Integrand size = 8, antiderivative size = 88 \[ \int \cos ^8(a+b x) \, dx=\frac {35 x}{128}+\frac {35 \cos (a+b x) \sin (a+b x)}{128 b}+\frac {35 \cos ^3(a+b x) \sin (a+b x)}{192 b}+\frac {7 \cos ^5(a+b x) \sin (a+b x)}{48 b}+\frac {\cos ^7(a+b x) \sin (a+b x)}{8 b} \]
35/128*x+35/128*cos(b*x+a)*sin(b*x+a)/b+35/192*cos(b*x+a)^3*sin(b*x+a)/b+7 /48*cos(b*x+a)^5*sin(b*x+a)/b+1/8*cos(b*x+a)^7*sin(b*x+a)/b
Time = 0.06 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.62 \[ \int \cos ^8(a+b x) \, dx=\frac {840 a+840 b x+672 \sin (2 (a+b x))+168 \sin (4 (a+b x))+32 \sin (6 (a+b x))+3 \sin (8 (a+b x))}{3072 b} \]
(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 = 0.38 (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.
\(\displaystyle \int \cos ^8(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin \left (a+b x+\frac {\pi }{2}\right )^8dx\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {7}{8} \int \cos ^6(a+b x)dx+\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {7}{8} \int \sin \left (a+b x+\frac {\pi }{2}\right )^6dx+\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {7}{8} \left (\frac {5}{6} \int \cos ^4(a+b x)dx+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )+\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {7}{8} \left (\frac {5}{6} \int \sin \left (a+b x+\frac {\pi }{2}\right )^4dx+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )+\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(a+b x)dx+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )+\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )+\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )+\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}+\frac {7}{8} \left (\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}+\frac {5}{6} \left (\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )\right )\right )\) |
(Cos[a + b*x]^7*Sin[a + b*x])/(8*b) + (7*((Cos[a + b*x]^5*Sin[a + b*x])/(6 *b) + (5*((Cos[a + b*x]^3*Sin[a + b*x])/(4*b) + (3*(x/2 + (Cos[a + b*x]*Si n[a + b*x])/(2*b)))/4))/6))/8
3.1.8.3.1 Defintions of rubi rules used
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 = 1.15 (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 | \(\frac {\frac {\left (\cos ^{7}\left (b x +a \right )+\frac {7 \left (\cos ^{5}\left (b x +a \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (b x +a \right )\right )}{24}+\frac {35 \cos \left (b x +a \right )}{16}\right ) \sin \left (b x +a \right )}{8}+\frac {35 b x}{128}+\frac {35 a}{128}}{b}\) | \(58\) |
default | \(\frac {\frac {\left (\cos ^{7}\left (b x +a \right )+\frac {7 \left (\cos ^{5}\left (b x +a \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (b x +a \right )\right )}{24}+\frac {35 \cos \left (b x +a \right )}{16}\right ) \sin \left (b x +a \right )}{8}+\frac {35 b x}{128}+\frac {35 a}{128}}{b}\) | \(58\) |
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 | \(\frac {\frac {35 x}{128}+\frac {93 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{64 b}+\frac {91 \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{192 b}+\frac {1799 \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{192 b}-\frac {1085 \left (\tan ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{192 b}+\frac {1085 \left (\tan ^{9}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{192 b}-\frac {1799 \left (\tan ^{11}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{192 b}-\frac {91 \left (\tan ^{13}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{192 b}-\frac {93 \left (\tan ^{15}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{64 b}+\frac {35 x \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{16}+\frac {245 x \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{32}+\frac {245 x \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{16}+\frac {1225 x \left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{64}+\frac {245 x \left (\tan ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{16}+\frac {245 x \left (\tan ^{12}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{32}+\frac {35 x \left (\tan ^{14}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{16}+\frac {35 x \left (\tan ^{16}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{128}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{8}}\) | \(259\) |
Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64 \[ \int \cos ^8(a+b x) \, dx=\frac {105 \, b x + {\left (48 \, \cos \left (b x + a\right )^{7} + 56 \, \cos \left (b x + a\right )^{5} + 70 \, \cos \left (b x + a\right )^{3} + 105 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{384 \, b} \]
1/384*(105*b*x + (48*cos(b*x + a)^7 + 56*cos(b*x + a)^5 + 70*cos(b*x + a)^ 3 + 105*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 = 0.71 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.09 \[ \int \cos ^8(a+b x) \, dx=\begin {cases} \frac {35 x \sin ^{8}{\left (a + b x \right )}}{128} + \frac {35 x \sin ^{6}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32} + \frac {105 x \sin ^{4}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{64} + \frac {35 x \sin ^{2}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{32} + \frac {35 x \cos ^{8}{\left (a + b x \right )}}{128} + \frac {35 \sin ^{7}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{128 b} + \frac {385 \sin ^{5}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{384 b} + \frac {511 \sin ^{3}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{384 b} + \frac {93 \sin {\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{128 b} & \text {for}\: b \neq 0 \\x \cos ^{8}{\left (a \right )} & \text {otherwise} \end {cases} \]
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 + 35*sin(a + b*x)**7*cos(a + b*x )/(128*b) + 385*sin(a + b*x)**5*cos(a + b*x)**3/(384*b) + 511*sin(a + b*x) **3*cos(a + b*x)**5/(384*b) + 93*sin(a + b*x)*cos(a + b*x)**7/(128*b), Ne( b, 0)), (x*cos(a)**8, True))
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.67 \[ \int \cos ^8(a+b x) \, dx=-\frac {128 \, \sin \left (2 \, b x + 2 \, a\right )^{3} - 840 \, b x - 840 \, a - 3 \, \sin \left (8 \, b x + 8 \, a\right ) - 168 \, \sin \left (4 \, b x + 4 \, a\right ) - 768 \, \sin \left (2 \, b x + 2 \, a\right )}{3072 \, b} \]
-1/3072*(128*sin(2*b*x + 2*a)^3 - 840*b*x - 840*a - 3*sin(8*b*x + 8*a) - 1 68*sin(4*b*x + 4*a) - 768*sin(2*b*x + 2*a))/b
Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.68 \[ \int \cos ^8(a+b x) \, dx=\frac {35}{128} \, x + \frac {\sin \left (8 \, b x + 8 \, a\right )}{1024 \, b} + \frac {\sin \left (6 \, b x + 6 \, a\right )}{96 \, b} + \frac {7 \, \sin \left (4 \, b x + 4 \, a\right )}{128 \, b} + \frac {7 \, \sin \left (2 \, b x + 2 \, a\right )}{32 \, b} \]
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 = 14.51 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.60 \[ \int \cos ^8(a+b x) \, dx=\frac {35\,x}{128}+\frac {\frac {7\,\sin \left (2\,a+2\,b\,x\right )}{32}+\frac {7\,\sin \left (4\,a+4\,b\,x\right )}{128}+\frac {\sin \left (6\,a+6\,b\,x\right )}{96}+\frac {\sin \left (8\,a+8\,b\,x\right )}{1024}}{b} \]