Integrand size = 19, antiderivative size = 87 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5 a x}{16}-\frac {a \cos ^7(c+d x)}{7 d}+\frac {5 a \cos (c+d x) \sin (c+d x)}{16 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{6 d} \]
5/16*a*x-1/7*a*cos(d*x+c)^7/d+5/16*a*cos(d*x+c)*sin(d*x+c)/d+5/24*a*cos(d* x+c)^3*sin(d*x+c)/d+1/6*a*cos(d*x+c)^5*sin(d*x+c)/d
Time = 0.17 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.66 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (-192 \cos ^7(c+d x)+7 (60 c+60 d x+45 \sin (2 (c+d x))+9 \sin (4 (c+d x))+\sin (6 (c+d x)))\right )}{1344 d} \]
(a*(-192*Cos[c + d*x]^7 + 7*(60*c + 60*d*x + 45*Sin[2*(c + d*x)] + 9*Sin[4 *(c + d*x)] + Sin[6*(c + d*x)])))/(1344*d)
Time = 0.43 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3042, 3148, 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.
\(\displaystyle \int \cos ^6(c+d x) (a \sin (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cos (c+d x)^6 (a \sin (c+d x)+a)dx\) |
\(\Big \downarrow \) 3148 |
\(\displaystyle a \int \cos ^6(c+d x)dx-\frac {a \cos ^7(c+d x)}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx-\frac {a \cos ^7(c+d x)}{7 d}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle a \left (\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle a \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle a \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle a \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {a \cos ^7(c+d x)}{7 d}\) |
-1/7*(a*Cos[c + d*x]^7)/d + a*((Cos[c + d*x]^5*Sin[c + d*x])/(6*d) + (5*(( Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (3*(x/2 + (Cos[c + d*x]*Sin[c + d*x]) /(2*d)))/4))/6)
3.1.2.3.1 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]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Time = 0.64 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{7}+a \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(62\) |
default | \(\frac {-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{7}+a \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(62\) |
parallelrisch | \(-\frac {a \left (-420 d x +3 \cos \left (7 d x +7 c \right )-7 \sin \left (6 d x +6 c \right )-63 \sin \left (4 d x +4 c \right )-315 \sin \left (2 d x +2 c \right )+21 \cos \left (5 d x +5 c \right )+63 \cos \left (3 d x +3 c \right )+105 \cos \left (d x +c \right )+192\right )}{1344 d}\) | \(87\) |
risch | \(\frac {5 a x}{16}-\frac {5 a \cos \left (d x +c \right )}{64 d}-\frac {a \cos \left (7 d x +7 c \right )}{448 d}+\frac {a \sin \left (6 d x +6 c \right )}{192 d}-\frac {a \cos \left (5 d x +5 c \right )}{64 d}+\frac {3 a \sin \left (4 d x +4 c \right )}{64 d}-\frac {3 a \cos \left (3 d x +3 c \right )}{64 d}+\frac {15 a \sin \left (2 d x +2 c \right )}{64 d}\) | \(108\) |
norman | \(\frac {\frac {5 a x}{16}-\frac {2 a}{7 d}+\frac {11 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {7 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {85 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {85 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {7 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {11 a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {35 a x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {105 a x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {175 a x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {175 a x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {105 a x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {5 a x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {6 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {10 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\) | \(284\) |
1/d*(-1/7*a*cos(d*x+c)^7+a*(1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d* x+c))*sin(d*x+c)+5/16*d*x+5/16*c))
Time = 0.30 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.71 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {48 \, a \cos \left (d x + c\right )^{7} - 105 \, a d x - 7 \, {\left (8 \, a \cos \left (d x + c\right )^{5} + 10 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{336 \, d} \]
-1/336*(48*a*cos(d*x + c)^7 - 105*a*d*x - 7*(8*a*cos(d*x + c)^5 + 10*a*cos (d*x + c)^3 + 15*a*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (82) = 164\).
Time = 0.51 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.98 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {5 a x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 a x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 a x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 a x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 a \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 a \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 a \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {a \cos ^{7}{\left (c + d x \right )}}{7 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((5*a*x*sin(c + d*x)**6/16 + 15*a*x*sin(c + d*x)**4*cos(c + d*x)* *2/16 + 15*a*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 5*a*x*cos(c + d*x)**6/ 16 + 5*a*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 5*a*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) + 11*a*sin(c + d*x)*cos(c + d*x)**5/(16*d) - a*cos(c + d*x) **7/(7*d), Ne(d, 0)), (x*(a*sin(c) + a)*cos(c)**6, True))
Time = 0.19 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {192 \, a \cos \left (d x + c\right )^{7} + 7 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a}{1344 \, d} \]
-1/1344*(192*a*cos(d*x + c)^7 + 7*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*a)/d
Time = 0.32 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.23 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5}{16} \, a x - \frac {a \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {a \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac {3 \, a \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {5 \, a \cos \left (d x + c\right )}{64 \, d} + \frac {a \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {3 \, a \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {15 \, a \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
5/16*a*x - 1/448*a*cos(7*d*x + 7*c)/d - 1/64*a*cos(5*d*x + 5*c)/d - 3/64*a *cos(3*d*x + 3*c)/d - 5/64*a*cos(d*x + c)/d + 1/192*a*sin(6*d*x + 6*c)/d + 3/64*a*sin(4*d*x + 4*c)/d + 15/64*a*sin(2*d*x + 2*c)/d
Time = 10.11 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.60 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5\,a\,x}{16}+\frac {-\frac {11\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{8}+\left (\frac {a\,\left (735\,c+735\,d\,x-672\right )}{336}-\frac {35\,a\,\left (c+d\,x\right )}{16}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{6}-\frac {85\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+\left (\frac {a\,\left (3675\,c+3675\,d\,x-3360\right )}{336}-\frac {175\,a\,\left (c+d\,x\right )}{16}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {85\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{24}+\left (\frac {a\,\left (2205\,c+2205\,d\,x-2016\right )}{336}-\frac {105\,a\,\left (c+d\,x\right )}{16}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6}+\frac {11\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {a\,\left (105\,c+105\,d\,x-96\right )}{336}-\frac {5\,a\,\left (c+d\,x\right )}{16}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]
(5*a*x)/16 + ((a*(105*c + 105*d*x - 96))/336 + (11*a*tan(c/2 + (d*x)/2))/8 - (5*a*(c + d*x))/16 + tan(c/2 + (d*x)/2)^12*((a*(735*c + 735*d*x - 672)) /336 - (35*a*(c + d*x))/16) + tan(c/2 + (d*x)/2)^4*((a*(2205*c + 2205*d*x - 2016))/336 - (105*a*(c + d*x))/16) + tan(c/2 + (d*x)/2)^8*((a*(3675*c + 3675*d*x - 3360))/336 - (175*a*(c + d*x))/16) + (7*a*tan(c/2 + (d*x)/2)^3) /6 + (85*a*tan(c/2 + (d*x)/2)^5)/24 - (85*a*tan(c/2 + (d*x)/2)^9)/24 - (7* a*tan(c/2 + (d*x)/2)^11)/6 - (11*a*tan(c/2 + (d*x)/2)^13)/8)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^7)