Integrand size = 11, antiderivative size = 46 \[ \int \left (a-a \sin ^2(x)\right )^3 \, dx=\frac {5 a^3 x}{16}+\frac {5}{16} a^3 \cos (x) \sin (x)+\frac {5}{24} a^3 \cos ^3(x) \sin (x)+\frac {1}{6} a^3 \cos ^5(x) \sin (x) \] Output:
5/16*a^3*x+5/16*a^3*cos(x)*sin(x)+5/24*a^3*cos(x)^3*sin(x)+1/6*a^3*cos(x)^ 5*sin(x)
Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.74 \[ \int \left (a-a \sin ^2(x)\right )^3 \, dx=a^3 \left (\frac {5 x}{16}+\frac {15}{64} \sin (2 x)+\frac {3}{64} \sin (4 x)+\frac {1}{192} \sin (6 x)\right ) \] Input:
Integrate[(a - a*Sin[x]^2)^3,x]
Output:
a^3*((5*x)/16 + (15*Sin[2*x])/64 + (3*Sin[4*x])/64 + Sin[6*x]/192)
Time = 0.33 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {3042, 3654, 3042, 3115, 3042, 3115, 3042, 3115, 24}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (a-a \sin ^2(x)\right )^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a-a \sin (x)^2\right )^3dx\) |
\(\Big \downarrow \) 3654 |
\(\displaystyle a^3 \int \cos ^6(x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^3 \int \sin \left (x+\frac {\pi }{2}\right )^6dx\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle a^3 \left (\frac {5}{6} \int \cos ^4(x)dx+\frac {1}{6} \sin (x) \cos ^5(x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^3 \left (\frac {5}{6} \int \sin \left (x+\frac {\pi }{2}\right )^4dx+\frac {1}{6} \sin (x) \cos ^5(x)\right )\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle a^3 \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(x)dx+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^3 \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (x+\frac {\pi }{2}\right )^2dx+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle a^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sin (x) \cos (x)\right )+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )\) |
\(\Big \downarrow \) 24 |
\(\displaystyle a^3 \left (\frac {1}{6} \sin (x) \cos ^5(x)+\frac {5}{6} \left (\frac {1}{4} \sin (x) \cos ^3(x)+\frac {3}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )\right )\right )\) |
Input:
Int[(a - a*Sin[x]^2)^3,x]
Output:
a^3*((Cos[x]^5*Sin[x])/6 + (5*((Cos[x]^3*Sin[x])/4 + (3*(x/2 + (Cos[x]*Sin [x])/2))/4))/6)
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[ a^p Int[ActivateTrig[u*cos[e + f*x]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]
Time = 2.52 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.57
method | result | size |
parallelrisch | \(\frac {a^{3} \left (45 \sin \left (2 x \right )+60 x +\sin \left (6 x \right )+9 \sin \left (4 x \right )\right )}{192}\) | \(26\) |
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 | \(-a^{3} \left (-\frac {\left (\sin \left (x \right )^{5}+\frac {5 \sin \left (x \right )^{3}}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )}{6}+\frac {5 x}{16}\right )+3 a^{3} \left (-\frac {\left (\sin \left (x \right )^{3}+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{4}+\frac {3 x}{8}\right )-3 a^{3} \left (-\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )+a^{3} x\) | \(72\) |
parts | \(-a^{3} \left (-\frac {\left (\sin \left (x \right )^{5}+\frac {5 \sin \left (x \right )^{3}}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )}{6}+\frac {5 x}{16}\right )+3 a^{3} \left (-\frac {\left (\sin \left (x \right )^{3}+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{4}+\frac {3 x}{8}\right )-3 a^{3} \left (-\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )+a^{3} x\) | \(72\) |
norman | \(\frac {\frac {5 a^{3} x}{16}+\frac {11 a^{3} \tan \left (\frac {x}{2}\right )}{8}-\frac {5 a^{3} \tan \left (\frac {x}{2}\right )^{3}}{24}+\frac {15 a^{3} \tan \left (\frac {x}{2}\right )^{5}}{4}-\frac {15 a^{3} \tan \left (\frac {x}{2}\right )^{7}}{4}+\frac {5 a^{3} \tan \left (\frac {x}{2}\right )^{9}}{24}-\frac {11 a^{3} \tan \left (\frac {x}{2}\right )^{11}}{8}+\frac {15 a^{3} x \tan \left (\frac {x}{2}\right )^{2}}{8}+\frac {75 a^{3} x \tan \left (\frac {x}{2}\right )^{4}}{16}+\frac {25 a^{3} x \tan \left (\frac {x}{2}\right )^{6}}{4}+\frac {75 a^{3} x \tan \left (\frac {x}{2}\right )^{8}}{16}+\frac {15 a^{3} x \tan \left (\frac {x}{2}\right )^{10}}{8}+\frac {5 a^{3} x \tan \left (\frac {x}{2}\right )^{12}}{16}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{6}}\) | \(155\) |
orering | \(\left (\frac {x}{2}+\frac {5}{75072}\right ) \left (a -a \sin \left (x \right )^{2}\right )^{3}+\frac {18863 \left (a -a \sin \left (x \right )^{2}\right )^{2} a \cos \left (x \right ) \sin \left (x \right )}{12512}+\left (\frac {49 x}{288}+\frac {5}{75072}\right ) \left (24 \left (a -a \sin \left (x \right )^{2}\right ) a^{2} \cos \left (x \right )^{2} \sin \left (x \right )^{2}+6 \left (a -a \sin \left (x \right )^{2}\right )^{2} a \sin \left (x \right )^{2}-6 \left (a -a \sin \left (x \right )^{2}\right )^{2} a \cos \left (x \right )^{2}\right )-\frac {4867 \cos \left (x \right )^{3} \sin \left (x \right )^{3} a^{3}}{9384}+\frac {801 \left (a -a \sin \left (x \right )^{2}\right ) a^{2} \cos \left (x \right ) \sin \left (x \right )^{3}}{782}-\frac {801 \left (a -a \sin \left (x \right )^{2}\right ) a^{2} \cos \left (x \right )^{3} \sin \left (x \right )}{782}+\left (\frac {7 x}{576}+\frac {5}{75072}\right ) \left (288 \cos \left (x \right )^{2} \sin \left (x \right )^{4} a^{3}-288 \cos \left (x \right )^{4} \sin \left (x \right )^{2} a^{3}+72 \left (a -a \sin \left (x \right )^{2}\right ) a^{2} \sin \left (x \right )^{4}-528 \left (a -a \sin \left (x \right )^{2}\right ) a^{2} \cos \left (x \right )^{2} \sin \left (x \right )^{2}+72 \left (a -a \sin \left (x \right )^{2}\right ) a^{2} \cos \left (x \right )^{4}-24 \left (a -a \sin \left (x \right )^{2}\right )^{2} a \sin \left (x \right )^{2}+24 \left (a -a \sin \left (x \right )^{2}\right )^{2} a \cos \left (x \right )^{2}\right )+\frac {2255 \cos \left (x \right ) \sin \left (x \right )^{5} a^{3}}{6256}+\frac {2255 a^{3} \cos \left (x \right )^{5} \sin \left (x \right )}{6256}+\left (\frac {x}{4608}-\frac {5}{4204032}\right ) \left (720 a^{3} \sin \left (x \right )^{6}-16560 \cos \left (x \right )^{2} \sin \left (x \right )^{4} a^{3}+16560 \cos \left (x \right )^{4} \sin \left (x \right )^{2} a^{3}-720 \cos \left (x \right )^{6} a^{3}-1440 \left (a -a \sin \left (x \right )^{2}\right ) a^{2} \sin \left (x \right )^{4}+9024 \left (a -a \sin \left (x \right )^{2}\right ) a^{2} \cos \left (x \right )^{2} \sin \left (x \right )^{2}-1440 \left (a -a \sin \left (x \right )^{2}\right ) a^{2} \cos \left (x \right )^{4}+96 \left (a -a \sin \left (x \right )^{2}\right )^{2} a \sin \left (x \right )^{2}-96 \left (a -a \sin \left (x \right )^{2}\right )^{2} a \cos \left (x \right )^{2}\right )\) | \(449\) |
Input:
int((a-a*sin(x)^2)^3,x,method=_RETURNVERBOSE)
Output:
1/192*a^3*(45*sin(2*x)+60*x+sin(6*x)+9*sin(4*x))
Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80 \[ \int \left (a-a \sin ^2(x)\right )^3 \, dx=\frac {5}{16} \, a^{3} x + \frac {1}{48} \, {\left (8 \, a^{3} \cos \left (x\right )^{5} + 10 \, a^{3} \cos \left (x\right )^{3} + 15 \, a^{3} \cos \left (x\right )\right )} \sin \left (x\right ) \] Input:
integrate((a-a*sin(x)^2)^3,x, algorithm="fricas")
Output:
5/16*a^3*x + 1/48*(8*a^3*cos(x)^5 + 10*a^3*cos(x)^3 + 15*a^3*cos(x))*sin(x )
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 \[ \int \left (a-a \sin ^2(x)\right )^3 \, dx=- \frac {5 a^{3} x \sin ^{6}{\left (x \right )}}{16} - \frac {15 a^{3} x \sin ^{4}{\left (x \right )} \cos ^{2}{\left (x \right )}}{16} + \frac {9 a^{3} x \sin ^{4}{\left (x \right )}}{8} - \frac {15 a^{3} x \sin ^{2}{\left (x \right )} \cos ^{4}{\left (x \right )}}{16} + \frac {9 a^{3} x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{4} - \frac {3 a^{3} x \sin ^{2}{\left (x \right )}}{2} - \frac {5 a^{3} x \cos ^{6}{\left (x \right )}}{16} + \frac {9 a^{3} x \cos ^{4}{\left (x \right )}}{8} - \frac {3 a^{3} x \cos ^{2}{\left (x \right )}}{2} + a^{3} x + \frac {11 a^{3} \sin ^{5}{\left (x \right )} \cos {\left (x \right )}}{16} + \frac {5 a^{3} \sin ^{3}{\left (x \right )} \cos ^{3}{\left (x \right )}}{6} - \frac {15 a^{3} \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{8} + \frac {5 a^{3} \sin {\left (x \right )} \cos ^{5}{\left (x \right )}}{16} - \frac {9 a^{3} \sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{8} + \frac {3 a^{3} \sin {\left (x \right )} \cos {\left (x \right )}}{2} \] Input:
integrate((a-a*sin(x)**2)**3,x)
Output:
-5*a**3*x*sin(x)**6/16 - 15*a**3*x*sin(x)**4*cos(x)**2/16 + 9*a**3*x*sin(x )**4/8 - 15*a**3*x*sin(x)**2*cos(x)**4/16 + 9*a**3*x*sin(x)**2*cos(x)**2/4 - 3*a**3*x*sin(x)**2/2 - 5*a**3*x*cos(x)**6/16 + 9*a**3*x*cos(x)**4/8 - 3 *a**3*x*cos(x)**2/2 + a**3*x + 11*a**3*sin(x)**5*cos(x)/16 + 5*a**3*sin(x) **3*cos(x)**3/6 - 15*a**3*sin(x)**3*cos(x)/8 + 5*a**3*sin(x)*cos(x)**5/16 - 9*a**3*sin(x)*cos(x)**3/8 + 3*a**3*sin(x)*cos(x)/2
Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.50 \[ \int \left (a-a \sin ^2(x)\right )^3 \, dx=-\frac {1}{192} \, {\left (4 \, \sin \left (2 \, x\right )^{3} + 60 \, x + 9 \, \sin \left (4 \, x\right ) - 48 \, \sin \left (2 \, x\right )\right )} a^{3} + \frac {3}{32} \, a^{3} {\left (12 \, x + \sin \left (4 \, x\right ) - 8 \, \sin \left (2 \, x\right )\right )} - \frac {3}{4} \, a^{3} {\left (2 \, x - \sin \left (2 \, x\right )\right )} + a^{3} x \] Input:
integrate((a-a*sin(x)^2)^3,x, algorithm="maxima")
Output:
-1/192*(4*sin(2*x)^3 + 60*x + 9*sin(4*x) - 48*sin(2*x))*a^3 + 3/32*a^3*(12 *x + sin(4*x) - 8*sin(2*x)) - 3/4*a^3*(2*x - sin(2*x)) + a^3*x
Time = 0.45 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.74 \[ \int \left (a-a \sin ^2(x)\right )^3 \, dx=\frac {5}{16} \, a^{3} x + \frac {1}{192} \, a^{3} \sin \left (6 \, x\right ) + \frac {3}{64} \, a^{3} \sin \left (4 \, x\right ) + \frac {15}{64} \, a^{3} \sin \left (2 \, x\right ) \] Input:
integrate((a-a*sin(x)^2)^3,x, algorithm="giac")
Output:
5/16*a^3*x + 1/192*a^3*sin(6*x) + 3/64*a^3*sin(4*x) + 15/64*a^3*sin(2*x)
Time = 36.19 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.91 \[ \int \left (a-a \sin ^2(x)\right )^3 \, dx=\frac {11\,a^3\,{\cos \left (x\right )}^5\,\sin \left (x\right )}{16}+\frac {5\,a^3\,{\cos \left (x\right )}^3\,{\sin \left (x\right )}^3}{6}+\frac {5\,a^3\,\cos \left (x\right )\,{\sin \left (x\right )}^5}{16}+\frac {5\,x\,a^3}{16} \] Input:
int((a - a*sin(x)^2)^3,x)
Output:
(5*a^3*x)/16 + (5*a^3*cos(x)*sin(x)^5)/16 + (11*a^3*cos(x)^5*sin(x))/16 + (5*a^3*cos(x)^3*sin(x)^3)/6
Time = 0.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.67 \[ \int \left (a-a \sin ^2(x)\right )^3 \, dx=\frac {a^{3} \left (8 \cos \left (x \right ) \sin \left (x \right )^{5}-26 \cos \left (x \right ) \sin \left (x \right )^{3}+33 \cos \left (x \right ) \sin \left (x \right )+15 x \right )}{48} \] Input:
int((a-a*sin(x)^2)^3,x)
Output:
(a**3*(8*cos(x)*sin(x)**5 - 26*cos(x)*sin(x)**3 + 33*cos(x)*sin(x) + 15*x) )/48