Integrand size = 11, antiderivative size = 59 \[ \int \left (a-a \sin ^2(x)\right )^4 \, dx=\frac {35 a^4 x}{128}+\frac {35}{128} a^4 \cos (x) \sin (x)+\frac {35}{192} a^4 \cos ^3(x) \sin (x)+\frac {7}{48} a^4 \cos ^5(x) \sin (x)+\frac {1}{8} a^4 \cos ^7(x) \sin (x) \] Output:
35/128*a^4*x+35/128*a^4*cos(x)*sin(x)+35/192*a^4*cos(x)^3*sin(x)+7/48*a^4* cos(x)^5*sin(x)+1/8*a^4*cos(x)^7*sin(x)
Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.71 \[ \int \left (a-a \sin ^2(x)\right )^4 \, dx=a^4 \left (\frac {35 x}{128}+\frac {7}{32} \sin (2 x)+\frac {7}{128} \sin (4 x)+\frac {1}{96} \sin (6 x)+\frac {\sin (8 x)}{1024}\right ) \] Input:
Integrate[(a - a*Sin[x]^2)^4,x]
Output:
a^4*((35*x)/128 + (7*Sin[2*x])/32 + (7*Sin[4*x])/128 + Sin[6*x]/96 + Sin[8 *x]/1024)
Time = 0.39 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3654, 3042, 3115, 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.
\(\displaystyle \int \left (a-a \sin ^2(x)\right )^4 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a-a \sin (x)^2\right )^4dx\) |
\(\Big \downarrow \) 3654 |
\(\displaystyle a^4 \int \cos ^8(x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^4 \int \sin \left (x+\frac {\pi }{2}\right )^8dx\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle a^4 \left (\frac {7}{8} \int \cos ^6(x)dx+\frac {1}{8} \sin (x) \cos ^7(x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^4 \left (\frac {7}{8} \int \sin \left (x+\frac {\pi }{2}\right )^6dx+\frac {1}{8} \sin (x) \cos ^7(x)\right )\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle a^4 \left (\frac {7}{8} \left (\frac {5}{6} \int \cos ^4(x)dx+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^4 \left (\frac {7}{8} \left (\frac {5}{6} \int \sin \left (x+\frac {\pi }{2}\right )^4dx+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle a^4 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(x)dx+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^4 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (x+\frac {\pi }{2}\right )^2dx+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle a^4 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sin (x) \cos (x)\right )+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )\) |
\(\Big \downarrow \) 24 |
\(\displaystyle a^4 \left (\frac {1}{8} \sin (x) \cos ^7(x)+\frac {7}{8} \left (\frac {1}{6} \sin (x) \cos ^5(x)+\frac {5}{6} \left (\frac {1}{4} \sin (x) \cos ^3(x)+\frac {3}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )\right )\right )\right )\) |
Input:
Int[(a - a*Sin[x]^2)^4,x]
Output:
a^4*((Cos[x]^7*Sin[x])/8 + (7*((Cos[x]^5*Sin[x])/6 + (5*((Cos[x]^3*Sin[x]) /4 + (3*(x/2 + (Cos[x]*Sin[x])/2))/4))/6))/8)
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 = 12.32 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.58
method | result | size |
parallelrisch | \(\frac {a^{4} \left (672 \sin \left (2 x \right )+840 x +3 \sin \left (8 x \right )+32 \sin \left (6 x \right )+168 \sin \left (4 x \right )\right )}{3072}\) | \(34\) |
risch | \(\frac {35 a^{4} x}{128}+\frac {a^{4} \sin \left (8 x \right )}{1024}+\frac {a^{4} \sin \left (6 x \right )}{96}+\frac {7 a^{4} \sin \left (4 x \right )}{128}+\frac {7 a^{4} \sin \left (2 x \right )}{32}\) | \(44\) |
default | \(a^{4} \left (-\frac {\left (\sin \left (x \right )^{7}+\frac {7 \sin \left (x \right )^{5}}{6}+\frac {35 \sin \left (x \right )^{3}}{24}+\frac {35 \sin \left (x \right )}{16}\right ) \cos \left (x \right )}{8}+\frac {35 x}{128}\right )-4 a^{4} \left (-\frac {\left (\sin \left (x \right )^{5}+\frac {5 \sin \left (x \right )^{3}}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )}{6}+\frac {5 x}{16}\right )+6 a^{4} \left (-\frac {\left (\sin \left (x \right )^{3}+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{4}+\frac {3 x}{8}\right )-4 a^{4} \left (-\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )+a^{4} x\) | \(105\) |
parts | \(a^{4} \left (-\frac {\left (\sin \left (x \right )^{7}+\frac {7 \sin \left (x \right )^{5}}{6}+\frac {35 \sin \left (x \right )^{3}}{24}+\frac {35 \sin \left (x \right )}{16}\right ) \cos \left (x \right )}{8}+\frac {35 x}{128}\right )-4 a^{4} \left (-\frac {\left (\sin \left (x \right )^{5}+\frac {5 \sin \left (x \right )^{3}}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )}{6}+\frac {5 x}{16}\right )+6 a^{4} \left (-\frac {\left (\sin \left (x \right )^{3}+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{4}+\frac {3 x}{8}\right )-4 a^{4} \left (-\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )+a^{4} x\) | \(105\) |
norman | \(\frac {\frac {35 a^{4} x}{128}+\frac {93 a^{4} \tan \left (\frac {x}{2}\right )}{64}+\frac {91 a^{4} \tan \left (\frac {x}{2}\right )^{3}}{192}+\frac {1799 a^{4} \tan \left (\frac {x}{2}\right )^{5}}{192}-\frac {1085 a^{4} \tan \left (\frac {x}{2}\right )^{7}}{192}+\frac {1085 a^{4} \tan \left (\frac {x}{2}\right )^{9}}{192}-\frac {1799 a^{4} \tan \left (\frac {x}{2}\right )^{11}}{192}-\frac {91 a^{4} \tan \left (\frac {x}{2}\right )^{13}}{192}-\frac {93 a^{4} \tan \left (\frac {x}{2}\right )^{15}}{64}+\frac {35 a^{4} x \tan \left (\frac {x}{2}\right )^{2}}{16}+\frac {245 a^{4} x \tan \left (\frac {x}{2}\right )^{4}}{32}+\frac {245 a^{4} x \tan \left (\frac {x}{2}\right )^{6}}{16}+\frac {1225 a^{4} x \tan \left (\frac {x}{2}\right )^{8}}{64}+\frac {245 a^{4} x \tan \left (\frac {x}{2}\right )^{10}}{16}+\frac {245 a^{4} x \tan \left (\frac {x}{2}\right )^{12}}{32}+\frac {35 a^{4} x \tan \left (\frac {x}{2}\right )^{14}}{16}+\frac {35 a^{4} x \tan \left (\frac {x}{2}\right )^{16}}{128}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{8}}\) | \(201\) |
Input:
int((a-a*sin(x)^2)^4,x,method=_RETURNVERBOSE)
Output:
1/3072*a^4*(672*sin(2*x)+840*x+3*sin(8*x)+32*sin(6*x)+168*sin(4*x))
Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.78 \[ \int \left (a-a \sin ^2(x)\right )^4 \, dx=\frac {35}{128} \, a^{4} x + \frac {1}{384} \, {\left (48 \, a^{4} \cos \left (x\right )^{7} + 56 \, a^{4} \cos \left (x\right )^{5} + 70 \, a^{4} \cos \left (x\right )^{3} + 105 \, a^{4} \cos \left (x\right )\right )} \sin \left (x\right ) \] Input:
integrate((a-a*sin(x)^2)^4,x, algorithm="fricas")
Output:
35/128*a^4*x + 1/384*(48*a^4*cos(x)^7 + 56*a^4*cos(x)^5 + 70*a^4*cos(x)^3 + 105*a^4*cos(x))*sin(x)
Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (65) = 130\).
Time = 0.53 (sec) , antiderivative size = 376, normalized size of antiderivative = 6.37 \[ \int \left (a-a \sin ^2(x)\right )^4 \, dx=\frac {35 a^{4} x \sin ^{8}{\left (x \right )}}{128} + \frac {35 a^{4} x \sin ^{6}{\left (x \right )} \cos ^{2}{\left (x \right )}}{32} - \frac {5 a^{4} x \sin ^{6}{\left (x \right )}}{4} + \frac {105 a^{4} x \sin ^{4}{\left (x \right )} \cos ^{4}{\left (x \right )}}{64} - \frac {15 a^{4} x \sin ^{4}{\left (x \right )} \cos ^{2}{\left (x \right )}}{4} + \frac {9 a^{4} x \sin ^{4}{\left (x \right )}}{4} + \frac {35 a^{4} x \sin ^{2}{\left (x \right )} \cos ^{6}{\left (x \right )}}{32} - \frac {15 a^{4} x \sin ^{2}{\left (x \right )} \cos ^{4}{\left (x \right )}}{4} + \frac {9 a^{4} x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{2} - 2 a^{4} x \sin ^{2}{\left (x \right )} + \frac {35 a^{4} x \cos ^{8}{\left (x \right )}}{128} - \frac {5 a^{4} x \cos ^{6}{\left (x \right )}}{4} + \frac {9 a^{4} x \cos ^{4}{\left (x \right )}}{4} - 2 a^{4} x \cos ^{2}{\left (x \right )} + a^{4} x - \frac {93 a^{4} \sin ^{7}{\left (x \right )} \cos {\left (x \right )}}{128} - \frac {511 a^{4} \sin ^{5}{\left (x \right )} \cos ^{3}{\left (x \right )}}{384} + \frac {11 a^{4} \sin ^{5}{\left (x \right )} \cos {\left (x \right )}}{4} - \frac {385 a^{4} \sin ^{3}{\left (x \right )} \cos ^{5}{\left (x \right )}}{384} + \frac {10 a^{4} \sin ^{3}{\left (x \right )} \cos ^{3}{\left (x \right )}}{3} - \frac {15 a^{4} \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{4} - \frac {35 a^{4} \sin {\left (x \right )} \cos ^{7}{\left (x \right )}}{128} + \frac {5 a^{4} \sin {\left (x \right )} \cos ^{5}{\left (x \right )}}{4} - \frac {9 a^{4} \sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{4} + 2 a^{4} \sin {\left (x \right )} \cos {\left (x \right )} \] Input:
integrate((a-a*sin(x)**2)**4,x)
Output:
35*a**4*x*sin(x)**8/128 + 35*a**4*x*sin(x)**6*cos(x)**2/32 - 5*a**4*x*sin( x)**6/4 + 105*a**4*x*sin(x)**4*cos(x)**4/64 - 15*a**4*x*sin(x)**4*cos(x)** 2/4 + 9*a**4*x*sin(x)**4/4 + 35*a**4*x*sin(x)**2*cos(x)**6/32 - 15*a**4*x* sin(x)**2*cos(x)**4/4 + 9*a**4*x*sin(x)**2*cos(x)**2/2 - 2*a**4*x*sin(x)** 2 + 35*a**4*x*cos(x)**8/128 - 5*a**4*x*cos(x)**6/4 + 9*a**4*x*cos(x)**4/4 - 2*a**4*x*cos(x)**2 + a**4*x - 93*a**4*sin(x)**7*cos(x)/128 - 511*a**4*si n(x)**5*cos(x)**3/384 + 11*a**4*sin(x)**5*cos(x)/4 - 385*a**4*sin(x)**3*co s(x)**5/384 + 10*a**4*sin(x)**3*cos(x)**3/3 - 15*a**4*sin(x)**3*cos(x)/4 - 35*a**4*sin(x)*cos(x)**7/128 + 5*a**4*sin(x)*cos(x)**5/4 - 9*a**4*sin(x)* cos(x)**3/4 + 2*a**4*sin(x)*cos(x)
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (49) = 98\).
Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.76 \[ \int \left (a-a \sin ^2(x)\right )^4 \, dx=\frac {1}{3072} \, {\left (128 \, \sin \left (2 \, x\right )^{3} + 840 \, x + 3 \, \sin \left (8 \, x\right ) + 168 \, \sin \left (4 \, x\right ) - 768 \, \sin \left (2 \, x\right )\right )} a^{4} - \frac {1}{48} \, {\left (4 \, \sin \left (2 \, x\right )^{3} + 60 \, x + 9 \, \sin \left (4 \, x\right ) - 48 \, \sin \left (2 \, x\right )\right )} a^{4} + \frac {3}{16} \, a^{4} {\left (12 \, x + \sin \left (4 \, x\right ) - 8 \, \sin \left (2 \, x\right )\right )} - a^{4} {\left (2 \, x - \sin \left (2 \, x\right )\right )} + a^{4} x \] Input:
integrate((a-a*sin(x)^2)^4,x, algorithm="maxima")
Output:
1/3072*(128*sin(2*x)^3 + 840*x + 3*sin(8*x) + 168*sin(4*x) - 768*sin(2*x)) *a^4 - 1/48*(4*sin(2*x)^3 + 60*x + 9*sin(4*x) - 48*sin(2*x))*a^4 + 3/16*a^ 4*(12*x + sin(4*x) - 8*sin(2*x)) - a^4*(2*x - sin(2*x)) + a^4*x
Time = 0.49 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.73 \[ \int \left (a-a \sin ^2(x)\right )^4 \, dx=\frac {35}{128} \, a^{4} x + \frac {1}{1024} \, a^{4} \sin \left (8 \, x\right ) + \frac {1}{96} \, a^{4} \sin \left (6 \, x\right ) + \frac {7}{128} \, a^{4} \sin \left (4 \, x\right ) + \frac {7}{32} \, a^{4} \sin \left (2 \, x\right ) \] Input:
integrate((a-a*sin(x)^2)^4,x, algorithm="giac")
Output:
35/128*a^4*x + 1/1024*a^4*sin(8*x) + 1/96*a^4*sin(6*x) + 7/128*a^4*sin(4*x ) + 7/32*a^4*sin(2*x)
Time = 36.44 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86 \[ \int \left (a-a \sin ^2(x)\right )^4 \, dx=\frac {\frac {35\,a^4\,{\mathrm {tan}\left (x\right )}^7}{128}+\frac {385\,a^4\,{\mathrm {tan}\left (x\right )}^5}{384}+\frac {511\,a^4\,{\mathrm {tan}\left (x\right )}^3}{384}+\frac {93\,a^4\,\mathrm {tan}\left (x\right )}{128}}{{\left ({\mathrm {tan}\left (x\right )}^2+1\right )}^4}+\frac {35\,a^4\,x}{128} \] Input:
int((a - a*sin(x)^2)^4,x)
Output:
((93*a^4*tan(x))/128 + (511*a^4*tan(x)^3)/384 + (385*a^4*tan(x)^5)/384 + ( 35*a^4*tan(x)^7)/128)/(tan(x)^2 + 1)^4 + (35*a^4*x)/128
Time = 0.17 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.66 \[ \int \left (a-a \sin ^2(x)\right )^4 \, dx=\frac {a^{4} \left (-48 \cos \left (x \right ) \sin \left (x \right )^{7}+200 \cos \left (x \right ) \sin \left (x \right )^{5}-326 \cos \left (x \right ) \sin \left (x \right )^{3}+279 \cos \left (x \right ) \sin \left (x \right )+105 x \right )}{384} \] Input:
int((a-a*sin(x)^2)^4,x)
Output:
(a**4*( - 48*cos(x)*sin(x)**7 + 200*cos(x)*sin(x)**5 - 326*cos(x)*sin(x)** 3 + 279*cos(x)*sin(x) + 105*x))/384