Integrand size = 24, antiderivative size = 79 \[ \int (c+d x)^2 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {(c+d x)^3}{24 d}-\frac {d (c+d x) \cos (4 a+4 b x)}{64 b^2}+\frac {d^2 \sin (4 a+4 b x)}{256 b^3}-\frac {(c+d x)^2 \sin (4 a+4 b x)}{32 b} \]
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Time = 0.14 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4491, 3377, 2717} \[ \int (c+d x)^2 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {d^2 \sin (4 a+4 b x)}{256 b^3}-\frac {d (c+d x) \cos (4 a+4 b x)}{64 b^2}-\frac {(c+d x)^2 \sin (4 a+4 b x)}{32 b}+\frac {(c+d x)^3}{24 d} \]
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Rule 2717
Rule 3377
Rule 4491
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{8} (c+d x)^2-\frac {1}{8} (c+d x)^2 \cos (4 a+4 b x)\right ) \, dx \\ & = \frac {(c+d x)^3}{24 d}-\frac {1}{8} \int (c+d x)^2 \cos (4 a+4 b x) \, dx \\ & = \frac {(c+d x)^3}{24 d}-\frac {(c+d x)^2 \sin (4 a+4 b x)}{32 b}+\frac {d \int (c+d x) \sin (4 a+4 b x) \, dx}{16 b} \\ & = \frac {(c+d x)^3}{24 d}-\frac {d (c+d x) \cos (4 a+4 b x)}{64 b^2}-\frac {(c+d x)^2 \sin (4 a+4 b x)}{32 b}+\frac {d^2 \int \cos (4 a+4 b x) \, dx}{64 b^2} \\ & = \frac {(c+d x)^3}{24 d}-\frac {d (c+d x) \cos (4 a+4 b x)}{64 b^2}+\frac {d^2 \sin (4 a+4 b x)}{256 b^3}-\frac {(c+d x)^2 \sin (4 a+4 b x)}{32 b} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.97 \[ \int (c+d x)^2 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {32 b^3 x \left (3 c^2+3 c d x+d^2 x^2\right )-12 b d (c+d x) \cos (4 (a+b x))-3 \left (-d^2+8 b^2 (c+d x)^2\right ) \sin (4 (a+b x))}{768 b^3} \]
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Time = 1.46 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(\frac {\left (-8 \left (d x +c \right )^{2} b^{2}+d^{2}\right ) \sin \left (4 x b +4 a \right )+32 b \left (-\frac {d \left (d x +c \right ) \cos \left (4 x b +4 a \right )}{8}+x \left (\frac {1}{3} x^{2} d^{2}+c d x +c^{2}\right ) b^{2}+\frac {c d}{8}\right )}{256 b^{3}}\) | \(79\) |
risch | \(\frac {d^{2} x^{3}}{24}+\frac {c d \,x^{2}}{8}+\frac {c^{2} x}{8}+\frac {c^{3}}{24 d}-\frac {d \left (d x +c \right ) \cos \left (4 x b +4 a \right )}{64 b^{2}}-\frac {\left (8 x^{2} d^{2} b^{2}+16 b^{2} c d x +8 b^{2} c^{2}-d^{2}\right ) \sin \left (4 x b +4 a \right )}{256 b^{3}}\) | \(98\) |
derivativedivides | \(\frac {\frac {a^{2} d^{2} \left (-\frac {\cos \left (x b +a \right )^{3} \sin \left (x b +a \right )}{4}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{8}+\frac {x b}{8}+\frac {a}{8}\right )}{b^{2}}-\frac {2 a c d \left (-\frac {\cos \left (x b +a \right )^{3} \sin \left (x b +a \right )}{4}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{8}+\frac {x b}{8}+\frac {a}{8}\right )}{b}-\frac {2 a \,d^{2} \left (\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}}{16}+\frac {\sin \left (x b +a \right )^{2}}{4}-\left (x b +a \right ) \left (-\frac {\left (\sin \left (x b +a \right )^{3}+\frac {3 \sin \left (x b +a \right )}{2}\right ) \cos \left (x b +a \right )}{4}+\frac {3 x b}{8}+\frac {3 a}{8}\right )-\frac {\left (2 \cos \left (x b +a \right )^{2}-5\right )^{2}}{64}\right )}{b^{2}}+c^{2} \left (-\frac {\cos \left (x b +a \right )^{3} \sin \left (x b +a \right )}{4}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{8}+\frac {x b}{8}+\frac {a}{8}\right )+\frac {2 c d \left (\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}}{16}+\frac {\sin \left (x b +a \right )^{2}}{4}-\left (x b +a \right ) \left (-\frac {\left (\sin \left (x b +a \right )^{3}+\frac {3 \sin \left (x b +a \right )}{2}\right ) \cos \left (x b +a \right )}{4}+\frac {3 x b}{8}+\frac {3 a}{8}\right )-\frac {\left (2 \cos \left (x b +a \right )^{2}-5\right )^{2}}{64}\right )}{b}+\frac {d^{2} \left (\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 )-\frac {\left (x b +a \right ) \cos \left (x b +a \right )^{2}}{8}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{16}+\frac {7 x b}{64}+\frac {7 a}{64}-\frac {\left (x b +a \right )^{3}}{12}-\left (x b +a \right )^{2} \left (-\frac {\left (\sin \left (x b +a \right )^{3}+\frac {3 \sin \left (x b +a \right )}{2}\right ) \cos \left (x b +a \right )}{4}+\frac {3 x b}{8}+\frac {3 a}{8}\right )-\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{4}}{8}-\frac {\left (\sin \left (x b +a \right )^{3}+\frac {3 \sin \left (x b +a \right )}{2}\right ) \cos \left (x b +a \right )}{32}\right )}{b^{2}}}{b}\) | \(531\) |
default | \(\frac {\frac {a^{2} d^{2} \left (-\frac {\cos \left (x b +a \right )^{3} \sin \left (x b +a \right )}{4}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{8}+\frac {x b}{8}+\frac {a}{8}\right )}{b^{2}}-\frac {2 a c d \left (-\frac {\cos \left (x b +a \right )^{3} \sin \left (x b +a \right )}{4}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{8}+\frac {x b}{8}+\frac {a}{8}\right )}{b}-\frac {2 a \,d^{2} \left (\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}}{16}+\frac {\sin \left (x b +a \right )^{2}}{4}-\left (x b +a \right ) \left (-\frac {\left (\sin \left (x b +a \right )^{3}+\frac {3 \sin \left (x b +a \right )}{2}\right ) \cos \left (x b +a \right )}{4}+\frac {3 x b}{8}+\frac {3 a}{8}\right )-\frac {\left (2 \cos \left (x b +a \right )^{2}-5\right )^{2}}{64}\right )}{b^{2}}+c^{2} \left (-\frac {\cos \left (x b +a \right )^{3} \sin \left (x b +a \right )}{4}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{8}+\frac {x b}{8}+\frac {a}{8}\right )+\frac {2 c d \left (\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}}{16}+\frac {\sin \left (x b +a \right )^{2}}{4}-\left (x b +a \right ) \left (-\frac {\left (\sin \left (x b +a \right )^{3}+\frac {3 \sin \left (x b +a \right )}{2}\right ) \cos \left (x b +a \right )}{4}+\frac {3 x b}{8}+\frac {3 a}{8}\right )-\frac {\left (2 \cos \left (x b +a \right )^{2}-5\right )^{2}}{64}\right )}{b}+\frac {d^{2} \left (\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 )-\frac {\left (x b +a \right ) \cos \left (x b +a \right )^{2}}{8}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{16}+\frac {7 x b}{64}+\frac {7 a}{64}-\frac {\left (x b +a \right )^{3}}{12}-\left (x b +a \right )^{2} \left (-\frac {\left (\sin \left (x b +a \right )^{3}+\frac {3 \sin \left (x b +a \right )}{2}\right ) \cos \left (x b +a \right )}{4}+\frac {3 x b}{8}+\frac {3 a}{8}\right )-\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{4}}{8}-\frac {\left (\sin \left (x b +a \right )^{3}+\frac {3 \sin \left (x b +a \right )}{2}\right ) \cos \left (x b +a \right )}{32}\right )}{b^{2}}}{b}\) | \(531\) |
norman | \(\frac {\frac {c d x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{7}}{2 b}+\frac {\left (8 b^{2} c^{2}+7 d^{2}\right ) x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{16 b^{2}}+\frac {\left (24 b^{2} c^{2}-35 d^{2}\right ) x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{32 b^{2}}+\frac {\left (8 b^{2} c^{2}+7 d^{2}\right ) x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{16 b^{2}}+\frac {\left (8 b^{2} c^{2}-d^{2}\right ) x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{8}}{64 b^{2}}-\frac {d^{2} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{4 b}+\frac {c d \,x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{2}-\frac {c d x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{2 b}+\frac {3 c d \,x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{4}-\frac {7 d^{2} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}}{4 b}+\frac {c d \,x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{2}+\frac {d^{2} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{7}}{4 b}+\frac {c d \,x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{8}}{8}-\frac {c d \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{b^{2}}+\frac {c d \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{2 b^{2}}+\frac {7 c d x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{2 b}-\frac {7 c d x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}}{2 b}+\frac {7 d^{2} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{4 b}+\frac {c d \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{2 b^{2}}+\frac {d^{2} x^{3} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{6}+\frac {d^{2} x^{3} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{4}+\frac {d^{2} x^{3} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{6}+\frac {d^{2} x^{3} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{8}}{24}+\frac {c d \,x^{2}}{8}-\frac {\left (8 b^{2} c^{2}-d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{32 b^{3}}+\frac {7 \left (8 b^{2} c^{2}-d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{32 b^{3}}-\frac {7 \left (8 b^{2} c^{2}-d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}}{32 b^{3}}+\frac {\left (8 b^{2} c^{2}-d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{7}}{32 b^{3}}+\frac {\left (8 b^{2} c^{2}-d^{2}\right ) x}{64 b^{2}}+\frac {d^{2} x^{3}}{24}}{\left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )^{4}}\) | \(657\) |
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Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (71) = 142\).
Time = 0.26 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.28 \[ \int (c+d x)^2 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {8 \, b^{3} d^{2} x^{3} + 24 \, b^{3} c d x^{2} - 24 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{4} + 24 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{2} + 3 \, {\left (8 \, b^{3} c^{2} - b d^{2}\right )} x - 3 \, {\left (2 \, {\left (8 \, b^{2} d^{2} x^{2} + 16 \, b^{2} c d x + 8 \, b^{2} c^{2} - d^{2}\right )} \cos \left (b x + a\right )^{3} - {\left (8 \, b^{2} d^{2} x^{2} + 16 \, b^{2} c d x + 8 \, b^{2} c^{2} - d^{2}\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{192 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (70) = 140\).
Time = 0.47 (sec) , antiderivative size = 493, normalized size of antiderivative = 6.24 \[ \int (c+d x)^2 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\begin {cases} \frac {c^{2} x \sin ^{4}{\left (a + b x \right )}}{8} + \frac {c^{2} x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} + \frac {c^{2} x \cos ^{4}{\left (a + b x \right )}}{8} + \frac {c d x^{2} \sin ^{4}{\left (a + b x \right )}}{8} + \frac {c d x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} + \frac {c d x^{2} \cos ^{4}{\left (a + b x \right )}}{8} + \frac {d^{2} x^{3} \sin ^{4}{\left (a + b x \right )}}{24} + \frac {d^{2} x^{3} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{12} + \frac {d^{2} x^{3} \cos ^{4}{\left (a + b x \right )}}{24} + \frac {c^{2} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} - \frac {c^{2} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} + \frac {c d x \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{4 b} - \frac {c d x \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{4 b} + \frac {d^{2} x^{2} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} - \frac {d^{2} x^{2} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} - \frac {c d \sin ^{4}{\left (a + b x \right )}}{16 b^{2}} - \frac {c d \cos ^{4}{\left (a + b x \right )}}{16 b^{2}} - \frac {d^{2} x \sin ^{4}{\left (a + b x \right )}}{64 b^{2}} + \frac {3 d^{2} x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32 b^{2}} - \frac {d^{2} x \cos ^{4}{\left (a + b x \right )}}{64 b^{2}} - \frac {d^{2} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{64 b^{3}} + \frac {d^{2} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{64 b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \sin ^{2}{\left (a \right )} \cos ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (71) = 142\).
Time = 0.24 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.94 \[ \int (c+d x)^2 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {24 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} c^{2} - \frac {48 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a c d}{b} + \frac {24 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a^{2} d^{2}}{b^{2}} + \frac {12 \, {\left (8 \, {\left (b x + a\right )}^{2} - 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )\right )} c d}{b} - \frac {12 \, {\left (8 \, {\left (b x + a\right )}^{2} - 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )\right )} a d^{2}}{b^{2}} + \frac {{\left (32 \, {\left (b x + a\right )}^{3} - 12 \, {\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) - 3 \, {\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (4 \, b x + 4 \, a\right )\right )} d^{2}}{b^{2}}}{768 \, b} \]
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Time = 0.31 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.19 \[ \int (c+d x)^2 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {1}{24} \, d^{2} x^{3} + \frac {1}{8} \, c d x^{2} + \frac {1}{8} \, c^{2} x - \frac {{\left (b d^{2} x + b c d\right )} \cos \left (4 \, b x + 4 \, a\right )}{64 \, b^{3}} - \frac {{\left (8 \, b^{2} d^{2} x^{2} + 16 \, b^{2} c d x + 8 \, b^{2} c^{2} - d^{2}\right )} \sin \left (4 \, b x + 4 \, a\right )}{256 \, b^{3}} \]
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Time = 23.13 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.27 \[ \int (c+d x)^2 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=x\,\left (\frac {c^2}{32}+\frac {3\,d^2}{256\,b^2}\right )+x\,\left (\frac {3\,c^2}{32}-\frac {3\,d^2}{256\,b^2}\right )+\frac {d^2\,x^3}{24}+\frac {\sin \left (4\,a+4\,b\,x\right )\,\left (d^2-8\,b^2\,c^2\right )}{256\,b^3}-\frac {x\,\cos \left (4\,a+4\,b\,x\right )\,\left (\frac {c^2}{4}+\frac {3\,d^2}{32\,b^2}\right )}{8}+\frac {x\,\cos \left (4\,a+4\,b\,x\right )\,\left (\frac {c^2}{8}-\frac {d^2}{64\,b^2}\right )}{4}+\frac {c\,d\,x^2}{8}-\frac {d^2\,x^2\,\sin \left (4\,a+4\,b\,x\right )}{32\,b}-\frac {c\,d\,\cos \left (4\,a+4\,b\,x\right )}{64\,b^2}-\frac {c\,d\,x\,\sin \left (4\,a+4\,b\,x\right )}{16\,b} \]
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