Integrand size = 10, antiderivative size = 42 \[ \int \left (4-5 \sin ^2(x)\right )^3 \, dx=\frac {279 x}{16}+\frac {1595}{48} \cos (x) \sin (x)-\frac {125}{8} \cos (x) \sin ^3(x)+\frac {5}{6} \cos (x) \sin (x) \left (4-5 \sin ^2(x)\right )^2 \] Output:
279/16*x+1595/48*cos(x)*sin(x)-125/8*cos(x)*sin(x)^3+5/6*cos(x)*sin(x)*(4- 5*sin(x)^2)^2
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int \left (4-5 \sin ^2(x)\right )^3 \, dx=\frac {279 x}{16}+\frac {915}{64} \sin (2 x)+\frac {225}{64} \sin (4 x)+\frac {125}{192} \sin (6 x) \] Input:
Integrate[(4 - 5*Sin[x]^2)^3,x]
Output:
(279*x)/16 + (915*Sin[2*x])/64 + (225*Sin[4*x])/64 + (125*Sin[6*x])/192
Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3659, 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.
\(\displaystyle \int \left (4-5 \sin ^2(x)\right )^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (4-5 \sin (x)^2\right )^3dx\) |
\(\Big \downarrow \) 3659 |
\(\displaystyle \frac {1}{6} \int \left (76-75 \sin ^2(x)\right ) \left (4-5 \sin ^2(x)\right )dx+\frac {5}{6} \sin (x) \left (4-5 \sin ^2(x)\right )^2 \cos (x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \int \left (76-75 \sin (x)^2\right ) \left (4-5 \sin (x)^2\right )dx+\frac {5}{6} \sin (x) \left (4-5 \sin ^2(x)\right )^2 \cos (x)\) |
\(\Big \downarrow \) 3648 |
\(\displaystyle \frac {1}{6} \left (\frac {837 x}{8}-\frac {375}{4} \sin ^3(x) \cos (x)+\frac {1595}{8} \sin (x) \cos (x)\right )+\frac {5}{6} \sin (x) \left (4-5 \sin ^2(x)\right )^2 \cos (x)\) |
Input:
Int[(4 - 5*Sin[x]^2)^3,x]
Output:
(5*Cos[x]*Sin[x]*(4 - 5*Sin[x]^2)^2)/6 + ((837*x)/8 + (1595*Cos[x]*Sin[x]) /8 - (375*Cos[x]*Sin[x]^3)/4)/6
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 = 2.39 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.55
method | result | size |
risch | \(\frac {279 x}{16}+\frac {125 \sin \left (6 x \right )}{192}+\frac {225 \sin \left (4 x \right )}{64}+\frac {915 \sin \left (2 x \right )}{64}\) | \(23\) |
parallelrisch | \(\frac {279 x}{16}+\frac {125 \sin \left (6 x \right )}{192}+\frac {225 \sin \left (4 x \right )}{64}+\frac {915 \sin \left (2 x \right )}{64}\) | \(23\) |
default | \(\frac {279 x}{16}+120 \cos \left (x \right ) \sin \left (x \right )-75 \left (\sin \left (x \right )^{3}+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )+\frac {125 \left (\sin \left (x \right )^{5}+\frac {5 \sin \left (x \right )^{3}}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )}{6}\) | \(43\) |
parts | \(\frac {279 x}{16}+120 \cos \left (x \right ) \sin \left (x \right )-75 \left (\sin \left (x \right )^{3}+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )+\frac {125 \left (\sin \left (x \right )^{5}+\frac {5 \sin \left (x \right )^{3}}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )}{6}\) | \(43\) |
norman | \(\frac {\frac {279 x}{16}-\frac {2695 \tan \left (\frac {x}{2}\right )^{3}}{24}+\frac {1845 \tan \left (\frac {x}{2}\right )^{5}}{4}-\frac {1845 \tan \left (\frac {x}{2}\right )^{7}}{4}+\frac {2695 \tan \left (\frac {x}{2}\right )^{9}}{24}-\frac {745 \tan \left (\frac {x}{2}\right )^{11}}{8}+\frac {837 x \tan \left (\frac {x}{2}\right )^{2}}{8}+\frac {4185 x \tan \left (\frac {x}{2}\right )^{4}}{16}+\frac {1395 x \tan \left (\frac {x}{2}\right )^{6}}{4}+\frac {4185 x \tan \left (\frac {x}{2}\right )^{8}}{16}+\frac {837 x \tan \left (\frac {x}{2}\right )^{10}}{8}+\frac {279 x \tan \left (\frac {x}{2}\right )^{12}}{16}+\frac {745 \tan \left (\frac {x}{2}\right )}{8}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{6}}\) | \(116\) |
orering | \(\left (\frac {x}{2}+\frac {93}{2670080}\right ) \left (4-5 \sin \left (x \right )^{2}\right )^{3}+\frac {2007861 \cos \left (x \right ) \sin \left (x \right ) \left (4-5 \sin \left (x \right )^{2}\right )^{2}}{267008}+\left (\frac {49 x}{288}+\frac {93}{2670080}\right ) \left (30 \sin \left (x \right )^{2} \left (4-5 \sin \left (x \right )^{2}\right )^{2}-30 \cos \left (x \right )^{2} \left (4-5 \sin \left (x \right )^{2}\right )^{2}+600 \cos \left (x \right )^{2} \sin \left (x \right )^{2} \left (4-5 \sin \left (x \right )^{2}\right )\right )+\frac {255645 \sin \left (x \right )^{3} \left (4-5 \sin \left (x \right )^{2}\right ) \cos \left (x \right )}{9536}-\frac {255645 \cos \left (x \right )^{3} \left (4-5 \sin \left (x \right )^{2}\right ) \sin \left (x \right )}{9536}-\frac {5136175 \cos \left (x \right )^{3} \sin \left (x \right )^{3}}{100128}+\left (\frac {7 x}{576}+\frac {93}{2670080}\right ) \left (-120 \sin \left (x \right )^{2} \left (4-5 \sin \left (x \right )^{2}\right )^{2}+120 \cos \left (x \right )^{2} \left (4-5 \sin \left (x \right )^{2}\right )^{2}-13200 \cos \left (x \right )^{2} \sin \left (x \right )^{2} \left (4-5 \sin \left (x \right )^{2}\right )+36000 \cos \left (x \right )^{2} \sin \left (x \right )^{4}+1800 \sin \left (x \right )^{4} \left (4-5 \sin \left (x \right )^{2}\right )-36000 \cos \left (x \right )^{4} \sin \left (x \right )^{2}+1800 \cos \left (x \right )^{4} \left (4-5 \sin \left (x \right )^{2}\right )\right )+\frac {1408375 \cos \left (x \right ) \sin \left (x \right )^{5}}{33376}+\frac {1408375 \cos \left (x \right )^{5} \sin \left (x \right )}{33376}+\left (\frac {x}{4608}-\frac {93}{149524480}\right ) \left (480 \sin \left (x \right )^{2} \left (4-5 \sin \left (x \right )^{2}\right )^{2}-480 \cos \left (x \right )^{2} \left (4-5 \sin \left (x \right )^{2}\right )^{2}+225600 \cos \left (x \right )^{2} \sin \left (x \right )^{2} \left (4-5 \sin \left (x \right )^{2}\right )-2070000 \cos \left (x \right )^{2} \sin \left (x \right )^{4}-36000 \sin \left (x \right )^{4} \left (4-5 \sin \left (x \right )^{2}\right )+2070000 \cos \left (x \right )^{4} \sin \left (x \right )^{2}-36000 \cos \left (x \right )^{4} \left (4-5 \sin \left (x \right )^{2}\right )+90000 \sin \left (x \right )^{6}-90000 \cos \left (x \right )^{6}\right )\) | \(371\) |
Input:
int((4-5*sin(x)^2)^3,x,method=_RETURNVERBOSE)
Output:
279/16*x+125/192*sin(6*x)+225/64*sin(4*x)+915/64*sin(2*x)
Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.60 \[ \int \left (4-5 \sin ^2(x)\right )^3 \, dx=\frac {5}{48} \, {\left (200 \, \cos \left (x\right )^{5} + 70 \, \cos \left (x\right )^{3} + 177 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {279}{16} \, x \] Input:
integrate((4-5*sin(x)^2)^3,x, algorithm="fricas")
Output:
5/48*(200*cos(x)^5 + 70*cos(x)^3 + 177*cos(x))*sin(x) + 279/16*x
Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (46) = 92\).
Time = 0.24 (sec) , antiderivative size = 173, normalized size of antiderivative = 4.12 \[ \int \left (4-5 \sin ^2(x)\right )^3 \, dx=- \frac {625 x \sin ^{6}{\left (x \right )}}{16} - \frac {1875 x \sin ^{4}{\left (x \right )} \cos ^{2}{\left (x \right )}}{16} + \frac {225 x \sin ^{4}{\left (x \right )}}{2} - \frac {1875 x \sin ^{2}{\left (x \right )} \cos ^{4}{\left (x \right )}}{16} + 225 x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )} - 120 x \sin ^{2}{\left (x \right )} - \frac {625 x \cos ^{6}{\left (x \right )}}{16} + \frac {225 x \cos ^{4}{\left (x \right )}}{2} - 120 x \cos ^{2}{\left (x \right )} + 64 x + \frac {1375 \sin ^{5}{\left (x \right )} \cos {\left (x \right )}}{16} + \frac {625 \sin ^{3}{\left (x \right )} \cos ^{3}{\left (x \right )}}{6} - \frac {375 \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{2} + \frac {625 \sin {\left (x \right )} \cos ^{5}{\left (x \right )}}{16} - \frac {225 \sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{2} + 120 \sin {\left (x \right )} \cos {\left (x \right )} \] Input:
integrate((4-5*sin(x)**2)**3,x)
Output:
-625*x*sin(x)**6/16 - 1875*x*sin(x)**4*cos(x)**2/16 + 225*x*sin(x)**4/2 - 1875*x*sin(x)**2*cos(x)**4/16 + 225*x*sin(x)**2*cos(x)**2 - 120*x*sin(x)** 2 - 625*x*cos(x)**6/16 + 225*x*cos(x)**4/2 - 120*x*cos(x)**2 + 64*x + 1375 *sin(x)**5*cos(x)/16 + 625*sin(x)**3*cos(x)**3/6 - 375*sin(x)**3*cos(x)/2 + 625*sin(x)*cos(x)**5/16 - 225*sin(x)*cos(x)**3/2 + 120*sin(x)*cos(x)
Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.57 \[ \int \left (4-5 \sin ^2(x)\right )^3 \, dx=-\frac {125}{48} \, \sin \left (2 \, x\right )^{3} + \frac {279}{16} \, x + \frac {225}{64} \, \sin \left (4 \, x\right ) + \frac {65}{4} \, \sin \left (2 \, x\right ) \] Input:
integrate((4-5*sin(x)^2)^3,x, algorithm="maxima")
Output:
-125/48*sin(2*x)^3 + 279/16*x + 225/64*sin(4*x) + 65/4*sin(2*x)
Time = 0.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.52 \[ \int \left (4-5 \sin ^2(x)\right )^3 \, dx=\frac {279}{16} \, x + \frac {125}{192} \, \sin \left (6 \, x\right ) + \frac {225}{64} \, \sin \left (4 \, x\right ) + \frac {915}{64} \, \sin \left (2 \, x\right ) \] Input:
integrate((4-5*sin(x)^2)^3,x, algorithm="giac")
Output:
279/16*x + 125/192*sin(6*x) + 225/64*sin(4*x) + 915/64*sin(2*x)
Time = 37.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int \left (4-5 \sin ^2(x)\right )^3 \, dx=\frac {745\,{\cos \left (x\right )}^5\,\sin \left (x\right )}{16}+\frac {265\,{\cos \left (x\right )}^3\,{\sin \left (x\right )}^3}{6}+\frac {295\,\cos \left (x\right )\,{\sin \left (x\right )}^5}{16}+\frac {279\,x}{16} \] Input:
int(-(5*sin(x)^2 - 4)^3,x)
Output:
(279*x)/16 + (295*cos(x)*sin(x)^5)/16 + (745*cos(x)^5*sin(x))/16 + (265*co s(x)^3*sin(x)^3)/6
Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.62 \[ \int \left (4-5 \sin ^2(x)\right )^3 \, dx=\frac {125 \cos \left (x \right ) \sin \left (x \right )^{5}}{6}-\frac {1175 \cos \left (x \right ) \sin \left (x \right )^{3}}{24}+\frac {745 \cos \left (x \right ) \sin \left (x \right )}{16}+\frac {279 x}{16} \] Input:
int((4-5*sin(x)^2)^3,x)
Output:
(1000*cos(x)*sin(x)**5 - 2350*cos(x)*sin(x)**3 + 2235*cos(x)*sin(x) + 837* x)/48