Integrand size = 11, antiderivative size = 46 \[ \int \left (a-a \sin ^2(x)\right )^3 \, dx=\frac {5 a^3 x}{16}+\frac {5}{16} a^3 \cos (x) \sin (x)+\frac {5}{24} a^3 \cos ^3(x) \sin (x)+\frac {1}{6} a^3 \cos ^5(x) \sin (x) \]
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Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3254, 2715, 8} \[ \int \left (a-a \sin ^2(x)\right )^3 \, dx=\frac {5 a^3 x}{16}+\frac {1}{6} a^3 \sin (x) \cos ^5(x)+\frac {5}{24} a^3 \sin (x) \cos ^3(x)+\frac {5}{16} a^3 \sin (x) \cos (x) \]
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Rule 8
Rule 2715
Rule 3254
Rubi steps \begin{align*} \text {integral}& = a^3 \int \cos ^6(x) \, dx \\ & = \frac {1}{6} a^3 \cos ^5(x) \sin (x)+\frac {1}{6} \left (5 a^3\right ) \int \cos ^4(x) \, dx \\ & = \frac {5}{24} a^3 \cos ^3(x) \sin (x)+\frac {1}{6} a^3 \cos ^5(x) \sin (x)+\frac {1}{8} \left (5 a^3\right ) \int \cos ^2(x) \, dx \\ & = \frac {5}{16} a^3 \cos (x) \sin (x)+\frac {5}{24} a^3 \cos ^3(x) \sin (x)+\frac {1}{6} a^3 \cos ^5(x) \sin (x)+\frac {1}{16} \left (5 a^3\right ) \int 1 \, dx \\ & = \frac {5 a^3 x}{16}+\frac {5}{16} a^3 \cos (x) \sin (x)+\frac {5}{24} a^3 \cos ^3(x) \sin (x)+\frac {1}{6} a^3 \cos ^5(x) \sin (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.74 \[ \int \left (a-a \sin ^2(x)\right )^3 \, dx=a^3 \left (\frac {5 x}{16}+\frac {15}{64} \sin (2 x)+\frac {3}{64} \sin (4 x)+\frac {1}{192} \sin (6 x)\right ) \]
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Time = 1.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.57
method | result | size |
parallelrisch | \(\frac {a^{3} \left (60 x +45 \sin \left (2 x \right )+\sin \left (6 x \right )+9 \sin \left (4 x \right )\right )}{192}\) | \(26\) |
risch | \(\frac {5 a^{3} x}{16}+\frac {a^{3} \sin \left (6 x \right )}{192}+\frac {3 a^{3} \sin \left (4 x \right )}{64}+\frac {15 a^{3} \sin \left (2 x \right )}{64}\) | \(35\) |
default | \(-a^{3} \left (-\frac {\left (\sin ^{5}\left (x \right )+\frac {5 \left (\sin ^{3}\left (x \right )\right )}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )}{6}+\frac {5 x}{16}\right )+3 a^{3} \left (-\frac {\left (\sin ^{3}\left (x \right )+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{4}+\frac {3 x}{8}\right )-3 a^{3} \left (-\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )+a^{3} x\) | \(72\) |
parts | \(-a^{3} \left (-\frac {\left (\sin ^{5}\left (x \right )+\frac {5 \left (\sin ^{3}\left (x \right )\right )}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )}{6}+\frac {5 x}{16}\right )+3 a^{3} \left (-\frac {\left (\sin ^{3}\left (x \right )+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{4}+\frac {3 x}{8}\right )-3 a^{3} \left (-\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )+a^{3} x\) | \(72\) |
norman | \(\frac {\frac {5 a^{3} x}{16}+\frac {11 a^{3} \tan \left (\frac {x}{2}\right )}{8}-\frac {5 a^{3} \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{24}+\frac {15 a^{3} \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{4}-\frac {15 a^{3} \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{4}+\frac {5 a^{3} \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{24}-\frac {11 a^{3} \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{8}+\frac {15 a^{3} x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {75 a^{3} x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{16}+\frac {25 a^{3} x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{4}+\frac {75 a^{3} x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{16}+\frac {15 a^{3} x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{8}+\frac {5 a^{3} x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{16}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{6}}\) | \(155\) |
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80 \[ \int \left (a-a \sin ^2(x)\right )^3 \, dx=\frac {5}{16} \, a^{3} x + \frac {1}{48} \, {\left (8 \, a^{3} \cos \left (x\right )^{5} + 10 \, a^{3} \cos \left (x\right )^{3} + 15 \, a^{3} \cos \left (x\right )\right )} \sin \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (49) = 98\).
Time = 0.26 (sec) , antiderivative size = 233, normalized size of antiderivative = 5.07 \[ \int \left (a-a \sin ^2(x)\right )^3 \, dx=- \frac {5 a^{3} x \sin ^{6}{\left (x \right )}}{16} - \frac {15 a^{3} x \sin ^{4}{\left (x \right )} \cos ^{2}{\left (x \right )}}{16} + \frac {9 a^{3} x \sin ^{4}{\left (x \right )}}{8} - \frac {15 a^{3} x \sin ^{2}{\left (x \right )} \cos ^{4}{\left (x \right )}}{16} + \frac {9 a^{3} x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{4} - \frac {3 a^{3} x \sin ^{2}{\left (x \right )}}{2} - \frac {5 a^{3} x \cos ^{6}{\left (x \right )}}{16} + \frac {9 a^{3} x \cos ^{4}{\left (x \right )}}{8} - \frac {3 a^{3} x \cos ^{2}{\left (x \right )}}{2} + a^{3} x + \frac {11 a^{3} \sin ^{5}{\left (x \right )} \cos {\left (x \right )}}{16} + \frac {5 a^{3} \sin ^{3}{\left (x \right )} \cos ^{3}{\left (x \right )}}{6} - \frac {15 a^{3} \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{8} + \frac {5 a^{3} \sin {\left (x \right )} \cos ^{5}{\left (x \right )}}{16} - \frac {9 a^{3} \sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{8} + \frac {3 a^{3} \sin {\left (x \right )} \cos {\left (x \right )}}{2} \]
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Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.50 \[ \int \left (a-a \sin ^2(x)\right )^3 \, dx=-\frac {1}{192} \, {\left (4 \, \sin \left (2 \, x\right )^{3} + 60 \, x + 9 \, \sin \left (4 \, x\right ) - 48 \, \sin \left (2 \, x\right )\right )} a^{3} + \frac {3}{32} \, a^{3} {\left (12 \, x + \sin \left (4 \, x\right ) - 8 \, \sin \left (2 \, x\right )\right )} - \frac {3}{4} \, a^{3} {\left (2 \, x - \sin \left (2 \, x\right )\right )} + a^{3} x \]
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Time = 0.31 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.74 \[ \int \left (a-a \sin ^2(x)\right )^3 \, dx=\frac {5}{16} \, a^{3} x + \frac {1}{192} \, a^{3} \sin \left (6 \, x\right ) + \frac {3}{64} \, a^{3} \sin \left (4 \, x\right ) + \frac {15}{64} \, a^{3} \sin \left (2 \, x\right ) \]
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Time = 14.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.91 \[ \int \left (a-a \sin ^2(x)\right )^3 \, dx=\frac {11\,a^3\,{\cos \left (x\right )}^5\,\sin \left (x\right )}{16}+\frac {5\,a^3\,{\cos \left (x\right )}^3\,{\sin \left (x\right )}^3}{6}+\frac {5\,a^3\,\cos \left (x\right )\,{\sin \left (x\right )}^5}{16}+\frac {5\,x\,a^3}{16} \]
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