Integrand size = 10, antiderivative size = 42 \[ \int \left (4-3 \sin ^2(x)\right )^3 \, dx=\frac {385 x}{16}+\frac {511}{16} \cos (x) \sin (x)-\frac {75}{8} \cos (x) \sin ^3(x)+\frac {1}{2} \cos (x) \sin (x) \left (4-3 \sin ^2(x)\right )^2 \] Output:
385/16*x+511/16*cos(x)*sin(x)-75/8*cos(x)*sin(x)^3+1/2*cos(x)*sin(x)*(4-3* sin(x)^2)^2
Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int \left (4-3 \sin ^2(x)\right )^3 \, dx=\frac {385 x}{16}+\frac {981}{64} \sin (2 x)+\frac {135}{64} \sin (4 x)+\frac {9}{64} \sin (6 x) \] Input:
Integrate[(4 - 3*Sin[x]^2)^3,x]
Output:
(385*x)/16 + (981*Sin[2*x])/64 + (135*Sin[4*x])/64 + (9*Sin[6*x])/64
Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3659, 27, 3042, 3648}
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.
\(\displaystyle \int \left (4-3 \sin ^2(x)\right )^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (4-3 \sin (x)^2\right )^3dx\) |
\(\Big \downarrow \) 3659 |
\(\displaystyle \frac {1}{6} \int 3 \left (28-25 \sin ^2(x)\right ) \left (4-3 \sin ^2(x)\right )dx+\frac {1}{2} \sin (x) \left (4-3 \sin ^2(x)\right )^2 \cos (x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \left (28-25 \sin ^2(x)\right ) \left (4-3 \sin ^2(x)\right )dx+\frac {1}{2} \sin (x) \left (4-3 \sin ^2(x)\right )^2 \cos (x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int \left (28-25 \sin (x)^2\right ) \left (4-3 \sin (x)^2\right )dx+\frac {1}{2} \sin (x) \left (4-3 \sin ^2(x)\right )^2 \cos (x)\) |
\(\Big \downarrow \) 3648 |
\(\displaystyle \frac {1}{2} \left (\frac {385 x}{8}-\frac {75}{4} \sin ^3(x) \cos (x)+\frac {511}{8} \sin (x) \cos (x)\right )+\frac {1}{2} \sin (x) \left (4-3 \sin ^2(x)\right )^2 \cos (x)\) |
Input:
Int[(4 - 3*Sin[x]^2)^3,x]
Output:
(Cos[x]*Sin[x]*(4 - 3*Sin[x]^2)^2)/2 + ((385*x)/8 + (511*Cos[x]*Sin[x])/8 - (75*Cos[x]*Sin[x]^3)/4)/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)*((A_.) + (B_.)*sin[(e_.) + (f_ .)*(x_)]^2), x_Symbol] :> Simp[(4*A*(2*a + b) + B*(4*a + 3*b))*(x/8), x] + (-Simp[b*B*Cos[e + f*x]*(Sin[e + f*x]^3/(4*f)), x] - Simp[(4*A*b + B*(4*a + 3*b))*Cos[e + f*x]*(Sin[e + f*x]/(8*f)), x]) /; FreeQ[{a, b, e, f, A, B}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*C os[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x]^2)^(p - 1)/(2*f*p)), x] + Sim p[1/(2*p) Int[(a + b*Sin[e + f*x]^2)^(p - 2)*Simp[a*(b + 2*a*p) + b*(2*a + b)*(2*p - 1)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[ a + b, 0] && GtQ[p, 1]
Time = 2.42 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.55
method | result | size |
risch | \(\frac {385 x}{16}+\frac {9 \sin \left (6 x \right )}{64}+\frac {135 \sin \left (4 x \right )}{64}+\frac {981 \sin \left (2 x \right )}{64}\) | \(23\) |
parallelrisch | \(\frac {385 x}{16}+\frac {9 \sin \left (6 x \right )}{64}+\frac {135 \sin \left (4 x \right )}{64}+\frac {981 \sin \left (2 x \right )}{64}\) | \(23\) |
default | \(\frac {385 x}{16}+72 \cos \left (x \right ) \sin \left (x \right )-27 \left (\sin \left (x \right )^{3}+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )+\frac {9 \left (\sin \left (x \right )^{5}+\frac {5 \sin \left (x \right )^{3}}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )}{2}\) | \(43\) |
parts | \(\frac {385 x}{16}+72 \cos \left (x \right ) \sin \left (x \right )-27 \left (\sin \left (x \right )^{3}+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )+\frac {9 \left (\sin \left (x \right )^{5}+\frac {5 \sin \left (x \right )^{3}}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )}{2}\) | \(43\) |
norman | \(\frac {\frac {385 x}{16}+\frac {549 \tan \left (\frac {x}{2}\right )^{3}}{8}+\frac {531 \tan \left (\frac {x}{2}\right )^{5}}{4}-\frac {531 \tan \left (\frac {x}{2}\right )^{7}}{4}-\frac {549 \tan \left (\frac {x}{2}\right )^{9}}{8}-\frac {639 \tan \left (\frac {x}{2}\right )^{11}}{8}+\frac {1155 x \tan \left (\frac {x}{2}\right )^{2}}{8}+\frac {5775 x \tan \left (\frac {x}{2}\right )^{4}}{16}+\frac {1925 x \tan \left (\frac {x}{2}\right )^{6}}{4}+\frac {5775 x \tan \left (\frac {x}{2}\right )^{8}}{16}+\frac {1155 x \tan \left (\frac {x}{2}\right )^{10}}{8}+\frac {385 x \tan \left (\frac {x}{2}\right )^{12}}{16}+\frac {639 \tan \left (\frac {x}{2}\right )}{8}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{6}}\) | \(116\) |
orering | \(\left (\frac {x}{2}+\frac {385}{2539008}\right ) \left (4-3 \sin \left (x \right )^{2}\right )^{3}+\frac {33793 \cos \left (x \right ) \sin \left (x \right ) \left (4-3 \sin \left (x \right )^{2}\right )^{2}}{7424}+\left (\frac {49 x}{288}+\frac {385}{2539008}\right ) \left (18 \sin \left (x \right )^{2} \left (4-3 \sin \left (x \right )^{2}\right )^{2}-18 \cos \left (x \right )^{2} \left (4-3 \sin \left (x \right )^{2}\right )^{2}+216 \cos \left (x \right )^{2} \sin \left (x \right )^{2} \left (4-3 \sin \left (x \right )^{2}\right )\right )+\frac {284283 \sin \left (x \right )^{3} \left (4-3 \sin \left (x \right )^{2}\right ) \cos \left (x \right )}{35264}-\frac {284283 \cos \left (x \right )^{3} \left (4-3 \sin \left (x \right )^{2}\right ) \sin \left (x \right )}{35264}-\frac {384867 \cos \left (x \right )^{3} \sin \left (x \right )^{3}}{17632}+\left (\frac {7 x}{576}+\frac {385}{2539008}\right ) \left (-72 \sin \left (x \right )^{2} \left (4-3 \sin \left (x \right )^{2}\right )^{2}+72 \cos \left (x \right )^{2} \left (4-3 \sin \left (x \right )^{2}\right )^{2}-4752 \cos \left (x \right )^{2} \sin \left (x \right )^{2} \left (4-3 \sin \left (x \right )^{2}\right )+7776 \cos \left (x \right )^{2} \sin \left (x \right )^{4}+648 \sin \left (x \right )^{4} \left (4-3 \sin \left (x \right )^{2}\right )-7776 \cos \left (x \right )^{4} \sin \left (x \right )^{2}+648 \cos \left (x \right )^{4} \left (4-3 \sin \left (x \right )^{2}\right )\right )+\frac {200745 \cos \left (x \right ) \sin \left (x \right )^{5}}{17632}+\frac {200745 \cos \left (x \right )^{5} \sin \left (x \right )}{17632}+\left (\frac {x}{4608}-\frac {55}{20312064}\right ) \left (288 \sin \left (x \right )^{2} \left (4-3 \sin \left (x \right )^{2}\right )^{2}-288 \cos \left (x \right )^{2} \left (4-3 \sin \left (x \right )^{2}\right )^{2}+81216 \cos \left (x \right )^{2} \sin \left (x \right )^{2} \left (4-3 \sin \left (x \right )^{2}\right )-447120 \cos \left (x \right )^{2} \sin \left (x \right )^{4}-12960 \sin \left (x \right )^{4} \left (4-3 \sin \left (x \right )^{2}\right )+447120 \cos \left (x \right )^{4} \sin \left (x \right )^{2}-12960 \cos \left (x \right )^{4} \left (4-3 \sin \left (x \right )^{2}\right )+19440 \sin \left (x \right )^{6}-19440 \cos \left (x \right )^{6}\right )\) | \(371\) |
Input:
int((4-3*sin(x)^2)^3,x,method=_RETURNVERBOSE)
Output:
385/16*x+9/64*sin(6*x)+135/64*sin(4*x)+981/64*sin(2*x)
Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.60 \[ \int \left (4-3 \sin ^2(x)\right )^3 \, dx=\frac {9}{16} \, {\left (8 \, \cos \left (x\right )^{5} + 22 \, \cos \left (x\right )^{3} + 41 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {385}{16} \, x \] Input:
integrate((4-3*sin(x)^2)^3,x, algorithm="fricas")
Output:
9/16*(8*cos(x)^5 + 22*cos(x)^3 + 41*cos(x))*sin(x) + 385/16*x
Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (44) = 88\).
Time = 0.25 (sec) , antiderivative size = 173, normalized size of antiderivative = 4.12 \[ \int \left (4-3 \sin ^2(x)\right )^3 \, dx=- \frac {135 x \sin ^{6}{\left (x \right )}}{16} - \frac {405 x \sin ^{4}{\left (x \right )} \cos ^{2}{\left (x \right )}}{16} + \frac {81 x \sin ^{4}{\left (x \right )}}{2} - \frac {405 x \sin ^{2}{\left (x \right )} \cos ^{4}{\left (x \right )}}{16} + 81 x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )} - 72 x \sin ^{2}{\left (x \right )} - \frac {135 x \cos ^{6}{\left (x \right )}}{16} + \frac {81 x \cos ^{4}{\left (x \right )}}{2} - 72 x \cos ^{2}{\left (x \right )} + 64 x + \frac {297 \sin ^{5}{\left (x \right )} \cos {\left (x \right )}}{16} + \frac {45 \sin ^{3}{\left (x \right )} \cos ^{3}{\left (x \right )}}{2} - \frac {135 \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{2} + \frac {135 \sin {\left (x \right )} \cos ^{5}{\left (x \right )}}{16} - \frac {81 \sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{2} + 72 \sin {\left (x \right )} \cos {\left (x \right )} \] Input:
integrate((4-3*sin(x)**2)**3,x)
Output:
-135*x*sin(x)**6/16 - 405*x*sin(x)**4*cos(x)**2/16 + 81*x*sin(x)**4/2 - 40 5*x*sin(x)**2*cos(x)**4/16 + 81*x*sin(x)**2*cos(x)**2 - 72*x*sin(x)**2 - 1 35*x*cos(x)**6/16 + 81*x*cos(x)**4/2 - 72*x*cos(x)**2 + 64*x + 297*sin(x)* *5*cos(x)/16 + 45*sin(x)**3*cos(x)**3/2 - 135*sin(x)**3*cos(x)/2 + 135*sin (x)*cos(x)**5/16 - 81*sin(x)*cos(x)**3/2 + 72*sin(x)*cos(x)
Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.57 \[ \int \left (4-3 \sin ^2(x)\right )^3 \, dx=-\frac {9}{16} \, \sin \left (2 \, x\right )^{3} + \frac {385}{16} \, x + \frac {135}{64} \, \sin \left (4 \, x\right ) + \frac {63}{4} \, \sin \left (2 \, x\right ) \] Input:
integrate((4-3*sin(x)^2)^3,x, algorithm="maxima")
Output:
-9/16*sin(2*x)^3 + 385/16*x + 135/64*sin(4*x) + 63/4*sin(2*x)
Time = 0.45 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.52 \[ \int \left (4-3 \sin ^2(x)\right )^3 \, dx=\frac {385}{16} \, x + \frac {9}{64} \, \sin \left (6 \, x\right ) + \frac {135}{64} \, \sin \left (4 \, x\right ) + \frac {981}{64} \, \sin \left (2 \, x\right ) \] Input:
integrate((4-3*sin(x)^2)^3,x, algorithm="giac")
Output:
385/16*x + 9/64*sin(6*x) + 135/64*sin(4*x) + 981/64*sin(2*x)
Time = 36.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int \left (4-3 \sin ^2(x)\right )^3 \, dx=\frac {639\,{\cos \left (x\right )}^5\,\sin \left (x\right )}{16}+\frac {117\,{\cos \left (x\right )}^3\,{\sin \left (x\right )}^3}{2}+\frac {369\,\cos \left (x\right )\,{\sin \left (x\right )}^5}{16}+\frac {385\,x}{16} \] Input:
int(-(3*sin(x)^2 - 4)^3,x)
Output:
(385*x)/16 + (369*cos(x)*sin(x)^5)/16 + (639*cos(x)^5*sin(x))/16 + (117*co s(x)^3*sin(x)^3)/2
Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.62 \[ \int \left (4-3 \sin ^2(x)\right )^3 \, dx=\frac {9 \cos \left (x \right ) \sin \left (x \right )^{5}}{2}-\frac {171 \cos \left (x \right ) \sin \left (x \right )^{3}}{8}+\frac {639 \cos \left (x \right ) \sin \left (x \right )}{16}+\frac {385 x}{16} \] Input:
int((4-3*sin(x)^2)^3,x)
Output:
(72*cos(x)*sin(x)**5 - 342*cos(x)*sin(x)**3 + 639*cos(x)*sin(x) + 385*x)/1 6