Integrand size = 29, antiderivative size = 133 \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \csc (c+d x)}{d}+\frac {3 a^3 \log (\sin (c+d x))}{d}+\frac {a^3 \sin (c+d x)}{d}-\frac {5 a^3 \sin ^2(c+d x)}{2 d}-\frac {5 a^3 \sin ^3(c+d x)}{3 d}+\frac {a^3 \sin ^4(c+d x)}{4 d}+\frac {3 a^3 \sin ^5(c+d x)}{5 d}+\frac {a^3 \sin ^6(c+d x)}{6 d} \] Output:
-a^3*csc(d*x+c)/d+3*a^3*ln(sin(d*x+c))/d+a^3*sin(d*x+c)/d-5/2*a^3*sin(d*x+ c)^2/d-5/3*a^3*sin(d*x+c)^3/d+1/4*a^3*sin(d*x+c)^4/d+3/5*a^3*sin(d*x+c)^5/ d+1/6*a^3*sin(d*x+c)^6/d
Time = 0.18 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.65 \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \left (60 \csc (c+d x)-180 \log (\sin (c+d x))-60 \sin (c+d x)+150 \sin ^2(c+d x)+100 \sin ^3(c+d x)-15 \sin ^4(c+d x)-36 \sin ^5(c+d x)-10 \sin ^6(c+d x)\right )}{60 d} \] Input:
Integrate[Cos[c + d*x]^3*Cot[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]
Output:
-1/60*(a^3*(60*Csc[c + d*x] - 180*Log[Sin[c + d*x]] - 60*Sin[c + d*x] + 15 0*Sin[c + d*x]^2 + 100*Sin[c + d*x]^3 - 15*Sin[c + d*x]^4 - 36*Sin[c + d*x ]^5 - 10*Sin[c + d*x]^6))/d
Time = 0.33 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.89, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3315, 27, 99, 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.
| \(\displaystyle \int \cos ^3(c+d x) \cot ^2(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^5 (a \sin (c+d x)+a)^3}{\sin (c+d x)^2}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {\int \csc ^2(c+d x) (a-a \sin (c+d x))^2 (\sin (c+d x) a+a)^5d(a \sin (c+d x))}{a^5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\csc ^2(c+d x) (a-a \sin (c+d x))^2 (\sin (c+d x) a+a)^5}{a^2}d(a \sin (c+d x))}{a^3 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (\sin ^5(c+d x) a^5+3 \sin ^4(c+d x) a^5+\sin ^3(c+d x) a^5+\csc ^2(c+d x) a^5-5 \sin ^2(c+d x) a^5+3 \csc (c+d x) a^5-5 \sin (c+d x) a^5+a^5\right )d(a \sin (c+d x))}{a^3 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{6} a^6 \sin ^6(c+d x)+\frac {3}{5} a^6 \sin ^5(c+d x)+\frac {1}{4} a^6 \sin ^4(c+d x)-\frac {5}{3} a^6 \sin ^3(c+d x)-\frac {5}{2} a^6 \sin ^2(c+d x)+a^6 \sin (c+d x)-a^6 \csc (c+d x)+3 a^6 \log (a \sin (c+d x))}{a^3 d}\) |
Input:
Int[Cos[c + d*x]^3*Cot[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]
Output:
(-(a^6*Csc[c + d*x]) + 3*a^6*Log[a*Sin[c + d*x]] + a^6*Sin[c + d*x] - (5*a ^6*Sin[c + d*x]^2)/2 - (5*a^6*Sin[c + d*x]^3)/3 + (a^6*Sin[c + d*x]^4)/4 + (3*a^6*Sin[c + d*x]^5)/5 + (a^6*Sin[c + d*x]^6)/6)/(a^3*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
Time = 9.10 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.01
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{3} \cos \left (d x +c \right )^{6}}{6}+\frac {3 a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 a^{3} \left (\frac {\cos \left (d x +c \right )^{4}}{4}+\frac {\cos \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )}{d}\) | \(134\) |
| default | \(\frac {-\frac {a^{3} \cos \left (d x +c \right )^{6}}{6}+\frac {3 a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 a^{3} \left (\frac {\cos \left (d x +c \right )^{4}}{4}+\frac {\cos \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )}{d}\) | \(134\) |
| risch | \(\frac {i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}-\frac {2 i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {67 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{128 d}+\frac {67 a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}-\frac {i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{16 d}-3 i a^{3} x -\frac {6 i a^{3} c}{d}+\frac {3 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}-\frac {a^{3} \cos \left (6 d x +6 c \right )}{192 d}+\frac {3 a^{3} \sin \left (5 d x +5 c \right )}{80 d}+\frac {a^{3} \cos \left (4 d x +4 c \right )}{16 d}+\frac {11 a^{3} \sin \left (3 d x +3 c \right )}{48 d}\) | \(208\) |
Input:
int(cos(d*x+c)^3*cot(d*x+c)^2*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/6*a^3*cos(d*x+c)^6+3/5*a^3*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin (d*x+c)+3*a^3*(1/4*cos(d*x+c)^4+1/2*cos(d*x+c)^2+ln(sin(d*x+c)))+a^3*(-1/s in(d*x+c)*cos(d*x+c)^6-(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)))
Time = 0.10 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.98 \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {144 \, a^{3} \cos \left (d x + c\right )^{6} - 32 \, a^{3} \cos \left (d x + c\right )^{4} - 128 \, a^{3} \cos \left (d x + c\right )^{2} - 720 \, a^{3} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 256 \, a^{3} + 5 \, {\left (8 \, a^{3} \cos \left (d x + c\right )^{6} - 36 \, a^{3} \cos \left (d x + c\right )^{4} - 72 \, a^{3} \cos \left (d x + c\right )^{2} + 47 \, a^{3}\right )} \sin \left (d x + c\right )}{240 \, d \sin \left (d x + c\right )} \] Input:
integrate(cos(d*x+c)^3*cot(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
-1/240*(144*a^3*cos(d*x + c)^6 - 32*a^3*cos(d*x + c)^4 - 128*a^3*cos(d*x + c)^2 - 720*a^3*log(1/2*sin(d*x + c))*sin(d*x + c) + 256*a^3 + 5*(8*a^3*co s(d*x + c)^6 - 36*a^3*cos(d*x + c)^4 - 72*a^3*cos(d*x + c)^2 + 47*a^3)*sin (d*x + c))/(d*sin(d*x + c))
\[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=a^{3} \left (\int \cos ^{3}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}\, dx + \int 3 \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cos(d*x+c)**3*cot(d*x+c)**2*(a+a*sin(d*x+c))**3,x)
Output:
a**3*(Integral(cos(c + d*x)**3*cot(c + d*x)**2, x) + Integral(3*sin(c + d* x)*cos(c + d*x)**3*cot(c + d*x)**2, x) + Integral(3*sin(c + d*x)**2*cos(c + d*x)**3*cot(c + d*x)**2, x) + Integral(sin(c + d*x)**3*cos(c + d*x)**3*c ot(c + d*x)**2, x))
Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.80 \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {10 \, a^{3} \sin \left (d x + c\right )^{6} + 36 \, a^{3} \sin \left (d x + c\right )^{5} + 15 \, a^{3} \sin \left (d x + c\right )^{4} - 100 \, a^{3} \sin \left (d x + c\right )^{3} - 150 \, a^{3} \sin \left (d x + c\right )^{2} + 180 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + 60 \, a^{3} \sin \left (d x + c\right ) - \frac {60 \, a^{3}}{\sin \left (d x + c\right )}}{60 \, d} \] Input:
integrate(cos(d*x+c)^3*cot(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
1/60*(10*a^3*sin(d*x + c)^6 + 36*a^3*sin(d*x + c)^5 + 15*a^3*sin(d*x + c)^ 4 - 100*a^3*sin(d*x + c)^3 - 150*a^3*sin(d*x + c)^2 + 180*a^3*log(sin(d*x + c)) + 60*a^3*sin(d*x + c) - 60*a^3/sin(d*x + c))/d
Time = 0.19 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.81 \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {10 \, a^{3} \sin \left (d x + c\right )^{6} + 36 \, a^{3} \sin \left (d x + c\right )^{5} + 15 \, a^{3} \sin \left (d x + c\right )^{4} - 100 \, a^{3} \sin \left (d x + c\right )^{3} - 150 \, a^{3} \sin \left (d x + c\right )^{2} + 180 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 60 \, a^{3} \sin \left (d x + c\right ) - \frac {60 \, a^{3}}{\sin \left (d x + c\right )}}{60 \, d} \] Input:
integrate(cos(d*x+c)^3*cot(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/60*(10*a^3*sin(d*x + c)^6 + 36*a^3*sin(d*x + c)^5 + 15*a^3*sin(d*x + c)^ 4 - 100*a^3*sin(d*x + c)^3 - 150*a^3*sin(d*x + c)^2 + 180*a^3*log(abs(sin( d*x + c))) + 60*a^3*sin(d*x + c) - 60*a^3/sin(d*x + c))/d
Time = 32.71 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.79 \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {14\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{d}-\frac {10\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{d}+\frac {8\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3\,d}-\frac {28\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{d}+\frac {32\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{d}-\frac {32\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3\,d}-\frac {3\,a^3\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}+\frac {3\,a^3\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {46\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {688\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {1064\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{15\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {288\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{5\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {96\,a^3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{5\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {3\,a^3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {a^3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )} \] Input:
int(cos(c + d*x)^3*cot(c + d*x)^2*(a + a*sin(c + d*x))^3,x)
Output:
(14*a^3*cos(c/2 + (d*x)/2)^4)/d - (10*a^3*cos(c/2 + (d*x)/2)^2)/d + (8*a^3 *cos(c/2 + (d*x)/2)^6)/(3*d) - (28*a^3*cos(c/2 + (d*x)/2)^8)/d + (32*a^3*c os(c/2 + (d*x)/2)^10)/d - (32*a^3*cos(c/2 + (d*x)/2)^12)/(3*d) - (3*a^3*lo g(1/cos(c/2 + (d*x)/2)^2))/d + (3*a^3*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d* x)/2)))/d - (46*a^3*cos(c/2 + (d*x)/2)^3)/(3*d*sin(c/2 + (d*x)/2)) + (688* a^3*cos(c/2 + (d*x)/2)^5)/(15*d*sin(c/2 + (d*x)/2)) - (1064*a^3*cos(c/2 + (d*x)/2)^7)/(15*d*sin(c/2 + (d*x)/2)) + (288*a^3*cos(c/2 + (d*x)/2)^9)/(5* d*sin(c/2 + (d*x)/2)) - (96*a^3*cos(c/2 + (d*x)/2)^11)/(5*d*sin(c/2 + (d*x )/2)) + (3*a^3*cos(c/2 + (d*x)/2))/(2*d*sin(c/2 + (d*x)/2)) - (a^3*sin(c/2 + (d*x)/2))/(2*d*cos(c/2 + (d*x)/2))
Time = 0.56 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.89 \[ \int \cos ^3(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (-180 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )+180 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )+10 \sin \left (d x +c \right )^{7}+36 \sin \left (d x +c \right )^{6}+15 \sin \left (d x +c \right )^{5}-100 \sin \left (d x +c \right )^{4}-150 \sin \left (d x +c \right )^{3}+60 \sin \left (d x +c \right )^{2}-60\right )}{60 \sin \left (d x +c \right ) d} \] Input:
int(cos(d*x+c)^3*cot(d*x+c)^2*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*( - 180*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x) + 180*log(tan((c + d*x)/2))*sin(c + d*x) + 10*sin(c + d*x)**7 + 36*sin(c + d*x)**6 + 15*sin( c + d*x)**5 - 100*sin(c + d*x)**4 - 150*sin(c + d*x)**3 + 60*sin(c + d*x)* *2 - 60))/(60*sin(c + d*x)*d)