Integrand size = 8, antiderivative size = 88 \[ \int \sin ^8(a+b x) \, dx=\frac {35 x}{128}-\frac {35 \cos (a+b x) \sin (a+b x)}{128 b}-\frac {35 \cos (a+b x) \sin ^3(a+b x)}{192 b}-\frac {7 \cos (a+b x) \sin ^5(a+b x)}{48 b}-\frac {\cos (a+b x) \sin ^7(a+b x)}{8 b} \] Output:
35/128*x-35/128*cos(b*x+a)*sin(b*x+a)/b-35/192*cos(b*x+a)*sin(b*x+a)^3/b-7 /48*cos(b*x+a)*sin(b*x+a)^5/b-1/8*cos(b*x+a)*sin(b*x+a)^7/b
Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.62 \[ \int \sin ^8(a+b x) \, dx=\frac {840 a+840 b x-672 \sin (2 (a+b x))+168 \sin (4 (a+b x))-32 \sin (6 (a+b x))+3 \sin (8 (a+b x))}{3072 b} \] Input:
Integrate[Sin[a + b*x]^8,x]
Output:
(840*a + 840*b*x - 672*Sin[2*(a + b*x)] + 168*Sin[4*(a + b*x)] - 32*Sin[6* (a + b*x)] + 3*Sin[8*(a + b*x)])/(3072*b)
Time = 0.68 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, 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.
\(\displaystyle \int \sin ^8(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (a+b x)^8dx\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {7}{8} \int \sin ^6(a+b x)dx-\frac {\sin ^7(a+b x) \cos (a+b x)}{8 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {7}{8} \int \sin (a+b x)^6dx-\frac {\sin ^7(a+b x) \cos (a+b x)}{8 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {7}{8} \left (\frac {5}{6} \int \sin ^4(a+b x)dx-\frac {\sin ^5(a+b x) \cos (a+b x)}{6 b}\right )-\frac {\sin ^7(a+b x) \cos (a+b x)}{8 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {7}{8} \left (\frac {5}{6} \int \sin (a+b x)^4dx-\frac {\sin ^5(a+b x) \cos (a+b x)}{6 b}\right )-\frac {\sin ^7(a+b x) \cos (a+b x)}{8 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin ^2(a+b x)dx-\frac {\sin ^3(a+b x) \cos (a+b x)}{4 b}\right )-\frac {\sin ^5(a+b x) \cos (a+b x)}{6 b}\right )-\frac {\sin ^7(a+b x) \cos (a+b x)}{8 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin (a+b x)^2dx-\frac {\sin ^3(a+b x) \cos (a+b x)}{4 b}\right )-\frac {\sin ^5(a+b x) \cos (a+b x)}{6 b}\right )-\frac {\sin ^7(a+b x) \cos (a+b x)}{8 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}-\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )-\frac {\sin ^3(a+b x) \cos (a+b x)}{4 b}\right )-\frac {\sin ^5(a+b x) \cos (a+b x)}{6 b}\right )-\frac {\sin ^7(a+b x) \cos (a+b x)}{8 b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2}-\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )-\frac {\sin ^3(a+b x) \cos (a+b x)}{4 b}\right )-\frac {\sin ^5(a+b x) \cos (a+b x)}{6 b}\right )-\frac {\sin ^7(a+b x) \cos (a+b x)}{8 b}\) |
Input:
Int[Sin[a + b*x]^8,x]
Output:
-1/8*(Cos[a + b*x]*Sin[a + b*x]^7)/b + (7*(-1/6*(Cos[a + b*x]*Sin[a + b*x] ^5)/b + (5*(-1/4*(Cos[a + b*x]*Sin[a + b*x]^3)/b + (3*(x/2 - (Cos[a + b*x] *Sin[a + b*x])/(2*b)))/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 = 14.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.62
method | result | size |
parallelrisch | \(\frac {840 b x +3 \sin \left (8 b x +8 a \right )-32 \sin \left (6 b x +6 a \right )+168 \sin \left (4 b x +4 a \right )-672 \sin \left (2 b x +2 a \right )}{3072 b}\) | \(55\) |
derivativedivides | \(\frac {-\frac {\left (\sin \left (b x +a \right )^{7}+\frac {7 \sin \left (b x +a \right )^{5}}{6}+\frac {35 \sin \left (b x +a \right )^{3}}{24}+\frac {35 \sin \left (b x +a \right )}{16}\right ) \cos \left (b x +a \right )}{8}+\frac {35 b x}{128}+\frac {35 a}{128}}{b}\) | \(58\) |
default | \(\frac {-\frac {\left (\sin \left (b x +a \right )^{7}+\frac {7 \sin \left (b x +a \right )^{5}}{6}+\frac {35 \sin \left (b x +a \right )^{3}}{24}+\frac {35 \sin \left (b x +a \right )}{16}\right ) \cos \left (b x +a \right )}{8}+\frac {35 b x}{128}+\frac {35 a}{128}}{b}\) | \(58\) |
risch | \(\frac {35 x}{128}+\frac {\sin \left (8 b x +8 a \right )}{1024 b}-\frac {\sin \left (6 b x +6 a \right )}{96 b}+\frac {7 \sin \left (4 b x +4 a \right )}{128 b}-\frac {7 \sin \left (2 b x +2 a \right )}{32 b}\) | \(61\) |
norman | \(\frac {\frac {35 x}{128}-\frac {35 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{64 b}-\frac {805 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{192 b}-\frac {2681 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}}{192 b}-\frac {5053 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{7}}{192 b}+\frac {5053 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{9}}{192 b}+\frac {2681 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{11}}{192 b}+\frac {805 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{13}}{192 b}+\frac {35 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{15}}{64 b}+\frac {35 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{16}+\frac {245 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}{32}+\frac {245 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}}{16}+\frac {1225 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{8}}{64}+\frac {245 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{10}}{16}+\frac {245 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{12}}{32}+\frac {35 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{14}}{16}+\frac {35 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{16}}{128}}{\left (1+\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right )^{8}}\) | \(259\) |
orering | \(x \sin \left (b x +a \right )^{8}-\frac {205 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{7}}{72 b}+\frac {205 x \left (56 \sin \left (b x +a \right )^{6} b^{2} \cos \left (b x +a \right )^{2}-8 \sin \left (b x +a \right )^{8} b^{2}\right )}{576 b^{2}}-\frac {91 \left (336 \sin \left (b x +a \right )^{5} b^{3} \cos \left (b x +a \right )^{3}-176 \sin \left (b x +a \right )^{7} b^{3} \cos \left (b x +a \right )\right )}{3072 b^{4}}+\frac {91 x \left (1680 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{4} b^{4}-2240 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{6} b^{4}+176 \sin \left (b x +a \right )^{8} b^{4}\right )}{3072 b^{4}}-\frac {5 \left (-20160 \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )^{5} b^{5}+6720 \cos \left (b x +a \right )^{5} \sin \left (b x +a \right )^{3} b^{5}+5888 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{7} b^{5}\right )}{6144 b^{6}}+\frac {5 x \left (101696 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{6} b^{6}-134400 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{4} b^{6}+20160 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{2} b^{6}-5888 b^{6} \sin \left (b x +a \right )^{8}\right )}{6144 b^{6}}-\frac {-250496 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{7} b^{7}+1147776 \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )^{5} b^{7}-658560 \cos \left (b x +a \right )^{5} \sin \left (b x +a \right )^{3} b^{7}+40320 \cos \left (b x +a \right )^{7} \sin \left (b x +a \right ) b^{7}}{147456 b^{8}}+\frac {x \left (250496 b^{8} \sin \left (b x +a \right )^{8}-5196800 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{6} b^{8}+9031680 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{4} b^{8}-2257920 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{2} b^{8}+40320 \cos \left (b x +a \right )^{8} b^{8}\right )}{147456 b^{8}}\) | \(512\) |
Input:
int(sin(b*x+a)^8,x,method=_RETURNVERBOSE)
Output:
1/3072*(840*b*x+3*sin(8*b*x+8*a)-32*sin(6*b*x+6*a)+168*sin(4*b*x+4*a)-672* sin(2*b*x+2*a))/b
Time = 0.10 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64 \[ \int \sin ^8(a+b x) \, dx=\frac {105 \, b x + {\left (48 \, \cos \left (b x + a\right )^{7} - 200 \, \cos \left (b x + a\right )^{5} + 326 \, \cos \left (b x + a\right )^{3} - 279 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{384 \, b} \] Input:
integrate(sin(b*x+a)^8,x, algorithm="fricas")
Output:
1/384*(105*b*x + (48*cos(b*x + a)^7 - 200*cos(b*x + a)^5 + 326*cos(b*x + a )^3 - 279*cos(b*x + a))*sin(b*x + a))/b
Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (82) = 164\).
Time = 0.64 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.09 \[ \int \sin ^8(a+b x) \, dx=\begin {cases} \frac {35 x \sin ^{8}{\left (a + b x \right )}}{128} + \frac {35 x \sin ^{6}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32} + \frac {105 x \sin ^{4}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{64} + \frac {35 x \sin ^{2}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{32} + \frac {35 x \cos ^{8}{\left (a + b x \right )}}{128} - \frac {93 \sin ^{7}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{128 b} - \frac {511 \sin ^{5}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{384 b} - \frac {385 \sin ^{3}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{384 b} - \frac {35 \sin {\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{128 b} & \text {for}\: b \neq 0 \\x \sin ^{8}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:
integrate(sin(b*x+a)**8,x)
Output:
Piecewise((35*x*sin(a + b*x)**8/128 + 35*x*sin(a + b*x)**6*cos(a + b*x)**2 /32 + 105*x*sin(a + b*x)**4*cos(a + b*x)**4/64 + 35*x*sin(a + b*x)**2*cos( a + b*x)**6/32 + 35*x*cos(a + b*x)**8/128 - 93*sin(a + b*x)**7*cos(a + b*x )/(128*b) - 511*sin(a + b*x)**5*cos(a + b*x)**3/(384*b) - 385*sin(a + b*x) **3*cos(a + b*x)**5/(384*b) - 35*sin(a + b*x)*cos(a + b*x)**7/(128*b), Ne( b, 0)), (x*sin(a)**8, True))
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.67 \[ \int \sin ^8(a+b x) \, dx=\frac {128 \, \sin \left (2 \, b x + 2 \, a\right )^{3} + 840 \, b x + 840 \, a + 3 \, \sin \left (8 \, b x + 8 \, a\right ) + 168 \, \sin \left (4 \, b x + 4 \, a\right ) - 768 \, \sin \left (2 \, b x + 2 \, a\right )}{3072 \, b} \] Input:
integrate(sin(b*x+a)^8,x, algorithm="maxima")
Output:
1/3072*(128*sin(2*b*x + 2*a)^3 + 840*b*x + 840*a + 3*sin(8*b*x + 8*a) + 16 8*sin(4*b*x + 4*a) - 768*sin(2*b*x + 2*a))/b
Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.68 \[ \int \sin ^8(a+b x) \, dx=\frac {35}{128} \, x + \frac {\sin \left (8 \, b x + 8 \, a\right )}{1024 \, b} - \frac {\sin \left (6 \, b x + 6 \, a\right )}{96 \, b} + \frac {7 \, \sin \left (4 \, b x + 4 \, a\right )}{128 \, b} - \frac {7 \, \sin \left (2 \, b x + 2 \, a\right )}{32 \, b} \] Input:
integrate(sin(b*x+a)^8,x, algorithm="giac")
Output:
35/128*x + 1/1024*sin(8*b*x + 8*a)/b - 1/96*sin(6*b*x + 6*a)/b + 7/128*sin (4*b*x + 4*a)/b - 7/32*sin(2*b*x + 2*a)/b
Time = 26.51 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.02 \[ \int \sin ^8(a+b x) \, dx=\frac {35\,x}{128}-\frac {\frac {93\,{\mathrm {tan}\left (a+b\,x\right )}^7}{128}+\frac {511\,{\mathrm {tan}\left (a+b\,x\right )}^5}{384}+\frac {385\,{\mathrm {tan}\left (a+b\,x\right )}^3}{384}+\frac {35\,\mathrm {tan}\left (a+b\,x\right )}{128}}{b\,\left ({\mathrm {tan}\left (a+b\,x\right )}^8+4\,{\mathrm {tan}\left (a+b\,x\right )}^6+6\,{\mathrm {tan}\left (a+b\,x\right )}^4+4\,{\mathrm {tan}\left (a+b\,x\right )}^2+1\right )} \] Input:
int(sin(a + b*x)^8,x)
Output:
(35*x)/128 - ((35*tan(a + b*x))/128 + (385*tan(a + b*x)^3)/384 + (511*tan( a + b*x)^5)/384 + (93*tan(a + b*x)^7)/128)/(b*(4*tan(a + b*x)^2 + 6*tan(a + b*x)^4 + 4*tan(a + b*x)^6 + tan(a + b*x)^8 + 1))
Time = 0.17 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.82 \[ \int \sin ^8(a+b x) \, dx=\frac {-48 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{7}-56 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{5}-70 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3}-105 \cos \left (b x +a \right ) \sin \left (b x +a \right )+105 b x}{384 b} \] Input:
int(sin(b*x+a)^8,x)
Output:
( - 48*cos(a + b*x)*sin(a + b*x)**7 - 56*cos(a + b*x)*sin(a + b*x)**5 - 70 *cos(a + b*x)*sin(a + b*x)**3 - 105*cos(a + b*x)*sin(a + b*x) + 105*b*x)/( 384*b)