Integrand size = 29, antiderivative size = 133 \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {3 a^3 \csc (c+d x)}{d}-\frac {a^3 \csc ^2(c+d x)}{2 d}+\frac {a^3 \log (\sin (c+d x))}{d}-\frac {5 a^3 \sin (c+d x)}{d}-\frac {5 a^3 \sin ^2(c+d x)}{2 d}+\frac {a^3 \sin ^3(c+d x)}{3 d}+\frac {3 a^3 \sin ^4(c+d x)}{4 d}+\frac {a^3 \sin ^5(c+d x)}{5 d} \] Output:
-3*a^3*csc(d*x+c)/d-1/2*a^3*csc(d*x+c)^2/d+a^3*ln(sin(d*x+c))/d-5*a^3*sin( d*x+c)/d-5/2*a^3*sin(d*x+c)^2/d+1/3*a^3*sin(d*x+c)^3/d+3/4*a^3*sin(d*x+c)^ 4/d+1/5*a^3*sin(d*x+c)^5/d
Time = 0.17 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.65 \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \left (180 \csc (c+d x)+30 \csc ^2(c+d x)-60 \log (\sin (c+d x))+300 \sin (c+d x)+150 \sin ^2(c+d x)-20 \sin ^3(c+d x)-45 \sin ^4(c+d x)-12 \sin ^5(c+d x)\right )}{60 d} \] Input:
Integrate[Cos[c + d*x]^2*Cot[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]
Output:
-1/60*(a^3*(180*Csc[c + d*x] + 30*Csc[c + d*x]^2 - 60*Log[Sin[c + d*x]] + 300*Sin[c + d*x] + 150*Sin[c + d*x]^2 - 20*Sin[c + d*x]^3 - 45*Sin[c + d*x ]^4 - 12*Sin[c + d*x]^5))/d
Time = 0.34 (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.
| \(\displaystyle \int \cos ^2(c+d x) \cot ^3(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^5 (a \sin (c+d x)+a)^3}{\sin (c+d x)^3}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {\int \csc ^3(c+d x) (a-a \sin (c+d x))^2 (\sin (c+d x) a+a)^5d(a \sin (c+d x))}{a^5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\csc ^3(c+d x) (a-a \sin (c+d x))^2 (\sin (c+d x) a+a)^5}{a^3}d(a \sin (c+d x))}{a^2 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (\sin ^4(c+d x) a^4+\csc ^3(c+d x) a^4+3 \sin ^3(c+d x) a^4+3 \csc ^2(c+d x) a^4+\sin ^2(c+d x) a^4+\csc (c+d x) a^4-5 \sin (c+d x) a^4-5 a^4\right )d(a \sin (c+d x))}{a^2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{5} a^5 \sin ^5(c+d x)+\frac {3}{4} a^5 \sin ^4(c+d x)+\frac {1}{3} a^5 \sin ^3(c+d x)-\frac {5}{2} a^5 \sin ^2(c+d x)-5 a^5 \sin (c+d x)-\frac {1}{2} a^5 \csc ^2(c+d x)-3 a^5 \csc (c+d x)+a^5 \log (a \sin (c+d x))}{a^2 d}\) |
Input:
Int[Cos[c + d*x]^2*Cot[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]
Output:
(-3*a^5*Csc[c + d*x] - (a^5*Csc[c + d*x]^2)/2 + a^5*Log[a*Sin[c + d*x]] - 5*a^5*Sin[c + d*x] - (5*a^5*Sin[c + d*x]^2)/2 + (a^5*Sin[c + d*x]^3)/3 + ( 3*a^5*Sin[c + d*x]^4)/4 + (a^5*Sin[c + d*x]^5)/5)/(a^2*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 = 7.38 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.31
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 a^{3} \left (\frac {\cos \left (d x +c \right )^{4}}{4}+\frac {\cos \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{6}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{4}}{2}-\cos \left (d x +c \right )^{2}-2 \ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(174\) |
| default | \(\frac {\frac {a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 a^{3} \left (\frac {\cos \left (d x +c \right )^{4}}{4}+\frac {\cos \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{6}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{4}}{2}-\cos \left (d x +c \right )^{2}-2 \ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(174\) |
| risch | \(-i a^{3} x +\frac {7 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{96 d}+\frac {7 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{16 d}+\frac {37 i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{16 d}-\frac {37 i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}+\frac {7 a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{16 d}-\frac {7 i a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{96 d}-\frac {2 i a^{3} c}{d}-\frac {2 i a^{3} \left (i {\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{3 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {a^{3} \sin \left (5 d x +5 c \right )}{80 d}+\frac {3 a^{3} \cos \left (4 d x +4 c \right )}{32 d}\) | \(235\) |
Input:
int(cos(d*x+c)^2*cot(d*x+c)^3*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(1/5*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)))+3*a^3*(-1/sin(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))+a^3*(-1/2/sin(d*x+c)^2* cos(d*x+c)^6-1/2*cos(d*x+c)^4-cos(d*x+c)^2-2*ln(sin(d*x+c))))
Time = 0.10 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.09 \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {360 \, a^{3} \cos \left (d x + c\right )^{6} + 120 \, a^{3} \cos \left (d x + c\right )^{4} - 855 \, a^{3} \cos \left (d x + c\right )^{2} + 615 \, a^{3} + 480 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 32 \, {\left (3 \, a^{3} \cos \left (d x + c\right )^{6} - 14 \, a^{3} \cos \left (d x + c\right )^{4} - 56 \, a^{3} \cos \left (d x + c\right )^{2} + 112 \, a^{3}\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(cos(d*x+c)^2*cot(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
1/480*(360*a^3*cos(d*x + c)^6 + 120*a^3*cos(d*x + c)^4 - 855*a^3*cos(d*x + c)^2 + 615*a^3 + 480*(a^3*cos(d*x + c)^2 - a^3)*log(1/2*sin(d*x + c)) + 3 2*(3*a^3*cos(d*x + c)^6 - 14*a^3*cos(d*x + c)^4 - 56*a^3*cos(d*x + c)^2 + 112*a^3)*sin(d*x + c))/(d*cos(d*x + c)^2 - d)
\[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=a^{3} \left (\int \cos ^{2}{\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx + \int 3 \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cos(d*x+c)**2*cot(d*x+c)**3*(a+a*sin(d*x+c))**3,x)
Output:
a**3*(Integral(cos(c + d*x)**2*cot(c + d*x)**3, x) + Integral(3*sin(c + d* x)*cos(c + d*x)**2*cot(c + d*x)**3, x) + Integral(3*sin(c + d*x)**2*cos(c + d*x)**2*cot(c + d*x)**3, x) + Integral(sin(c + d*x)**3*cos(c + d*x)**2*c ot(c + d*x)**3, x))
Time = 0.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.80 \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {12 \, a^{3} \sin \left (d x + c\right )^{5} + 45 \, a^{3} \sin \left (d x + c\right )^{4} + 20 \, a^{3} \sin \left (d x + c\right )^{3} - 150 \, a^{3} \sin \left (d x + c\right )^{2} + 60 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) - 300 \, a^{3} \sin \left (d x + c\right ) - \frac {30 \, {\left (6 \, a^{3} \sin \left (d x + c\right ) + a^{3}\right )}}{\sin \left (d x + c\right )^{2}}}{60 \, d} \] Input:
integrate(cos(d*x+c)^2*cot(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
1/60*(12*a^3*sin(d*x + c)^5 + 45*a^3*sin(d*x + c)^4 + 20*a^3*sin(d*x + c)^ 3 - 150*a^3*sin(d*x + c)^2 + 60*a^3*log(sin(d*x + c)) - 300*a^3*sin(d*x + c) - 30*(6*a^3*sin(d*x + c) + a^3)/sin(d*x + c)^2)/d
Time = 0.17 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.80 \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {12 \, a^{3} \sin \left (d x + c\right )^{5} + 45 \, a^{3} \sin \left (d x + c\right )^{4} + 20 \, a^{3} \sin \left (d x + c\right )^{3} - 150 \, a^{3} \sin \left (d x + c\right )^{2} + 60 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 300 \, a^{3} \sin \left (d x + c\right ) - \frac {30 \, {\left (6 \, a^{3} \sin \left (d x + c\right ) + a^{3}\right )}}{\sin \left (d x + c\right )^{2}}}{60 \, d} \] Input:
integrate(cos(d*x+c)^2*cot(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/60*(12*a^3*sin(d*x + c)^5 + 45*a^3*sin(d*x + c)^4 + 20*a^3*sin(d*x + c)^ 3 - 150*a^3*sin(d*x + c)^2 + 60*a^3*log(abs(sin(d*x + c))) - 300*a^3*sin(d *x + c) - 30*(6*a^3*sin(d*x + c) + a^3)/sin(d*x + c)^2)/d
Time = 33.83 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.57 \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {46\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\frac {81\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{2}+\frac {538\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{3}+\frac {149\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2}+\frac {3796\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{15}+77\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {628\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+45\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+70\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+6\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{2}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {a^3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \] Input:
int(cos(c + d*x)^2*cot(c + d*x)^3*(a + a*sin(c + d*x))^3,x)
Output:
(a^3*log(tan(c/2 + (d*x)/2)))/d - (a^3*tan(c/2 + (d*x)/2)^2)/(8*d) - (3*a^ 3*tan(c/2 + (d*x)/2))/(2*d) - ((5*a^3*tan(c/2 + (d*x)/2)^2)/2 + 70*a^3*tan (c/2 + (d*x)/2)^3 + 45*a^3*tan(c/2 + (d*x)/2)^4 + (628*a^3*tan(c/2 + (d*x) /2)^5)/3 + 77*a^3*tan(c/2 + (d*x)/2)^6 + (3796*a^3*tan(c/2 + (d*x)/2)^7)/1 5 + (149*a^3*tan(c/2 + (d*x)/2)^8)/2 + (538*a^3*tan(c/2 + (d*x)/2)^9)/3 + (81*a^3*tan(c/2 + (d*x)/2)^10)/2 + 46*a^3*tan(c/2 + (d*x)/2)^11 + a^3/2 + 6*a^3*tan(c/2 + (d*x)/2))/(d*(4*tan(c/2 + (d*x)/2)^2 + 20*tan(c/2 + (d*x)/ 2)^4 + 40*tan(c/2 + (d*x)/2)^6 + 40*tan(c/2 + (d*x)/2)^8 + 20*tan(c/2 + (d *x)/2)^10 + 4*tan(c/2 + (d*x)/2)^12)) - (a^3*log(tan(c/2 + (d*x)/2)^2 + 1) )/d
Time = 0.47 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.98 \[ \int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (-120 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{2}+120 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2}+24 \sin \left (d x +c \right )^{7}+90 \sin \left (d x +c \right )^{6}+40 \sin \left (d x +c \right )^{5}-300 \sin \left (d x +c \right )^{4}-600 \sin \left (d x +c \right )^{3}+105 \sin \left (d x +c \right )^{2}-360 \sin \left (d x +c \right )-60\right )}{120 \sin \left (d x +c \right )^{2} d} \] Input:
int(cos(d*x+c)^2*cot(d*x+c)^3*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*( - 120*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2 + 120*log(tan(( c + d*x)/2))*sin(c + d*x)**2 + 24*sin(c + d*x)**7 + 90*sin(c + d*x)**6 + 4 0*sin(c + d*x)**5 - 300*sin(c + d*x)**4 - 600*sin(c + d*x)**3 + 105*sin(c + d*x)**2 - 360*sin(c + d*x) - 60))/(120*sin(c + d*x)**2*d)