Integrand size = 4, antiderivative size = 44 \[ \int \sin ^8(x) \, dx=\frac {35 x}{128}-\frac {35}{128} \cos (x) \sin (x)-\frac {35}{192} \cos (x) \sin ^3(x)-\frac {7}{48} \cos (x) \sin ^5(x)-\frac {1}{8} \cos (x) \sin ^7(x) \] Output:
35/128*x-35/128*cos(x)*sin(x)-35/192*cos(x)*sin(x)^3-7/48*cos(x)*sin(x)^5- 1/8*cos(x)*sin(x)^7
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.86 \[ \int \sin ^8(x) \, dx=\frac {35 x}{128}-\frac {7}{32} \sin (2 x)+\frac {7}{128} \sin (4 x)-\frac {1}{96} \sin (6 x)+\frac {\sin (8 x)}{1024} \] Input:
Integrate[Sin[x]^8,x]
Output:
(35*x)/128 - (7*Sin[2*x])/32 + (7*Sin[4*x])/128 - Sin[6*x]/96 + Sin[8*x]/1 024
Time = 0.29 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.34, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.250, Rules used = {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.
\(\displaystyle \int \sin ^8(x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (x)^8dx\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {7}{8} \int \sin ^6(x)dx-\frac {1}{8} \sin ^7(x) \cos (x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {7}{8} \int \sin (x)^6dx-\frac {1}{8} \sin ^7(x) \cos (x)\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {7}{8} \left (\frac {5}{6} \int \sin ^4(x)dx-\frac {1}{6} \sin ^5(x) \cos (x)\right )-\frac {1}{8} \sin ^7(x) \cos (x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {7}{8} \left (\frac {5}{6} \int \sin (x)^4dx-\frac {1}{6} \sin ^5(x) \cos (x)\right )-\frac {1}{8} \sin ^7(x) \cos (x)\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin ^2(x)dx-\frac {1}{4} \sin ^3(x) \cos (x)\right )-\frac {1}{6} \sin ^5(x) \cos (x)\right )-\frac {1}{8} \sin ^7(x) \cos (x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin (x)^2dx-\frac {1}{4} \sin ^3(x) \cos (x)\right )-\frac {1}{6} \sin ^5(x) \cos (x)\right )-\frac {1}{8} \sin ^7(x) \cos (x)\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin ^3(x) \cos (x)\right )-\frac {1}{6} \sin ^5(x) \cos (x)\right )-\frac {1}{8} \sin ^7(x) \cos (x)\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin ^3(x) \cos (x)\right )-\frac {1}{6} \sin ^5(x) \cos (x)\right )-\frac {1}{8} \sin ^7(x) \cos (x)\) |
Input:
Int[Sin[x]^8,x]
Output:
-1/8*(Cos[x]*Sin[x]^7) + (7*(-1/6*(Cos[x]*Sin[x]^5) + (5*(-1/4*(Cos[x]*Sin [x]^3) + (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]
Time = 1.98 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.66
method | result | size |
risch | \(\frac {35 x}{128}+\frac {\sin \left (8 x \right )}{1024}-\frac {\sin \left (6 x \right )}{96}+\frac {7 \sin \left (4 x \right )}{128}-\frac {7 \sin \left (2 x \right )}{32}\) | \(29\) |
parallelrisch | \(\frac {35 x}{128}+\frac {\sin \left (8 x \right )}{1024}-\frac {\sin \left (6 x \right )}{96}+\frac {7 \sin \left (4 x \right )}{128}-\frac {7 \sin \left (2 x \right )}{32}\) | \(29\) |
default | \(-\frac {\left (\sin \left (x \right )^{7}+\frac {7 \sin \left (x \right )^{5}}{6}+\frac {35 \sin \left (x \right )^{3}}{24}+\frac {35 \sin \left (x \right )}{16}\right ) \cos \left (x \right )}{8}+\frac {35 x}{128}\) | \(30\) |
norman | \(\frac {\frac {35 x}{128}-\frac {805 \tan \left (\frac {x}{2}\right )^{3}}{192}-\frac {2681 \tan \left (\frac {x}{2}\right )^{5}}{192}-\frac {5053 \tan \left (\frac {x}{2}\right )^{7}}{192}+\frac {5053 \tan \left (\frac {x}{2}\right )^{9}}{192}+\frac {2681 \tan \left (\frac {x}{2}\right )^{11}}{192}+\frac {805 \tan \left (\frac {x}{2}\right )^{13}}{192}+\frac {35 \tan \left (\frac {x}{2}\right )^{15}}{64}+\frac {35 x \tan \left (\frac {x}{2}\right )^{2}}{16}+\frac {245 x \tan \left (\frac {x}{2}\right )^{4}}{32}+\frac {245 x \tan \left (\frac {x}{2}\right )^{6}}{16}+\frac {1225 x \tan \left (\frac {x}{2}\right )^{8}}{64}+\frac {245 x \tan \left (\frac {x}{2}\right )^{10}}{16}+\frac {245 x \tan \left (\frac {x}{2}\right )^{12}}{32}+\frac {35 x \tan \left (\frac {x}{2}\right )^{14}}{16}+\frac {35 x \tan \left (\frac {x}{2}\right )^{16}}{128}-\frac {35 \tan \left (\frac {x}{2}\right )}{64}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{8}}\) | \(150\) |
orering | \(x \sin \left (x \right )^{8}-\frac {93 \cos \left (x \right ) \sin \left (x \right )^{7}}{128}+\frac {205 x \left (-8 \sin \left (x \right )^{8}+56 \cos \left (x \right )^{2} \sin \left (x \right )^{6}\right )}{576}-\frac {511 \cos \left (x \right )^{3} \sin \left (x \right )^{5}}{384}+\frac {91 x \left (176 \sin \left (x \right )^{8}-2240 \cos \left (x \right )^{2} \sin \left (x \right )^{6}+1680 \cos \left (x \right )^{4} \sin \left (x \right )^{4}\right )}{3072}-\frac {385 \cos \left (x \right )^{5} \sin \left (x \right )^{3}}{384}+\frac {5 x \left (-5888 \sin \left (x \right )^{8}+101696 \cos \left (x \right )^{2} \sin \left (x \right )^{6}-134400 \cos \left (x \right )^{4} \sin \left (x \right )^{4}+20160 \cos \left (x \right )^{6} \sin \left (x \right )^{2}\right )}{6144}-\frac {35 \sin \left (x \right ) \cos \left (x \right )^{7}}{128}+\frac {x \left (250496 \sin \left (x \right )^{8}-5196800 \cos \left (x \right )^{2} \sin \left (x \right )^{6}+9031680 \cos \left (x \right )^{4} \sin \left (x \right )^{4}-2257920 \cos \left (x \right )^{6} \sin \left (x \right )^{2}+40320 \cos \left (x \right )^{8}\right )}{147456}\) | \(180\) |
Input:
int(sin(x)^8,x,method=_RETURNVERBOSE)
Output:
35/128*x+1/1024*sin(8*x)-1/96*sin(6*x)+7/128*sin(4*x)-7/32*sin(2*x)
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.70 \[ \int \sin ^8(x) \, dx=\frac {1}{384} \, {\left (48 \, \cos \left (x\right )^{7} - 200 \, \cos \left (x\right )^{5} + 326 \, \cos \left (x\right )^{3} - 279 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {35}{128} \, x \] Input:
integrate(sin(x)^8,x, algorithm="fricas")
Output:
1/384*(48*cos(x)^7 - 200*cos(x)^5 + 326*cos(x)^3 - 279*cos(x))*sin(x) + 35 /128*x
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.09 \[ \int \sin ^8(x) \, dx=\frac {35 x}{128} - \frac {\sin ^{7}{\left (x \right )} \cos {\left (x \right )}}{8} - \frac {7 \sin ^{5}{\left (x \right )} \cos {\left (x \right )}}{48} - \frac {35 \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{192} - \frac {35 \sin {\left (x \right )} \cos {\left (x \right )}}{128} \] Input:
integrate(sin(x)**8,x)
Output:
35*x/128 - sin(x)**7*cos(x)/8 - 7*sin(x)**5*cos(x)/48 - 35*sin(x)**3*cos(x )/192 - 35*sin(x)*cos(x)/128
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.68 \[ \int \sin ^8(x) \, dx=\frac {1}{24} \, \sin \left (2 \, x\right )^{3} + \frac {35}{128} \, x + \frac {1}{1024} \, \sin \left (8 \, x\right ) + \frac {7}{128} \, \sin \left (4 \, x\right ) - \frac {1}{4} \, \sin \left (2 \, x\right ) \] Input:
integrate(sin(x)^8,x, algorithm="maxima")
Output:
1/24*sin(2*x)^3 + 35/128*x + 1/1024*sin(8*x) + 7/128*sin(4*x) - 1/4*sin(2* x)
Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.64 \[ \int \sin ^8(x) \, dx=\frac {35}{128} \, x + \frac {1}{1024} \, \sin \left (8 \, x\right ) - \frac {1}{96} \, \sin \left (6 \, x\right ) + \frac {7}{128} \, \sin \left (4 \, x\right ) - \frac {7}{32} \, \sin \left (2 \, x\right ) \] Input:
integrate(sin(x)^8,x, algorithm="giac")
Output:
35/128*x + 1/1024*sin(8*x) - 1/96*sin(6*x) + 7/128*sin(4*x) - 7/32*sin(2*x )
Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.64 \[ \int \sin ^8(x) \, dx=\frac {35\,x}{128}-\frac {7\,\sin \left (2\,x\right )}{32}+\frac {7\,\sin \left (4\,x\right )}{128}-\frac {\sin \left (6\,x\right )}{96}+\frac {\sin \left (8\,x\right )}{1024} \] Input:
int(sin(x)^8,x)
Output:
(35*x)/128 - (7*sin(2*x))/32 + (7*sin(4*x))/128 - sin(6*x)/96 + sin(8*x)/1 024
Time = 0.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.77 \[ \int \sin ^8(x) \, dx=-\frac {\cos \left (x \right ) \sin \left (x \right )^{7}}{8}-\frac {7 \cos \left (x \right ) \sin \left (x \right )^{5}}{48}-\frac {35 \cos \left (x \right ) \sin \left (x \right )^{3}}{192}-\frac {35 \cos \left (x \right ) \sin \left (x \right )}{128}+\frac {35 x}{128} \] Input:
int(sin(x)^8,x)
Output:
( - 48*cos(x)*sin(x)**7 - 56*cos(x)*sin(x)**5 - 70*cos(x)*sin(x)**3 - 105* cos(x)*sin(x) + 105*x)/384