Integrand size = 10, antiderivative size = 87 \[ \int \left (a+b \sin ^2(x)\right )^3 \, dx=\frac {1}{16} (2 a+b) \left (8 a^2+8 a b+5 b^2\right ) x-\frac {1}{48} b \left (64 a^2+54 a b+15 b^2\right ) \cos (x) \sin (x)-\frac {5}{24} b^2 (2 a+b) \cos (x) \sin ^3(x)-\frac {1}{6} b \cos (x) \sin (x) \left (a+b \sin ^2(x)\right )^2 \]
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Time = 0.05 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3259, 3248} \[ \int \left (a+b \sin ^2(x)\right )^3 \, dx=\frac {1}{16} x (2 a+b) \left (8 a^2+8 a b+5 b^2\right )-\frac {1}{48} b \left (64 a^2+54 a b+15 b^2\right ) \sin (x) \cos (x)-\frac {5}{24} b^2 (2 a+b) \sin ^3(x) \cos (x)-\frac {1}{6} b \sin (x) \cos (x) \left (a+b \sin ^2(x)\right )^2 \]
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Rule 3248
Rule 3259
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{6} b \cos (x) \sin (x) \left (a+b \sin ^2(x)\right )^2+\frac {1}{6} \int \left (a+b \sin ^2(x)\right ) \left (a (6 a+b)+5 b (2 a+b) \sin ^2(x)\right ) \, dx \\ & = \frac {1}{16} (2 a+b) \left (8 a^2+8 a b+5 b^2\right ) x-\frac {1}{48} b \left (64 a^2+54 a b+15 b^2\right ) \cos (x) \sin (x)-\frac {5}{24} b^2 (2 a+b) \cos (x) \sin ^3(x)-\frac {1}{6} b \cos (x) \sin (x) \left (a+b \sin ^2(x)\right )^2 \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.92 \[ \int \left (a+b \sin ^2(x)\right )^3 \, dx=\frac {1}{192} \left (12 (2 a+b) \left (8 a^2+8 a b+5 b^2\right ) x+9 i b (4 i a+(1+2 i) b) (4 a+(2+i) b) \sin (2 x)+9 b^2 (2 a+b) \sin (4 x)-b^3 \sin (6 x)\right ) \]
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Time = 1.01 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.80
method | result | size |
parallelrisch | \(\frac {3 \left (-16 a^{2} b -16 a \,b^{2}-5 b^{3}\right ) \sin \left (2 x \right )}{64}+\frac {3 \left (2 a \,b^{2}+b^{3}\right ) \sin \left (4 x \right )}{64}-\frac {b^{3} \sin \left (6 x \right )}{192}+\left (a^{2}+a b +\frac {5}{8} b^{2}\right ) \left (a +\frac {b}{2}\right ) x\) | \(70\) |
default | \(b^{3} \left (-\frac {\left (\sin ^{5}\left (x \right )+\frac {5 \left (\sin ^{3}\left (x \right )\right )}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )}{6}+\frac {5 x}{16}\right )+3 a \,b^{2} \left (-\frac {\left (\sin ^{3}\left (x \right )+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{4}+\frac {3 x}{8}\right )+3 a^{2} b \left (-\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )+a^{3} x\) | \(73\) |
parts | \(b^{3} \left (-\frac {\left (\sin ^{5}\left (x \right )+\frac {5 \left (\sin ^{3}\left (x \right )\right )}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )}{6}+\frac {5 x}{16}\right )+3 a \,b^{2} \left (-\frac {\left (\sin ^{3}\left (x \right )+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{4}+\frac {3 x}{8}\right )+3 a^{2} b \left (-\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )+a^{3} x\) | \(73\) |
risch | \(a^{3} x +\frac {3 a^{2} b x}{2}+\frac {9 a \,b^{2} x}{8}+\frac {5 b^{3} x}{16}-\frac {b^{3} \sin \left (6 x \right )}{192}+\frac {3 \sin \left (4 x \right ) a \,b^{2}}{32}+\frac {3 \sin \left (4 x \right ) b^{3}}{64}-\frac {3 \sin \left (2 x \right ) a^{2} b}{4}-\frac {3 \sin \left (2 x \right ) a \,b^{2}}{4}-\frac {15 \sin \left (2 x \right ) b^{3}}{64}\) | \(84\) |
norman | \(\frac {\left (-9 a^{2} b -\frac {51}{4} a \,b^{2}-\frac {85}{24} b^{3}\right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )+\left (-6 a^{2} b -\frac {21}{2} a \,b^{2}-\frac {33}{4} b^{3}\right ) \left (\tan ^{5}\left (\frac {x}{2}\right )\right )+\left (-3 a^{2} b -\frac {9}{4} a \,b^{2}-\frac {5}{8} b^{3}\right ) \tan \left (\frac {x}{2}\right )+\left (3 a^{2} b +\frac {9}{4} a \,b^{2}+\frac {5}{8} b^{3}\right ) \left (\tan ^{11}\left (\frac {x}{2}\right )\right )+\left (6 a^{2} b +\frac {21}{2} a \,b^{2}+\frac {33}{4} b^{3}\right ) \left (\tan ^{7}\left (\frac {x}{2}\right )\right )+\left (9 a^{2} b +\frac {51}{4} a \,b^{2}+\frac {85}{24} b^{3}\right ) \left (\tan ^{9}\left (\frac {x}{2}\right )\right )+\left (a^{3}+\frac {3}{2} a^{2} b +\frac {9}{8} a \,b^{2}+\frac {5}{16} b^{3}\right ) x +\left (a^{3}+\frac {3}{2} a^{2} b +\frac {9}{8} a \,b^{2}+\frac {5}{16} b^{3}\right ) x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )+\left (6 a^{3}+9 a^{2} b +\frac {27}{4} a \,b^{2}+\frac {15}{8} b^{3}\right ) x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\left (6 a^{3}+9 a^{2} b +\frac {27}{4} a \,b^{2}+\frac {15}{8} b^{3}\right ) x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )+\left (15 a^{3}+\frac {45}{2} a^{2} b +\frac {135}{8} a \,b^{2}+\frac {75}{16} b^{3}\right ) x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+\left (15 a^{3}+\frac {45}{2} a^{2} b +\frac {135}{8} a \,b^{2}+\frac {75}{16} b^{3}\right ) x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )+\left (20 a^{3}+30 a^{2} b +\frac {45}{2} a \,b^{2}+\frac {25}{4} b^{3}\right ) x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{6}}\) | \(368\) |
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Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.93 \[ \int \left (a+b \sin ^2(x)\right )^3 \, dx=\frac {1}{16} \, {\left (16 \, a^{3} + 24 \, a^{2} b + 18 \, a b^{2} + 5 \, b^{3}\right )} x - \frac {1}{48} \, {\left (8 \, b^{3} \cos \left (x\right )^{5} - 2 \, {\left (18 \, a b^{2} + 13 \, b^{3}\right )} \cos \left (x\right )^{3} + 3 \, {\left (24 \, a^{2} b + 30 \, a b^{2} + 11 \, b^{3}\right )} \cos \left (x\right )\right )} \sin \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (88) = 176\).
Time = 0.27 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.83 \[ \int \left (a+b \sin ^2(x)\right )^3 \, dx=a^{3} x + \frac {3 a^{2} b x \sin ^{2}{\left (x \right )}}{2} + \frac {3 a^{2} b x \cos ^{2}{\left (x \right )}}{2} - \frac {3 a^{2} b \sin {\left (x \right )} \cos {\left (x \right )}}{2} + \frac {9 a b^{2} x \sin ^{4}{\left (x \right )}}{8} + \frac {9 a b^{2} x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{4} + \frac {9 a b^{2} x \cos ^{4}{\left (x \right )}}{8} - \frac {15 a b^{2} \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{8} - \frac {9 a b^{2} \sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{8} + \frac {5 b^{3} x \sin ^{6}{\left (x \right )}}{16} + \frac {15 b^{3} x \sin ^{4}{\left (x \right )} \cos ^{2}{\left (x \right )}}{16} + \frac {15 b^{3} x \sin ^{2}{\left (x \right )} \cos ^{4}{\left (x \right )}}{16} + \frac {5 b^{3} x \cos ^{6}{\left (x \right )}}{16} - \frac {11 b^{3} \sin ^{5}{\left (x \right )} \cos {\left (x \right )}}{16} - \frac {5 b^{3} \sin ^{3}{\left (x \right )} \cos ^{3}{\left (x \right )}}{6} - \frac {5 b^{3} \sin {\left (x \right )} \cos ^{5}{\left (x \right )}}{16} \]
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Time = 0.24 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.82 \[ \int \left (a+b \sin ^2(x)\right )^3 \, dx=\frac {1}{192} \, {\left (4 \, \sin \left (2 \, x\right )^{3} + 60 \, x + 9 \, \sin \left (4 \, x\right ) - 48 \, \sin \left (2 \, x\right )\right )} b^{3} + \frac {3}{32} \, a b^{2} {\left (12 \, x + \sin \left (4 \, x\right ) - 8 \, \sin \left (2 \, x\right )\right )} + \frac {3}{4} \, a^{2} b {\left (2 \, x - \sin \left (2 \, x\right )\right )} + a^{3} x \]
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Time = 0.38 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87 \[ \int \left (a+b \sin ^2(x)\right )^3 \, dx=-\frac {1}{192} \, b^{3} \sin \left (6 \, x\right ) + \frac {1}{16} \, {\left (16 \, a^{3} + 24 \, a^{2} b + 18 \, a b^{2} + 5 \, b^{3}\right )} x + \frac {3}{64} \, {\left (2 \, a b^{2} + b^{3}\right )} \sin \left (4 \, x\right ) - \frac {3}{64} \, {\left (16 \, a^{2} b + 16 \, a b^{2} + 5 \, b^{3}\right )} \sin \left (2 \, x\right ) \]
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Time = 13.98 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.36 \[ \int \left (a+b \sin ^2(x)\right )^3 \, dx=a^3\,x+\frac {5\,b^3\,x}{16}-\frac {\left (72\,a^2\,b+90\,a\,b^2+33\,b^3\right )\,{\mathrm {tan}\left (x\right )}^5+\left (144\,a^2\,b+144\,a\,b^2+40\,b^3\right )\,{\mathrm {tan}\left (x\right )}^3+\left (72\,a^2\,b+54\,a\,b^2+15\,b^3\right )\,\mathrm {tan}\left (x\right )}{48\,{\mathrm {tan}\left (x\right )}^6+144\,{\mathrm {tan}\left (x\right )}^4+144\,{\mathrm {tan}\left (x\right )}^2+48}+\frac {9\,a\,b^2\,x}{8}+\frac {3\,a^2\,b\,x}{2} \]
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