Integrand size = 19, antiderivative size = 87 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x)) \, dx=\frac {8 (a+a \sin (c+d x))^5}{5 a^4 d}-\frac {2 (a+a \sin (c+d x))^6}{a^5 d}+\frac {6 (a+a \sin (c+d x))^7}{7 a^6 d}-\frac {(a+a \sin (c+d x))^8}{8 a^7 d} \]
8/5*(a+a*sin(d*x+c))^5/a^4/d-2*(a+a*sin(d*x+c))^6/a^5/d+6/7*(a+a*sin(d*x+c ))^7/a^6/d-1/8*(a+a*sin(d*x+c))^8/a^7/d
Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.85 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos ^8(c+d x)}{8 d}+\frac {a \sin (c+d x)}{d}-\frac {a \sin ^3(c+d x)}{d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^7(c+d x)}{7 d} \]
-1/8*(a*Cos[c + d*x]^8)/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/d + (3 *a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^7)/(7*d)
Time = 0.25 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3042, 3146, 49, 2009}
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.
\(\displaystyle \int \cos ^7(c+d x) (a \sin (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cos (c+d x)^7 (a \sin (c+d x)+a)dx\) |
\(\Big \downarrow \) 3146 |
\(\displaystyle \frac {\int (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^4d(a \sin (c+d x))}{a^7 d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (-(\sin (c+d x) a+a)^7+6 a (\sin (c+d x) a+a)^6-12 a^2 (\sin (c+d x) a+a)^5+8 a^3 (\sin (c+d x) a+a)^4\right )d(a \sin (c+d x))}{a^7 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {8}{5} a^3 (a \sin (c+d x)+a)^5-2 a^2 (a \sin (c+d x)+a)^6-\frac {1}{8} (a \sin (c+d x)+a)^8+\frac {6}{7} a (a \sin (c+d x)+a)^7}{a^7 d}\) |
((8*a^3*(a + a*Sin[c + d*x])^5)/5 - 2*a^2*(a + a*Sin[c + d*x])^6 + (6*a*(a + a*Sin[c + d*x])^7)/7 - (a + a*Sin[c + d*x])^8/8)/(a^7*d)
3.1.1.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x )^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && I ntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/ 2])
Time = 0.44 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}-\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\sin ^{3}\left (d x +c \right )-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\sin \left (d x +c \right )\right )}{d}\) | \(84\) |
default | \(-\frac {a \left (\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}-\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\sin ^{3}\left (d x +c \right )-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\sin \left (d x +c \right )\right )}{d}\) | \(84\) |
risch | \(\frac {35 a \sin \left (d x +c \right )}{64 d}-\frac {a \cos \left (8 d x +8 c \right )}{1024 d}+\frac {a \sin \left (7 d x +7 c \right )}{448 d}-\frac {a \cos \left (6 d x +6 c \right )}{128 d}+\frac {7 a \sin \left (5 d x +5 c \right )}{320 d}-\frac {7 a \cos \left (4 d x +4 c \right )}{256 d}+\frac {7 a \sin \left (3 d x +3 c \right )}{64 d}-\frac {7 a \cos \left (2 d x +2 c \right )}{128 d}\) | \(119\) |
parallelrisch | \(\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )+\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )+3 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {53 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+7 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {513 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35}+\frac {513 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35}+7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {53 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}\) | \(168\) |
norman | \(\frac {\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {6 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {106 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {1026 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {1026 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {106 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {6 a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {14 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {14 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}\) | \(220\) |
-a/d*(1/8*sin(d*x+c)^8+1/7*sin(d*x+c)^7-1/2*sin(d*x+c)^6-3/5*sin(d*x+c)^5+ 3/4*sin(d*x+c)^4+sin(d*x+c)^3-1/2*sin(d*x+c)^2-sin(d*x+c))
Time = 0.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.71 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {35 \, a \cos \left (d x + c\right )^{8} - 8 \, {\left (5 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} + 8 \, a \cos \left (d x + c\right )^{2} + 16 \, a\right )} \sin \left (d x + c\right )}{280 \, d} \]
-1/280*(35*a*cos(d*x + c)^8 - 8*(5*a*cos(d*x + c)^6 + 6*a*cos(d*x + c)^4 + 8*a*cos(d*x + c)^2 + 16*a)*sin(d*x + c))/d
Time = 0.70 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.21 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {16 a \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 a \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 a \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {a \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {a \cos ^{8}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \cos ^{7}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((16*a*sin(c + d*x)**7/(35*d) + 8*a*sin(c + d*x)**5*cos(c + d*x)* *2/(5*d) + 2*a*sin(c + d*x)**3*cos(c + d*x)**4/d + a*sin(c + d*x)*cos(c + d*x)**6/d - a*cos(c + d*x)**8/(8*d), Ne(d, 0)), (x*(a*sin(c) + a)*cos(c)** 7, True))
Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.06 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {35 \, a \sin \left (d x + c\right )^{8} + 40 \, a \sin \left (d x + c\right )^{7} - 140 \, a \sin \left (d x + c\right )^{6} - 168 \, a \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} + 280 \, a \sin \left (d x + c\right )^{3} - 140 \, a \sin \left (d x + c\right )^{2} - 280 \, a \sin \left (d x + c\right )}{280 \, d} \]
-1/280*(35*a*sin(d*x + c)^8 + 40*a*sin(d*x + c)^7 - 140*a*sin(d*x + c)^6 - 168*a*sin(d*x + c)^5 + 210*a*sin(d*x + c)^4 + 280*a*sin(d*x + c)^3 - 140* a*sin(d*x + c)^2 - 280*a*sin(d*x + c))/d
Time = 0.33 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.36 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a \cos \left (6 \, d x + 6 \, c\right )}{128 \, d} - \frac {7 \, a \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {7 \, a \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {a \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {7 \, a \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {7 \, a \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} + \frac {35 \, a \sin \left (d x + c\right )}{64 \, d} \]
-1/1024*a*cos(8*d*x + 8*c)/d - 1/128*a*cos(6*d*x + 6*c)/d - 7/256*a*cos(4* d*x + 4*c)/d - 7/128*a*cos(2*d*x + 2*c)/d + 1/448*a*sin(7*d*x + 7*c)/d + 7 /320*a*sin(5*d*x + 5*c)/d + 7/64*a*sin(3*d*x + 3*c)/d + 35/64*a*sin(d*x + c)/d
Time = 5.92 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.03 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x)) \, dx=\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^8}{8}-\frac {a\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a\,{\sin \left (c+d\,x\right )}^6}{2}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^5}{5}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^4}{4}-a\,{\sin \left (c+d\,x\right )}^3+\frac {a\,{\sin \left (c+d\,x\right )}^2}{2}+a\,\sin \left (c+d\,x\right )}{d} \]