Integrand size = 22, antiderivative size = 103 \[ \int (c+d x)^2 \cos (a+b x) \sin ^2(a+b x) \, dx=\frac {4 d (c+d x) \cos (a+b x)}{9 b^2}-\frac {4 d^2 \sin (a+b x)}{9 b^3}+\frac {2 d (c+d x) \cos (a+b x) \sin ^2(a+b x)}{9 b^2}-\frac {2 d^2 \sin ^3(a+b x)}{27 b^3}+\frac {(c+d x)^2 \sin ^3(a+b x)}{3 b} \]
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Time = 0.10 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4489, 3391, 3377, 2717} \[ \int (c+d x)^2 \cos (a+b x) \sin ^2(a+b x) \, dx=-\frac {2 d^2 \sin ^3(a+b x)}{27 b^3}-\frac {4 d^2 \sin (a+b x)}{9 b^3}+\frac {4 d (c+d x) \cos (a+b x)}{9 b^2}+\frac {2 d (c+d x) \sin ^2(a+b x) \cos (a+b x)}{9 b^2}+\frac {(c+d x)^2 \sin ^3(a+b x)}{3 b} \]
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Rule 2717
Rule 3377
Rule 3391
Rule 4489
Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^2 \sin ^3(a+b x)}{3 b}-\frac {(2 d) \int (c+d x) \sin ^3(a+b x) \, dx}{3 b} \\ & = \frac {2 d (c+d x) \cos (a+b x) \sin ^2(a+b x)}{9 b^2}-\frac {2 d^2 \sin ^3(a+b x)}{27 b^3}+\frac {(c+d x)^2 \sin ^3(a+b x)}{3 b}-\frac {(4 d) \int (c+d x) \sin (a+b x) \, dx}{9 b} \\ & = \frac {4 d (c+d x) \cos (a+b x)}{9 b^2}+\frac {2 d (c+d x) \cos (a+b x) \sin ^2(a+b x)}{9 b^2}-\frac {2 d^2 \sin ^3(a+b x)}{27 b^3}+\frac {(c+d x)^2 \sin ^3(a+b x)}{3 b}-\frac {\left (4 d^2\right ) \int \cos (a+b x) \, dx}{9 b^2} \\ & = \frac {4 d (c+d x) \cos (a+b x)}{9 b^2}-\frac {4 d^2 \sin (a+b x)}{9 b^3}+\frac {2 d (c+d x) \cos (a+b x) \sin ^2(a+b x)}{9 b^2}-\frac {2 d^2 \sin ^3(a+b x)}{27 b^3}+\frac {(c+d x)^2 \sin ^3(a+b x)}{3 b} \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.90 \[ \int (c+d x)^2 \cos (a+b x) \sin ^2(a+b x) \, dx=\frac {54 b d (c+d x) \cos (a+b x)-6 b d (c+d x) \cos (3 (a+b x))-2 \left (26 d^2-9 b^2 (c+d x)^2+\left (-2 d^2+9 b^2 (c+d x)^2\right ) \cos (2 (a+b x))\right ) \sin (a+b x)}{108 b^3} \]
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Time = 1.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.24
method | result | size |
risch | \(\frac {d \left (d x +c \right ) \cos \left (x b +a \right )}{2 b^{2}}+\frac {\left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}-2 d^{2}\right ) \sin \left (x b +a \right )}{4 b^{3}}-\frac {d \left (d x +c \right ) \cos \left (3 x b +3 a \right )}{18 b^{2}}-\frac {\left (9 x^{2} d^{2} b^{2}+18 b^{2} c d x +9 b^{2} c^{2}-2 d^{2}\right ) \sin \left (3 x b +3 a \right )}{108 b^{3}}\) | \(128\) |
parallelrisch | \(\frac {-12 d^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6} x b -24 d^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}-36 d^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4} x b +\left (\left (72 x^{2} b^{2}-64\right ) d^{2}+144 b^{2} c d x +72 b^{2} c^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}+72 b d \left (\frac {d x}{2}+c \right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}-24 d^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+24 b d \left (\frac {d x}{2}+c \right )}{27 b^{3} \left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )^{3}}\) | \(162\) |
derivativedivides | \(\frac {\frac {a^{2} d^{2} \sin \left (x b +a \right )^{3}}{3 b^{2}}-\frac {2 a c d \sin \left (x b +a \right )^{3}}{3 b}-\frac {2 a \,d^{2} \left (\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{3}}{3}+\frac {\left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}\right )}{b^{2}}+\frac {c^{2} \sin \left (x b +a \right )^{3}}{3}+\frac {2 c d \left (\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{3}}{3}+\frac {\left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}\right )}{b}+\frac {d^{2} \left (\frac {\left (x b +a \right )^{2} \sin \left (x b +a \right )^{3}}{3}+\frac {2 \left (x b +a \right ) \left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}-\frac {2 \sin \left (x b +a \right )^{3}}{27}-\frac {4 \sin \left (x b +a \right )}{9}\right )}{b^{2}}}{b}\) | \(204\) |
default | \(\frac {\frac {a^{2} d^{2} \sin \left (x b +a \right )^{3}}{3 b^{2}}-\frac {2 a c d \sin \left (x b +a \right )^{3}}{3 b}-\frac {2 a \,d^{2} \left (\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{3}}{3}+\frac {\left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}\right )}{b^{2}}+\frac {c^{2} \sin \left (x b +a \right )^{3}}{3}+\frac {2 c d \left (\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{3}}{3}+\frac {\left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}\right )}{b}+\frac {d^{2} \left (\frac {\left (x b +a \right )^{2} \sin \left (x b +a \right )^{3}}{3}+\frac {2 \left (x b +a \right ) \left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}-\frac {2 \sin \left (x b +a \right )^{3}}{27}-\frac {4 \sin \left (x b +a \right )}{9}\right )}{b^{2}}}{b}\) | \(204\) |
norman | \(\frac {\frac {8 c d}{9 b^{2}}-\frac {8 d^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{9 b^{3}}-\frac {8 d^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}}{9 b^{3}}+\frac {4 d^{2} x}{9 b^{2}}+\frac {8 \left (9 b^{2} c^{2}-8 d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{27 b^{3}}+\frac {8 c d \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{3 b^{2}}+\frac {4 d^{2} x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{3 b^{2}}-\frac {4 d^{2} x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{3 b^{2}}-\frac {4 d^{2} x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{9 b^{2}}+\frac {8 d^{2} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{3 b}+\frac {16 c d x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{3 b}}{\left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )^{3}}\) | \(219\) |
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Time = 0.26 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.26 \[ \int (c+d x)^2 \cos (a+b x) \sin ^2(a+b x) \, dx=-\frac {6 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{3} - 18 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) - {\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} - {\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} - 2 \, d^{2}\right )} \cos \left (b x + a\right )^{2} - 14 \, d^{2}\right )} \sin \left (b x + a\right )}{27 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (102) = 204\).
Time = 0.32 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.10 \[ \int (c+d x)^2 \cos (a+b x) \sin ^2(a+b x) \, dx=\begin {cases} \frac {c^{2} \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac {2 c d x \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac {d^{2} x^{2} \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac {2 c d \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{3 b^{2}} + \frac {4 c d \cos ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {2 d^{2} x \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{3 b^{2}} + \frac {4 d^{2} x \cos ^{3}{\left (a + b x \right )}}{9 b^{2}} - \frac {14 d^{2} \sin ^{3}{\left (a + b x \right )}}{27 b^{3}} - \frac {4 d^{2} \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{9 b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \sin ^{2}{\left (a \right )} \cos {\left (a \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (93) = 186\).
Time = 0.21 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.33 \[ \int (c+d x)^2 \cos (a+b x) \sin ^2(a+b x) \, dx=\frac {36 \, c^{2} \sin \left (b x + a\right )^{3} - \frac {72 \, a c d \sin \left (b x + a\right )^{3}}{b} + \frac {36 \, a^{2} d^{2} \sin \left (b x + a\right )^{3}}{b^{2}} - \frac {6 \, {\left (3 \, {\left (b x + a\right )} \sin \left (3 \, b x + 3 \, a\right ) - 9 \, {\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (3 \, b x + 3 \, a\right ) - 9 \, \cos \left (b x + a\right )\right )} c d}{b} + \frac {6 \, {\left (3 \, {\left (b x + a\right )} \sin \left (3 \, b x + 3 \, a\right ) - 9 \, {\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (3 \, b x + 3 \, a\right ) - 9 \, \cos \left (b x + a\right )\right )} a d^{2}}{b^{2}} - \frac {{\left (6 \, {\left (b x + a\right )} \cos \left (3 \, b x + 3 \, a\right ) - 54 \, {\left (b x + a\right )} \cos \left (b x + a\right ) + {\left (9 \, {\left (b x + a\right )}^{2} - 2\right )} \sin \left (3 \, b x + 3 \, a\right ) - 27 \, {\left ({\left (b x + a\right )}^{2} - 2\right )} \sin \left (b x + a\right )\right )} d^{2}}{b^{2}}}{108 \, b} \]
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Time = 0.30 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.33 \[ \int (c+d x)^2 \cos (a+b x) \sin ^2(a+b x) \, dx=-\frac {{\left (b d^{2} x + b c d\right )} \cos \left (3 \, b x + 3 \, a\right )}{18 \, b^{3}} + \frac {{\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )}{2 \, b^{3}} - \frac {{\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} - 2 \, d^{2}\right )} \sin \left (3 \, b x + 3 \, a\right )}{108 \, b^{3}} + \frac {{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, d^{2}\right )} \sin \left (b x + a\right )}{4 \, b^{3}} \]
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Time = 22.97 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.56 \[ \int (c+d x)^2 \cos (a+b x) \sin ^2(a+b x) \, dx=\frac {4\,d^2\,x\,{\cos \left (a+b\,x\right )}^3}{9\,b^2}-\frac {4\,d^2\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{9\,b^3}-\frac {{\sin \left (a+b\,x\right )}^3\,\left (14\,d^2-9\,b^2\,c^2\right )}{27\,b^3}+\frac {d^2\,x^2\,{\sin \left (a+b\,x\right )}^3}{3\,b}+\frac {4\,c\,d\,{\cos \left (a+b\,x\right )}^3}{9\,b^2}+\frac {2\,c\,d\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{3\,b^2}+\frac {2\,c\,d\,x\,{\sin \left (a+b\,x\right )}^3}{3\,b}+\frac {2\,d^2\,x\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{3\,b^2} \]
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