Integrand size = 17, antiderivative size = 88 \[ \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx=\frac {5 x}{128}+\frac {5 \cos (a+b x) \sin (a+b x)}{128 b}+\frac {5 \cos ^3(a+b x) \sin (a+b x)}{192 b}+\frac {\cos ^5(a+b x) \sin (a+b x)}{48 b}-\frac {\cos ^7(a+b x) \sin (a+b x)}{8 b} \]
5/128*x+5/128*cos(b*x+a)*sin(b*x+a)/b+5/192*cos(b*x+a)^3*sin(b*x+a)/b+1/48 *cos(b*x+a)^5*sin(b*x+a)/b-1/8*cos(b*x+a)^7*sin(b*x+a)/b
Time = 0.12 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.59 \[ \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx=\frac {120 b x+48 \sin (2 (a+b x))-24 \sin (4 (a+b x))-16 \sin (6 (a+b x))-3 \sin (8 (a+b x))}{3072 b} \]
(120*b*x + 48*Sin[2*(a + b*x)] - 24*Sin[4*(a + b*x)] - 16*Sin[6*(a + b*x)] - 3*Sin[8*(a + b*x)])/(3072*b)
Time = 0.40 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {3042, 3048, 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.
\(\displaystyle \int \sin ^2(a+b x) \cos ^6(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (a+b x)^2 \cos (a+b x)^6dx\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle \frac {1}{8} \int \cos ^6(a+b x)dx-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{8} \int \sin \left (a+b x+\frac {\pi }{2}\right )^6dx-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {1}{8} \left (\frac {5}{6} \int \cos ^4(a+b x)dx+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{8} \left (\frac {5}{6} \int \sin \left (a+b x+\frac {\pi }{2}\right )^4dx+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(a+b x)dx+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {1}{8} \left (\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}+\frac {5}{6} \left (\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )\right )\right )-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\) |
-1/8*(Cos[a + b*x]^7*Sin[a + b*x])/b + ((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]*Sin[ a + b*x])/(2*b)))/4))/6)/8
3.1.58.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
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 = 0.36 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.62
method | result | size |
parallelrisch | \(\frac {120 b x -3 \sin \left (8 b x +8 a \right )-16 \sin \left (6 b x +6 a \right )-24 \sin \left (4 b x +4 a \right )+48 \sin \left (2 b x +2 a \right )}{3072 b}\) | \(55\) |
risch | \(\frac {5 x}{128}-\frac {\sin \left (8 b x +8 a \right )}{1024 b}-\frac {\sin \left (6 b x +6 a \right )}{192 b}-\frac {\sin \left (4 b x +4 a \right )}{128 b}+\frac {\sin \left (2 b x +2 a \right )}{64 b}\) | \(61\) |
derivativedivides | \(\frac {-\frac {\left (\cos ^{7}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{8}+\frac {\left (\cos ^{5}\left (b x +a \right )+\frac {5 \left (\cos ^{3}\left (b x +a \right )\right )}{4}+\frac {15 \cos \left (b x +a \right )}{8}\right ) \sin \left (b x +a \right )}{48}+\frac {5 b x}{128}+\frac {5 a}{128}}{b}\) | \(64\) |
default | \(\frac {-\frac {\left (\cos ^{7}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{8}+\frac {\left (\cos ^{5}\left (b x +a \right )+\frac {5 \left (\cos ^{3}\left (b x +a \right )\right )}{4}+\frac {15 \cos \left (b x +a \right )}{8}\right ) \sin \left (b x +a \right )}{48}+\frac {5 b x}{128}+\frac {5 a}{128}}{b}\) | \(64\) |
norman | \(\frac {\frac {5 x}{128}-\frac {5 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{64 b}+\frac {397 \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{192 b}-\frac {895 \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{192 b}+\frac {1765 \left (\tan ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{192 b}-\frac {1765 \left (\tan ^{9}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{192 b}+\frac {895 \left (\tan ^{11}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{192 b}-\frac {397 \left (\tan ^{13}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{192 b}+\frac {5 \left (\tan ^{15}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{64 b}+\frac {5 x \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{16}+\frac {35 x \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{32}+\frac {35 x \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{16}+\frac {175 x \left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{64}+\frac {35 x \left (\tan ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{16}+\frac {35 x \left (\tan ^{12}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{32}+\frac {5 x \left (\tan ^{14}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{16}+\frac {5 x \left (\tan ^{16}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{128}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{8}}\) | \(259\) |
Time = 0.31 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.65 \[ \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx=\frac {15 \, b x - {\left (48 \, \cos \left (b x + a\right )^{7} - 8 \, \cos \left (b x + a\right )^{5} - 10 \, \cos \left (b x + a\right )^{3} - 15 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{384 \, b} \]
1/384*(15*b*x - (48*cos(b*x + a)^7 - 8*cos(b*x + a)^5 - 10*cos(b*x + a)^3 - 15*cos(b*x + a))*sin(b*x + a))/b
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (80) = 160\).
Time = 0.65 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.15 \[ \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx=\begin {cases} \frac {5 x \sin ^{8}{\left (a + b x \right )}}{128} + \frac {5 x \sin ^{6}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32} + \frac {15 x \sin ^{4}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{64} + \frac {5 x \sin ^{2}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{32} + \frac {5 x \cos ^{8}{\left (a + b x \right )}}{128} + \frac {5 \sin ^{7}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{128 b} + \frac {55 \sin ^{5}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{384 b} + \frac {73 \sin ^{3}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{384 b} - \frac {5 \sin {\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{128 b} & \text {for}\: b \neq 0 \\x \sin ^{2}{\left (a \right )} \cos ^{6}{\left (a \right )} & \text {otherwise} \end {cases} \]
Piecewise((5*x*sin(a + b*x)**8/128 + 5*x*sin(a + b*x)**6*cos(a + b*x)**2/3 2 + 15*x*sin(a + b*x)**4*cos(a + b*x)**4/64 + 5*x*sin(a + b*x)**2*cos(a + b*x)**6/32 + 5*x*cos(a + b*x)**8/128 + 5*sin(a + b*x)**7*cos(a + b*x)/(128 *b) + 55*sin(a + b*x)**5*cos(a + b*x)**3/(384*b) + 73*sin(a + b*x)**3*cos( a + b*x)**5/(384*b) - 5*sin(a + b*x)*cos(a + b*x)**7/(128*b), Ne(b, 0)), ( x*sin(a)**2*cos(a)**6, True))
Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.55 \[ \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx=\frac {64 \, \sin \left (2 \, b x + 2 \, a\right )^{3} + 120 \, b x + 120 \, a - 3 \, \sin \left (8 \, b x + 8 \, a\right ) - 24 \, \sin \left (4 \, b x + 4 \, a\right )}{3072 \, b} \]
1/3072*(64*sin(2*b*x + 2*a)^3 + 120*b*x + 120*a - 3*sin(8*b*x + 8*a) - 24* sin(4*b*x + 4*a))/b
Time = 0.30 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.68 \[ \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx=\frac {5}{128} \, x - \frac {\sin \left (8 \, b x + 8 \, a\right )}{1024 \, b} - \frac {\sin \left (6 \, b x + 6 \, a\right )}{192 \, b} - \frac {\sin \left (4 \, b x + 4 \, a\right )}{128 \, b} + \frac {\sin \left (2 \, b x + 2 \, a\right )}{64 \, b} \]
5/128*x - 1/1024*sin(8*b*x + 8*a)/b - 1/192*sin(6*b*x + 6*a)/b - 1/128*sin (4*b*x + 4*a)/b + 1/64*sin(2*b*x + 2*a)/b
Time = 0.76 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.01 \[ \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx=\frac {5\,x}{128}+\frac {\frac {5\,{\mathrm {tan}\left (a+b\,x\right )}^7}{128}+\frac {55\,{\mathrm {tan}\left (a+b\,x\right )}^5}{384}+\frac {73\,{\mathrm {tan}\left (a+b\,x\right )}^3}{384}-\frac {5\,\mathrm {tan}\left (a+b\,x\right )}{128}}{b\,\left ({\mathrm {tan}\left (a+b\,x\right )}^8+4\,{\mathrm {tan}\left (a+b\,x\right )}^6+6\,{\mathrm {tan}\left (a+b\,x\right )}^4+4\,{\mathrm {tan}\left (a+b\,x\right )}^2+1\right )} \]