Integrand size = 27, antiderivative size = 143 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {19 a^3 x}{16}-\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{3 d}-\frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {19 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {19 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d} \] Output:
19/16*a^3*x-a^3*arctanh(cos(d*x+c))/d+a^3*cos(d*x+c)/d+1/3*a^3*cos(d*x+c)^ 3/d-3/5*a^3*cos(d*x+c)^5/d+19/16*a^3*cos(d*x+c)*sin(d*x+c)/d+19/24*a^3*cos (d*x+c)^3*sin(d*x+c)/d-1/6*a^3*cos(d*x+c)^5*sin(d*x+c)/d
Time = 6.16 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.71 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \left (1140 c+1140 d x+840 \cos (c+d x)-100 \cos (3 (c+d x))-36 \cos (5 (c+d x))-960 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+960 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+735 \sin (2 (c+d x))+75 \sin (4 (c+d x))-5 \sin (6 (c+d x))\right )}{960 d} \] Input:
Integrate[Cos[c + d*x]^3*Cot[c + d*x]*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*(1140*c + 1140*d*x + 840*Cos[c + d*x] - 100*Cos[3*(c + d*x)] - 36*Cos [5*(c + d*x)] - 960*Log[Cos[(c + d*x)/2]] + 960*Log[Sin[(c + d*x)/2]] + 73 5*Sin[2*(c + d*x)] + 75*Sin[4*(c + d*x)] - 5*Sin[6*(c + d*x)]))/(960*d)
Time = 0.45 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3042, 3352, 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.
| \(\displaystyle \int \cos ^3(c+d x) \cot (c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a \sin (c+d x)+a)^3}{\sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (3 a^3 \cos ^4(c+d x)+a^3 \sin ^2(c+d x) \cos ^4(c+d x)+3 a^3 \sin (c+d x) \cos ^4(c+d x)+a^3 \cos ^3(c+d x) \cot (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}-\frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {19 a^3 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {19 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {19 a^3 x}{16}\) |
Input:
Int[Cos[c + d*x]^3*Cot[c + d*x]*(a + a*Sin[c + d*x])^3,x]
Output:
(19*a^3*x)/16 - (a^3*ArcTanh[Cos[c + d*x]])/d + (a^3*Cos[c + d*x])/d + (a^ 3*Cos[c + d*x]^3)/(3*d) - (3*a^3*Cos[c + d*x]^5)/(5*d) + (19*a^3*Cos[c + d *x]*Sin[c + d*x])/(16*d) + (19*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - ( a^3*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Time = 5.52 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.03
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{5}}{6}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {3 a^{3} \cos \left (d x +c \right )^{5}}{5}+3 a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} \left (\frac {\cos \left (d x +c \right )^{3}}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(147\) |
| default | \(\frac {a^{3} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{5}}{6}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {3 a^{3} \cos \left (d x +c \right )^{5}}{5}+3 a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} \left (\frac {\cos \left (d x +c \right )^{3}}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(147\) |
| risch | \(\frac {19 a^{3} x}{16}+\frac {7 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{16 d}+\frac {7 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {a^{3} \sin \left (6 d x +6 c \right )}{192 d}-\frac {3 a^{3} \cos \left (5 d x +5 c \right )}{80 d}+\frac {5 a^{3} \sin \left (4 d x +4 c \right )}{64 d}-\frac {5 a^{3} \cos \left (3 d x +3 c \right )}{48 d}+\frac {49 a^{3} \sin \left (2 d x +2 c \right )}{64 d}\) | \(166\) |
Input:
int(cos(d*x+c)^3*cot(d*x+c)*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*(-1/6*sin(d*x+c)*cos(d*x+c)^5+1/24*(cos(d*x+c)^3+3/2*cos(d*x+c))* sin(d*x+c)+1/16*d*x+1/16*c)-3/5*a^3*cos(d*x+c)^5+3*a^3*(1/4*(cos(d*x+c)^3+ 3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+a^3*(1/3*cos(d*x+c)^3+cos(d*x+c) +ln(csc(d*x+c)-cot(d*x+c))))
Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.90 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {144 \, a^{3} \cos \left (d x + c\right )^{5} - 80 \, a^{3} \cos \left (d x + c\right )^{3} - 285 \, a^{3} d x - 240 \, a^{3} \cos \left (d x + c\right ) + 120 \, a^{3} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 120 \, a^{3} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 5 \, {\left (8 \, a^{3} \cos \left (d x + c\right )^{5} - 38 \, a^{3} \cos \left (d x + c\right )^{3} - 57 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \] Input:
integrate(cos(d*x+c)^3*cot(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
-1/240*(144*a^3*cos(d*x + c)^5 - 80*a^3*cos(d*x + c)^3 - 285*a^3*d*x - 240 *a^3*cos(d*x + c) + 120*a^3*log(1/2*cos(d*x + c) + 1/2) - 120*a^3*log(-1/2 *cos(d*x + c) + 1/2) + 5*(8*a^3*cos(d*x + c)^5 - 38*a^3*cos(d*x + c)^3 - 5 7*a^3*cos(d*x + c))*sin(d*x + c))/d
\[ \int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=a^{3} \left (\int \cos ^{3}{\left (c + d x \right )} \cot {\left (c + d x \right )}\, dx + \int 3 \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )} \cot {\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )} \cot {\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )} \cot {\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cos(d*x+c)**3*cot(d*x+c)*(a+a*sin(d*x+c))**3,x)
Output:
a**3*(Integral(cos(c + d*x)**3*cot(c + d*x), x) + Integral(3*sin(c + d*x)* cos(c + d*x)**3*cot(c + d*x), x) + Integral(3*sin(c + d*x)**2*cos(c + d*x) **3*cot(c + d*x), x) + Integral(sin(c + d*x)**3*cos(c + d*x)**3*cot(c + d* x), x))
Time = 0.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.94 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {576 \, a^{3} \cos \left (d x + c\right )^{5} - 160 \, {\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3}}{960 \, d} \] Input:
integrate(cos(d*x+c)^3*cot(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
-1/960*(576*a^3*cos(d*x + c)^5 - 160*(2*cos(d*x + c)^3 + 6*cos(d*x + c) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1))*a^3 - 5*(4*sin(2*d*x + 2*c)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4*c))*a^3 - 90*(12*d*x + 12*c + sin (4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*a^3)/d
Time = 0.23 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.60 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {285 \, {\left (d x + c\right )} a^{3} + 240 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, {\left (435 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 240 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 865 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1200 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 210 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1760 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 210 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1440 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 865 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1296 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 435 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 176 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \] Input:
integrate(cos(d*x+c)^3*cot(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/240*(285*(d*x + c)*a^3 + 240*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 2*(435 *a^3*tan(1/2*d*x + 1/2*c)^11 + 240*a^3*tan(1/2*d*x + 1/2*c)^10 + 865*a^3*t an(1/2*d*x + 1/2*c)^9 - 1200*a^3*tan(1/2*d*x + 1/2*c)^8 - 210*a^3*tan(1/2* d*x + 1/2*c)^7 - 1760*a^3*tan(1/2*d*x + 1/2*c)^6 + 210*a^3*tan(1/2*d*x + 1 /2*c)^5 - 1440*a^3*tan(1/2*d*x + 1/2*c)^4 - 865*a^3*tan(1/2*d*x + 1/2*c)^3 - 1296*a^3*tan(1/2*d*x + 1/2*c)^2 - 435*a^3*tan(1/2*d*x + 1/2*c) - 176*a^ 3)/(tan(1/2*d*x + 1/2*c)^2 + 1)^6)/d
Time = 18.92 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.48 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {19\,a^3\,\mathrm {atan}\left (\frac {361\,a^6}{64\,\left (\frac {19\,a^6}{4}-\frac {361\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}+\frac {19\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {19\,a^6}{4}-\frac {361\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}\right )}{8\,d}+\frac {-\frac {29\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}-2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {173\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+10\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {44\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}-\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+12\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {173\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {54\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {29\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {22\,a^3}{15}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \] Input:
int(cos(c + d*x)^3*cot(c + d*x)*(a + a*sin(c + d*x))^3,x)
Output:
(a^3*log(tan(c/2 + (d*x)/2)))/d + (19*a^3*atan((361*a^6)/(64*((19*a^6)/4 - (361*a^6*tan(c/2 + (d*x)/2))/64)) + (19*a^6*tan(c/2 + (d*x)/2))/(4*((19*a ^6)/4 - (361*a^6*tan(c/2 + (d*x)/2))/64))))/(8*d) + ((54*a^3*tan(c/2 + (d* x)/2)^2)/5 + (173*a^3*tan(c/2 + (d*x)/2)^3)/24 + 12*a^3*tan(c/2 + (d*x)/2) ^4 - (7*a^3*tan(c/2 + (d*x)/2)^5)/4 + (44*a^3*tan(c/2 + (d*x)/2)^6)/3 + (7 *a^3*tan(c/2 + (d*x)/2)^7)/4 + 10*a^3*tan(c/2 + (d*x)/2)^8 - (173*a^3*tan( c/2 + (d*x)/2)^9)/24 - 2*a^3*tan(c/2 + (d*x)/2)^10 - (29*a^3*tan(c/2 + (d* x)/2)^11)/8 + (22*a^3)/15 + (29*a^3*tan(c/2 + (d*x)/2))/8)/(d*(6*tan(c/2 + (d*x)/2)^2 + 15*tan(c/2 + (d*x)/2)^4 + 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c /2 + (d*x)/2)^8 + 6*tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1))
Time = 0.17 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.80 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (-40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-144 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-110 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+208 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+435 \cos \left (d x +c \right ) \sin \left (d x +c \right )+176 \cos \left (d x +c \right )+240 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+285 c +285 d x -176\right )}{240 d} \] Input:
int(cos(d*x+c)^3*cot(d*x+c)*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*( - 40*cos(c + d*x)*sin(c + d*x)**5 - 144*cos(c + d*x)*sin(c + d*x)* *4 - 110*cos(c + d*x)*sin(c + d*x)**3 + 208*cos(c + d*x)*sin(c + d*x)**2 + 435*cos(c + d*x)*sin(c + d*x) + 176*cos(c + d*x) + 240*log(tan((c + d*x)/ 2)) + 285*c + 285*d*x - 176))/(240*d)