Integrand size = 17, antiderivative size = 88 \[ \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx=\frac {5 x}{128}+\frac {5 \cos (a+b x) \sin (a+b x)}{128 b}+\frac {5 \cos ^3(a+b x) \sin (a+b x)}{192 b}+\frac {\cos ^5(a+b x) \sin (a+b x)}{48 b}-\frac {\cos ^7(a+b x) \sin (a+b x)}{8 b} \] Output:
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 = 0.19 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.59 \[ \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx=\frac {120 b x+48 \sin (2 (a+b x))-24 \sin (4 (a+b x))-16 \sin (6 (a+b x))-3 \sin (8 (a+b x))}{3072 b} \] Input:
Integrate[Cos[a + b*x]^6*Sin[a + b*x]^2,x]
Output:
(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 = 0.71 (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 |
| \(\displaystyle \frac {1}{8} \left (\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}+\frac {5}{6} \left (\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )\right )\right )-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\) |
Input:
Int[Cos[a + b*x]^6*Sin[a + b*x]^2,x]
Output:
-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
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 = 15.71 (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 | \(\frac {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )^{7}}{8}+\frac {\left (\cos \left (b x +a \right )^{5}+\frac {5 \cos \left (b x +a \right )^{3}}{4}+\frac {15 \cos \left (b x +a \right )}{8}\right ) \sin \left (b x +a \right )}{48}+\frac {5 b x}{128}+\frac {5 a}{128}}{b}\) | \(64\) |
| default | \(\frac {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )^{7}}{8}+\frac {\left (\cos \left (b x +a \right )^{5}+\frac {5 \cos \left (b x +a \right )^{3}}{4}+\frac {15 \cos \left (b x +a \right )}{8}\right ) \sin \left (b x +a \right )}{48}+\frac {5 b x}{128}+\frac {5 a}{128}}{b}\) | \(64\) |
| norman | \(\frac {\frac {5 x}{128}-\frac {5 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{64 b}+\frac {397 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{192 b}-\frac {895 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}}{192 b}+\frac {1765 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{7}}{192 b}-\frac {1765 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{9}}{192 b}+\frac {895 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{11}}{192 b}-\frac {397 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{13}}{192 b}+\frac {5 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{15}}{64 b}+\frac {5 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{16}+\frac {35 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}{32}+\frac {35 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}}{16}+\frac {175 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{8}}{64}+\frac {35 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{10}}{16}+\frac {35 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{12}}{32}+\frac {5 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{14}}{16}+\frac {5 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{16}}{128}}{\left (1+\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right )^{8}}\) | \(259\) |
| orering | \(x \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{2}-\frac {205 \left (-6 \cos \left (b x +a \right )^{5} \sin \left (b x +a \right )^{3} b +2 \cos \left (b x +a \right )^{7} \sin \left (b x +a \right ) b \right )}{576 b^{2}}+\frac {205 x \left (30 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{4} b^{2}-32 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{2} b^{2}+2 \cos \left (b x +a \right )^{8} b^{2}\right )}{576 b^{2}}-\frac {91 \left (-120 \sin \left (b x +a \right )^{5} b^{3} \cos \left (b x +a \right )^{3}+312 \cos \left (b x +a \right )^{5} \sin \left (b x +a \right )^{3} b^{3}-80 \cos \left (b x +a \right )^{7} \sin \left (b x +a \right ) b^{3}\right )}{3072 b^{4}}+\frac {91 x \left (-2160 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{4} b^{4}+360 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{6} b^{4}+1496 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{2} b^{4}-80 \cos \left (b x +a \right )^{8} b^{4}\right )}{3072 b^{4}}-\frac {5 \left (10800 \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )^{5} b^{5}-17616 \cos \left (b x +a \right )^{5} \sin \left (b x +a \right )^{3} b^{5}-720 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{7} b^{5}+3632 \cos \left (b x +a \right )^{7} \sin \left (b x +a \right ) b^{5}\right )}{6144 b^{6}}+\frac {5 x \left (-37440 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{6} b^{6}+142080 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{4} b^{6}-78272 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{2} b^{6}+720 b^{6} \sin \left (b x +a \right )^{8}+3632 \cos \left (b x +a \right )^{8} b^{6}\right )}{6144 b^{6}}-\frac {80640 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{7} b^{7}-792960 \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )^{5} b^{7}+1037952 \cos \left (b x +a \right )^{5} \sin \left (b x +a \right )^{3} b^{7}-185600 \cos \left (b x +a \right )^{7} \sin \left (b x +a \right ) b^{7}}{147456 b^{8}}+\frac {x \left (-80640 b^{8} \sin \left (b x +a \right )^{8}+2943360 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{6} b^{8}-9154560 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{4} b^{8}+4413056 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{2} b^{8}-185600 \cos \left (b x +a \right )^{8} b^{8}\right )}{147456 b^{8}}\) | \(638\) |
Input:
int(cos(b*x+a)^6*sin(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/3072*(120*b*x-3*sin(8*b*x+8*a)-16*sin(6*b*x+6*a)-24*sin(4*b*x+4*a)+48*si n(2*b*x+2*a))/b
Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.65 \[ \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx=\frac {15 \, b x - {\left (48 \, \cos \left (b x + a\right )^{7} - 8 \, \cos \left (b x + a\right )^{5} - 10 \, \cos \left (b x + a\right )^{3} - 15 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{384 \, b} \] Input:
integrate(cos(b*x+a)^6*sin(b*x+a)^2,x, algorithm="fricas")
Output:
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 \[ \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx=\begin {cases} \frac {5 x \sin ^{8}{\left (a + b x \right )}}{128} + \frac {5 x \sin ^{6}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32} + \frac {15 x \sin ^{4}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{64} + \frac {5 x \sin ^{2}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{32} + \frac {5 x \cos ^{8}{\left (a + b x \right )}}{128} + \frac {5 \sin ^{7}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{128 b} + \frac {55 \sin ^{5}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{384 b} + \frac {73 \sin ^{3}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{384 b} - \frac {5 \sin {\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{128 b} & \text {for}\: b \neq 0 \\x \sin ^{2}{\left (a \right )} \cos ^{6}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(b*x+a)**6*sin(b*x+a)**2,x)
Output:
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 = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.55 \[ \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx=\frac {64 \, \sin \left (2 \, b x + 2 \, a\right )^{3} + 120 \, b x + 120 \, a - 3 \, \sin \left (8 \, b x + 8 \, a\right ) - 24 \, \sin \left (4 \, b x + 4 \, a\right )}{3072 \, b} \] Input:
integrate(cos(b*x+a)^6*sin(b*x+a)^2,x, algorithm="maxima")
Output:
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 = 0.13 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.68 \[ \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx=\frac {5}{128} \, x - \frac {\sin \left (8 \, b x + 8 \, a\right )}{1024 \, b} - \frac {\sin \left (6 \, b x + 6 \, a\right )}{192 \, b} - \frac {\sin \left (4 \, b x + 4 \, a\right )}{128 \, b} + \frac {\sin \left (2 \, b x + 2 \, a\right )}{64 \, b} \] Input:
integrate(cos(b*x+a)^6*sin(b*x+a)^2,x, algorithm="giac")
Output:
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 = 26.73 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.01 \[ \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx=\frac {5\,x}{128}+\frac {\frac {5\,{\mathrm {tan}\left (a+b\,x\right )}^7}{128}+\frac {55\,{\mathrm {tan}\left (a+b\,x\right )}^5}{384}+\frac {73\,{\mathrm {tan}\left (a+b\,x\right )}^3}{384}-\frac {5\,\mathrm {tan}\left (a+b\,x\right )}{128}}{b\,\left ({\mathrm {tan}\left (a+b\,x\right )}^8+4\,{\mathrm {tan}\left (a+b\,x\right )}^6+6\,{\mathrm {tan}\left (a+b\,x\right )}^4+4\,{\mathrm {tan}\left (a+b\,x\right )}^2+1\right )} \] Input:
int(cos(a + b*x)^6*sin(a + b*x)^2,x)
Output:
(5*x)/128 + ((73*tan(a + b*x)^3)/384 - (5*tan(a + b*x))/128 + (55*tan(a + b*x)^5)/384 + (5*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 = 0.15 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.82 \[ \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx=\frac {48 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{7}-136 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{5}+118 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3}-15 \cos \left (b x +a \right ) \sin \left (b x +a \right )+15 b x}{384 b} \] Input:
int(cos(b*x+a)^6*sin(b*x+a)^2,x)
Output:
(48*cos(a + b*x)*sin(a + b*x)**7 - 136*cos(a + b*x)*sin(a + b*x)**5 + 118* cos(a + b*x)*sin(a + b*x)**3 - 15*cos(a + b*x)*sin(a + b*x) + 15*b*x)/(384 *b)