4} d^{4}}{b^{4}} + \frac {160 \, {\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^{3} d}{b} - \frac {480 \, {\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^{2} d^{2}}{b^{2}} + \frac {480 \, {\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} c d^{3}}{b^{3}} - \frac {160 \, {\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^{3} d^{4}}{b^{4}} + \frac {40 \, {\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^{2} d^{2}}{b^{2}} - \frac {80 \, {\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 c d^{3}}{b^{3}} + \frac {40 \, {\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^{2} d^{4}}{b^{4}} + \frac {20 \, {\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 )} c d^{3}}{b^{3}} - \frac {20 \, {\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 )} a d^{4}}{b^{4}} + \frac {{\left (128 \, {\left (b x + a\right )}^{5} - 20 \, {\left (8 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \cos \left (4 \, b x + 4 \, a\right ) - 5 \, {\left (32 \, {\left (b x + a\right )}^{4} - 24 \, {\left (b x + a\right )}^{2} + 3\right )} \sin \left (4 \, b x + 4 \, a\right )\right )} d^{4}}{b^{4}}}{5120 \, b} \]
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Time = 0.32 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.71 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {1}{40} \, d^{4} x^{5} + \frac {1}{8} \, c d^{3} x^{4} + \frac {1}{4} \, c^{2} d^{2} x^{3} + \frac {1}{4} \, c^{3} d x^{2} + \frac {1}{8} \, c^{4} x - \frac {{\left (8 \, b^{3} d^{4} x^{3} + 24 \, b^{3} c d^{3} x^{2} + 24 \, b^{3} c^{2} d^{2} x + 8 \, b^{3} c^{3} d - 3 \, b d^{4} x - 3 \, b c d^{3}\right )} \cos \left (4 \, b x + 4 \, a\right )}{256 \, b^{5}} - \frac {{\left (32 \, b^{4} d^{4} x^{4} + 128 \, b^{4} c d^{3} x^{3} + 192 \, b^{4} c^{2} d^{2} x^{2} + 128 \, b^{4} c^{3} d x + 32 \, b^{4} c^{4} - 24 \, b^{2} d^{4} x^{2} - 48 \, b^{2} c d^{3} x - 24 \, b^{2} c^{2} d^{2} + 3 \, d^{4}\right )} \sin \left (4 \, b x + 4 \, a\right )}{1024 \, b^{5}} \]
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Time = 23.66 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.66 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=-\frac {15\,d^4\,\sin \left (4\,a+4\,b\,x\right )-640\,b^5\,c^4\,x+160\,b^4\,c^4\,\sin \left (4\,a+4\,b\,x\right )-128\,b^5\,d^4\,x^5+160\,b^3\,c^3\,d\,\cos \left (4\,a+4\,b\,x\right )-1280\,b^5\,c^3\,d\,x^2-640\,b^5\,c\,d^3\,x^4-120\,b^2\,c^2\,d^2\,\sin \left (4\,a+4\,b\,x\right )+160\,b^3\,d^4\,x^3\,\cos \left (4\,a+4\,b\,x\right )-1280\,b^5\,c^2\,d^2\,x^3-120\,b^2\,d^4\,x^2\,\sin \left (4\,a+4\,b\,x\right )+160\,b^4\,d^4\,x^4\,\sin \left (4\,a+4\,b\,x\right )-60\,b\,c\,d^3\,\cos \left (4\,a+4\,b\,x\right )-60\,b\,d^4\,x\,\cos \left (4\,a+4\,b\,x\right )+960\,b^4\,c^2\,d^2\,x^2\,\sin \left (4\,a+4\,b\,x\right )-240\,b^2\,c\,d^3\,x\,\sin \left (4\,a+4\,b\,x\right )+640\,b^4\,c^3\,d\,x\,\sin \left (4\,a+4\,b\,x\right )+480\,b^3\,c^2\,d^2\,x\,\cos \left (4\,a+4\,b\,x\right )+480\,b^3\,c\,d^3\,x^2\,\cos \left (4\,a+4\,b\,x\right )+640\,b^4\,c\,d^3\,x^3\,\sin \left (4\,a+4\,b\,x\right )}{5120\,b^5} \]
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