Integrand size = 20, antiderivative size = 120 \[ \int (c+d x)^3 \cos (a+b x) \sin (a+b x) \, dx=\frac {3 d^3 x}{8 b^3}-\frac {(c+d x)^3}{4 b}-\frac {3 d^3 \cos (a+b x) \sin (a+b x)}{8 b^4}+\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{4 b^2}-\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}+\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b} \]
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Time = 0.10 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4489, 3392, 32, 2715, 8} \[ \int (c+d x)^3 \cos (a+b x) \sin (a+b x) \, dx=-\frac {3 d^3 \sin (a+b x) \cos (a+b x)}{8 b^4}-\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}+\frac {3 d (c+d x)^2 \sin (a+b x) \cos (a+b x)}{4 b^2}+\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}+\frac {3 d^3 x}{8 b^3}-\frac {(c+d x)^3}{4 b} \]
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Rule 8
Rule 32
Rule 2715
Rule 3392
Rule 4489
Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {(3 d) \int (c+d x)^2 \sin ^2(a+b x) \, dx}{2 b} \\ & = \frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{4 b^2}-\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}+\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {(3 d) \int (c+d x)^2 \, dx}{4 b}+\frac {\left (3 d^3\right ) \int \sin ^2(a+b x) \, dx}{4 b^3} \\ & = -\frac {(c+d x)^3}{4 b}-\frac {3 d^3 \cos (a+b x) \sin (a+b x)}{8 b^4}+\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{4 b^2}-\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}+\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}+\frac {\left (3 d^3\right ) \int 1 \, dx}{8 b^3} \\ & = \frac {3 d^3 x}{8 b^3}-\frac {(c+d x)^3}{4 b}-\frac {3 d^3 \cos (a+b x) \sin (a+b x)}{8 b^4}+\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{4 b^2}-\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}+\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.59 \[ \int (c+d x)^3 \cos (a+b x) \sin (a+b x) \, dx=\frac {-2 b (c+d x) \left (-3 d^2+2 b^2 (c+d x)^2\right ) \cos (2 (a+b x))+3 d \left (-d^2+2 b^2 (c+d x)^2\right ) \sin (2 (a+b x))}{16 b^4} \]
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Time = 0.44 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.98
method | result | size |
risch | \(-\frac {\left (2 b^{2} d^{3} x^{3}+6 b^{2} c \,d^{2} x^{2}+6 b^{2} c^{2} d x +2 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \cos \left (2 x b +2 a \right )}{8 b^{3}}+\frac {3 d \left (2 x^{2} d^{2} b^{2}+4 b^{2} c d x +2 b^{2} c^{2}-d^{2}\right ) \sin \left (2 x b +2 a \right )}{16 b^{4}}\) | \(118\) |
parallelrisch | \(\frac {-3 b x d \left (\left (\frac {1}{3} x^{2} d^{2}+c d x +c^{2}\right ) b^{2}-\frac {d^{2}}{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}-6 d \left (\left (d x +c \right )^{2} b^{2}-\frac {d^{2}}{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}+8 b \left (\left (\frac {3}{4} d^{3} x^{3}+\frac {9}{4} c \,d^{2} x^{2}+\frac {9}{4} c^{2} d x +c^{3}\right ) b^{2}-\frac {3 \left (\frac {3 d x}{4}+c \right ) d^{2}}{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}+6 d \left (\left (d x +c \right )^{2} b^{2}-\frac {d^{2}}{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )-3 b x d \left (\left (\frac {1}{3} x^{2} d^{2}+c d x +c^{2}\right ) b^{2}-\frac {d^{2}}{2}\right )}{4 b^{4} \left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )^{2}}\) | \(213\) |
norman | \(\frac {\frac {\left (2 b^{2} c^{3}-3 c \,d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{b^{3}}-\frac {d^{3} x^{3}}{4 b}-\frac {3 c \,d^{2} x^{2}}{4 b}+\frac {3 d \left (2 b^{2} c^{2}-d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{4 b^{4}}-\frac {3 d \left (2 b^{2} c^{2}-d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{4 b^{4}}-\frac {3 d \left (2 b^{2} c^{2}-d^{2}\right ) x}{8 b^{3}}+\frac {3 d^{3} x^{3} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{2 b}-\frac {d^{3} x^{3} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{4 b}+\frac {3 d^{3} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{2 b^{2}}-\frac {3 d^{3} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{2 b^{2}}+\frac {9 c \,d^{2} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{2 b}-\frac {3 c \,d^{2} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{4 b}+\frac {3 c \,d^{2} x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{b^{2}}-\frac {3 c \,d^{2} x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{b^{2}}+\frac {9 d \left (2 b^{2} c^{2}-d^{2}\right ) x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{4 b^{3}}-\frac {3 d \left (2 b^{2} c^{2}-d^{2}\right ) x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{8 b^{3}}}{\left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )^{2}}\) | \(388\) |
derivativedivides | \(\frac {\frac {a^{3} d^{3} \cos \left (x b +a \right )^{2}}{2 b^{3}}-\frac {3 a^{2} c \,d^{2} \cos \left (x b +a \right )^{2}}{2 b^{2}}+\frac {3 a^{2} d^{3} \left (-\frac {\left (x b +a \right ) \cos \left (x b +a \right )^{2}}{2}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{4}+\frac {x b}{4}+\frac {a}{4}\right )}{b^{3}}+\frac {3 a \,c^{2} d \cos \left (x b +a \right )^{2}}{2 b}-\frac {6 a c \,d^{2} \left (-\frac {\left (x b +a \right ) \cos \left (x b +a \right )^{2}}{2}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{4}+\frac {x b}{4}+\frac {a}{4}\right )}{b^{2}}-\frac {3 a \,d^{3} \left (-\frac {\left (x b +a \right )^{2} \cos \left (x b +a \right )^{2}}{2}+\left (x b +a \right ) \left (\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{2}+\frac {x b}{2}+\frac {a}{2}\right )-\frac {\left (x b +a \right )^{2}}{4}-\frac {\sin \left (x b +a \right )^{2}}{4}\right )}{b^{3}}-\frac {c^{3} \cos \left (x b +a \right )^{2}}{2}+\frac {3 c^{2} d \left (-\frac {\left (x b +a \right ) \cos \left (x b +a \right )^{2}}{2}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{4}+\frac {x b}{4}+\frac {a}{4}\right )}{b}+\frac {3 c \,d^{2} \left (-\frac {\left (x b +a \right )^{2} \cos \left (x b +a \right )^{2}}{2}+\left (x b +a \right ) \left (\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{2}+\frac {x b}{2}+\frac {a}{2}\right )-\frac {\left (x b +a \right )^{2}}{4}-\frac {\sin \left (x b +a \right )^{2}}{4}\right )}{b^{2}}+\frac {d^{3} \left (-\frac {\left (x b +a \right )^{3} \cos \left (x b +a \right )^{2}}{2}+\frac {3 \left (x b +a \right )^{2} \left (\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{2}+\frac {x b}{2}+\frac {a}{2}\right )}{2}+\frac {3 \left (x b +a \right ) \cos \left (x b +a \right )^{2}}{4}-\frac {3 \cos \left (x b +a \right ) \sin \left (x b +a \right )}{8}-\frac {3 x b}{8}-\frac {3 a}{8}-\frac {\left (x b +a \right )^{3}}{2}\right )}{b^{3}}}{b}\) | \(466\) |
default | \(\frac {\frac {a^{3} d^{3} \cos \left (x b +a \right )^{2}}{2 b^{3}}-\frac {3 a^{2} c \,d^{2} \cos \left (x b +a \right )^{2}}{2 b^{2}}+\frac {3 a^{2} d^{3} \left (-\frac {\left (x b +a \right ) \cos \left (x b +a \right )^{2}}{2}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{4}+\frac {x b}{4}+\frac {a}{4}\right )}{b^{3}}+\frac {3 a \,c^{2} d \cos \left (x b +a \right )^{2}}{2 b}-\frac {6 a c \,d^{2} \left (-\frac {\left (x b +a \right ) \cos \left (x b +a \right )^{2}}{2}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{4}+\frac {x b}{4}+\frac {a}{4}\right )}{b^{2}}-\frac {3 a \,d^{3} \left (-\frac {\left (x b +a \right )^{2} \cos \left (x b +a \right )^{2}}{2}+\left (x b +a \right ) \left (\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{2}+\frac {x b}{2}+\frac {a}{2}\right )-\frac {\left (x b +a \right )^{2}}{4}-\frac {\sin \left (x b +a \right )^{2}}{4}\right )}{b^{3}}-\frac {c^{3} \cos \left (x b +a \right )^{2}}{2}+\frac {3 c^{2} d \left (-\frac {\left (x b +a \right ) \cos \left (x b +a \right )^{2}}{2}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{4}+\frac {x b}{4}+\frac {a}{4}\right )}{b}+\frac {3 c \,d^{2} \left (-\frac {\left (x b +a \right )^{2} \cos \left (x b +a \right )^{2}}{2}+\left (x b +a \right ) \left (\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{2}+\frac {x b}{2}+\frac {a}{2}\right )-\frac {\left (x b +a \right )^{2}}{4}-\frac {\sin \left (x b +a \right )^{2}}{4}\right )}{b^{2}}+\frac {d^{3} \left (-\frac {\left (x b +a \right )^{3} \cos \left (x b +a \right )^{2}}{2}+\frac {3 \left (x b +a \right )^{2} \left (\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{2}+\frac {x b}{2}+\frac {a}{2}\right )}{2}+\frac {3 \left (x b +a \right ) \cos \left (x b +a \right )^{2}}{4}-\frac {3 \cos \left (x b +a \right ) \sin \left (x b +a \right )}{8}-\frac {3 x b}{8}-\frac {3 a}{8}-\frac {\left (x b +a \right )^{3}}{2}\right )}{b^{3}}}{b}\) | \(466\) |
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Time = 0.24 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.38 \[ \int (c+d x)^3 \cos (a+b x) \sin (a+b x) \, dx=\frac {2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} - 2 \, {\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 2 \, b^{3} c^{3} - 3 \, b c d^{2} + 3 \, {\left (2 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{2} + 3 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 3 \, {\left (2 \, b^{3} c^{2} d - b d^{3}\right )} x}{8 \, b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (116) = 232\).
Time = 0.34 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.85 \[ \int (c+d x)^3 \cos (a+b x) \sin (a+b x) \, dx=\begin {cases} \frac {c^{3} \sin ^{2}{\left (a + b x \right )}}{2 b} + \frac {3 c^{2} d x \sin ^{2}{\left (a + b x \right )}}{4 b} - \frac {3 c^{2} d x \cos ^{2}{\left (a + b x \right )}}{4 b} + \frac {3 c d^{2} x^{2} \sin ^{2}{\left (a + b x \right )}}{4 b} - \frac {3 c d^{2} x^{2} \cos ^{2}{\left (a + b x \right )}}{4 b} + \frac {d^{3} x^{3} \sin ^{2}{\left (a + b x \right )}}{4 b} - \frac {d^{3} x^{3} \cos ^{2}{\left (a + b x \right )}}{4 b} + \frac {3 c^{2} d \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{4 b^{2}} + \frac {3 c d^{2} x \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{2 b^{2}} + \frac {3 d^{3} x^{2} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{4 b^{2}} - \frac {3 c d^{2} \sin ^{2}{\left (a + b x \right )}}{4 b^{3}} - \frac {3 d^{3} x \sin ^{2}{\left (a + b x \right )}}{8 b^{3}} + \frac {3 d^{3} x \cos ^{2}{\left (a + b x \right )}}{8 b^{3}} - \frac {3 d^{3} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \sin {\left (a \right )} \cos {\left (a \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (108) = 216\).
Time = 0.24 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.85 \[ \int (c+d x)^3 \cos (a+b x) \sin (a+b x) \, dx=-\frac {8 \, c^{3} \cos \left (b x + a\right )^{2} - \frac {24 \, a c^{2} d \cos \left (b x + a\right )^{2}}{b} + \frac {24 \, a^{2} c d^{2} \cos \left (b x + a\right )^{2}}{b^{2}} - \frac {8 \, a^{3} d^{3} \cos \left (b x + a\right )^{2}}{b^{3}} + \frac {6 \, {\left (2 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right )\right )} c^{2} d}{b} - \frac {12 \, {\left (2 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right )\right )} a c d^{2}}{b^{2}} + \frac {6 \, {\left (2 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right )\right )} a^{2} d^{3}}{b^{3}} + \frac {6 \, {\left ({\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} c d^{2}}{b^{2}} - \frac {6 \, {\left ({\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} a d^{3}}{b^{3}} + \frac {{\left (2 \, {\left (2 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \cos \left (2 \, b x + 2 \, a\right ) - 3 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} d^{3}}{b^{3}}}{16 \, b} \]
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Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.01 \[ \int (c+d x)^3 \cos (a+b x) \sin (a+b x) \, dx=-\frac {{\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 6 \, b^{3} c^{2} d x + 2 \, b^{3} c^{3} - 3 \, b d^{3} x - 3 \, b c d^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )}{8 \, b^{4}} + \frac {3 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \sin \left (2 \, b x + 2 \, a\right )}{16 \, b^{4}} \]
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Time = 22.51 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.38 \[ \int (c+d x)^3 \cos (a+b x) \sin (a+b x) \, dx=\frac {\cos \left (2\,a+2\,b\,x\right )\,\left (\frac {3\,c\,d^2}{4}-\frac {b^2\,c^3}{2}\right )}{2\,b^3}-\frac {3\,\sin \left (2\,a+2\,b\,x\right )\,\left (d^3-2\,b^2\,c^2\,d\right )}{16\,b^4}-\frac {d^3\,x^3\,\cos \left (2\,a+2\,b\,x\right )}{4\,b}+\frac {3\,d^3\,x^2\,\sin \left (2\,a+2\,b\,x\right )}{8\,b^2}+\frac {3\,x\,\cos \left (2\,a+2\,b\,x\right )\,\left (d^3-2\,b^2\,c^2\,d\right )}{8\,b^3}+\frac {3\,c\,d^2\,x\,\sin \left (2\,a+2\,b\,x\right )}{4\,b^2}-\frac {3\,c\,d^2\,x^2\,\cos \left (2\,a+2\,b\,x\right )}{4\,b} \]
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