Integrand size = 27, antiderivative size = 126 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {6 \csc (c+d x)}{a^3 d}+\frac {3 \csc ^2(c+d x)}{2 a^3 d}-\frac {\csc ^3(c+d x)}{3 a^3 d}-\frac {10 \log (\sin (c+d x))}{a^3 d}+\frac {10 \log (1+\sin (c+d x))}{a^3 d}-\frac {1}{2 a d (a+a \sin (c+d x))^2}-\frac {4}{d \left (a^3+a^3 \sin (c+d x)\right )} \] Output:
-6*csc(d*x+c)/a^3/d+3/2*csc(d*x+c)^2/a^3/d-1/3*csc(d*x+c)^3/a^3/d-10*ln(si n(d*x+c))/a^3/d+10*ln(1+sin(d*x+c))/a^3/d-1/2/a/d/(a+a*sin(d*x+c))^2-4/d/( a^3+a^3*sin(d*x+c))
Time = 1.65 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.64 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {36 \csc (c+d x)-9 \csc ^2(c+d x)+2 \csc ^3(c+d x)+60 \log (\sin (c+d x))-60 \log (1+\sin (c+d x))+\frac {3 (9+8 \sin (c+d x))}{(1+\sin (c+d x))^2}}{6 a^3 d} \] Input:
Integrate[(Cot[c + d*x]*Csc[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]
Output:
-1/6*(36*Csc[c + d*x] - 9*Csc[c + d*x]^2 + 2*Csc[c + d*x]^3 + 60*Log[Sin[c + d*x]] - 60*Log[1 + Sin[c + d*x]] + (3*(9 + 8*Sin[c + d*x]))/(1 + Sin[c + d*x])^2)/(a^3*d)
Time = 0.33 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.91, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3312, 27, 54, 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.
| \(\displaystyle \int \frac {\cot (c+d x) \csc ^3(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)}{\sin (c+d x)^4 (a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3312 |
| \(\displaystyle \frac {\int \frac {\csc ^4(c+d x)}{(\sin (c+d x) a+a)^3}d(a \sin (c+d x))}{a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^3 \int \frac {\csc ^4(c+d x)}{a^4 (\sin (c+d x) a+a)^3}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {a^3 \int \left (\frac {\csc ^4(c+d x)}{a^7}-\frac {3 \csc ^3(c+d x)}{a^7}+\frac {6 \csc ^2(c+d x)}{a^7}-\frac {10 \csc (c+d x)}{a^7}+\frac {10}{a^6 (\sin (c+d x) a+a)}+\frac {4}{a^5 (\sin (c+d x) a+a)^2}+\frac {1}{a^4 (\sin (c+d x) a+a)^3}\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 \left (-\frac {\csc ^3(c+d x)}{3 a^6}+\frac {3 \csc ^2(c+d x)}{2 a^6}-\frac {6 \csc (c+d x)}{a^6}-\frac {10 \log (a \sin (c+d x))}{a^6}+\frac {10 \log (a \sin (c+d x)+a)}{a^6}-\frac {4}{a^5 (a \sin (c+d x)+a)}-\frac {1}{2 a^4 (a \sin (c+d x)+a)^2}\right )}{d}\) |
Input:
Int[(Cot[c + d*x]*Csc[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*((-6*Csc[c + d*x])/a^6 + (3*Csc[c + d*x]^2)/(2*a^6) - Csc[c + d*x]^3/ (3*a^6) - (10*Log[a*Sin[c + d*x]])/a^6 + (10*Log[a + a*Sin[c + d*x]])/a^6 - 1/(2*a^4*(a + a*Sin[c + d*x])^2) - 4/(a^5*(a + a*Sin[c + d*x]))))/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_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*(( c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b*f) Su bst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 3.61 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.58
| method | result | size |
| derivativedivides | \(-\frac {\frac {\csc \left (d x +c \right )^{3}}{3}-\frac {3 \csc \left (d x +c \right )^{2}}{2}+6 \csc \left (d x +c \right )+\frac {1}{2 \left (1+\csc \left (d x +c \right )\right )^{2}}-10 \ln \left (1+\csc \left (d x +c \right )\right )-\frac {5}{1+\csc \left (d x +c \right )}}{d \,a^{3}}\) | \(73\) |
| default | \(-\frac {\frac {\csc \left (d x +c \right )^{3}}{3}-\frac {3 \csc \left (d x +c \right )^{2}}{2}+6 \csc \left (d x +c \right )+\frac {1}{2 \left (1+\csc \left (d x +c \right )\right )^{2}}-10 \ln \left (1+\csc \left (d x +c \right )\right )-\frac {5}{1+\csc \left (d x +c \right )}}{d \,a^{3}}\) | \(73\) |
| risch | \(-\frac {4 i \left (45 i {\mathrm e}^{8 i \left (d x +c \right )}+15 \,{\mathrm e}^{9 i \left (d x +c \right )}-125 i {\mathrm e}^{6 i \left (d x +c \right )}-80 \,{\mathrm e}^{7 i \left (d x +c \right )}+125 i {\mathrm e}^{4 i \left (d x +c \right )}+138 \,{\mathrm e}^{5 i \left (d x +c \right )}-45 i {\mathrm e}^{2 i \left (d x +c \right )}-80 \,{\mathrm e}^{3 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{4} d \,a^{3}}+\frac {20 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,a^{3}}-\frac {10 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d \,a^{3}}\) | \(183\) |
Input:
int(cot(d*x+c)*csc(d*x+c)^3/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-1/d/a^3*(1/3*csc(d*x+c)^3-3/2*csc(d*x+c)^2+6*csc(d*x+c)+1/2/(1+csc(d*x+c) )^2-10*ln(1+csc(d*x+c))-5/(1+csc(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (120) = 240\).
Time = 0.09 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.92 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {60 \, \cos \left (d x + c\right )^{4} - 140 \, \cos \left (d x + c\right )^{2} + 60 \, {\left (2 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{2} + 2\right )} \sin \left (d x + c\right ) + 2\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 60 \, {\left (2 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{2} + 2\right )} \sin \left (d x + c\right ) + 2\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 5 \, {\left (18 \, \cos \left (d x + c\right )^{2} - 17\right )} \sin \left (d x + c\right ) + 82}{6 \, {\left (2 \, a^{3} d \cos \left (d x + c\right )^{4} - 4 \, a^{3} d \cos \left (d x + c\right )^{2} + 2 \, a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{4} - 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 2 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
-1/6*(60*cos(d*x + c)^4 - 140*cos(d*x + c)^2 + 60*(2*cos(d*x + c)^4 - 4*co s(d*x + c)^2 + (cos(d*x + c)^4 - 3*cos(d*x + c)^2 + 2)*sin(d*x + c) + 2)*l og(1/2*sin(d*x + c)) - 60*(2*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + (cos(d*x + c)^4 - 3*cos(d*x + c)^2 + 2)*sin(d*x + c) + 2)*log(sin(d*x + c) + 1) - 5 *(18*cos(d*x + c)^2 - 17)*sin(d*x + c) + 82)/(2*a^3*d*cos(d*x + c)^4 - 4*a ^3*d*cos(d*x + c)^2 + 2*a^3*d + (a^3*d*cos(d*x + c)^4 - 3*a^3*d*cos(d*x + c)^2 + 2*a^3*d)*sin(d*x + c))
\[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\cot {\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)**3/(a+a*sin(d*x+c))**3,x)
Output:
Integral(cot(c + d*x)*csc(c + d*x)**3/(sin(c + d*x)**3 + 3*sin(c + d*x)**2 + 3*sin(c + d*x) + 1), x)/a**3
Time = 0.03 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.90 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {60 \, \sin \left (d x + c\right )^{4} + 90 \, \sin \left (d x + c\right )^{3} + 20 \, \sin \left (d x + c\right )^{2} - 5 \, \sin \left (d x + c\right ) + 2}{a^{3} \sin \left (d x + c\right )^{5} + 2 \, a^{3} \sin \left (d x + c\right )^{4} + a^{3} \sin \left (d x + c\right )^{3}} - \frac {60 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {60 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}}}{6 \, d} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
-1/6*((60*sin(d*x + c)^4 + 90*sin(d*x + c)^3 + 20*sin(d*x + c)^2 - 5*sin(d *x + c) + 2)/(a^3*sin(d*x + c)^5 + 2*a^3*sin(d*x + c)^4 + a^3*sin(d*x + c) ^3) - 60*log(sin(d*x + c) + 1)/a^3 + 60*log(sin(d*x + c))/a^3)/d
Time = 0.17 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.80 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {10 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} d} - \frac {10 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3} d} - \frac {60 \, \sin \left (d x + c\right )^{4} + 90 \, \sin \left (d x + c\right )^{3} + 20 \, \sin \left (d x + c\right )^{2} - 5 \, \sin \left (d x + c\right ) + 2}{6 \, a^{3} d {\left (\sin \left (d x + c\right ) + 1\right )}^{2} \sin \left (d x + c\right )^{3}} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
10*log(abs(sin(d*x + c) + 1))/(a^3*d) - 10*log(abs(sin(d*x + c)))/(a^3*d) - 1/6*(60*sin(d*x + c)^4 + 90*sin(d*x + c)^3 + 20*sin(d*x + c)^2 - 5*sin(d *x + c) + 2)/(a^3*d*(sin(d*x + c) + 1)^2*sin(d*x + c)^3)
Time = 18.10 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.06 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^3\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^3\,d}-\frac {10\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}+\frac {20\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^3\,d}-\frac {25\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a^3\,d}-\frac {-55\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-47\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {175\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {250\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {1}{3}}{d\,\left (8\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+32\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+48\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+32\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )} \] Input:
int(cot(c + d*x)/(sin(c + d*x)^3*(a + a*sin(c + d*x))^3),x)
Output:
(3*tan(c/2 + (d*x)/2)^2)/(8*a^3*d) - tan(c/2 + (d*x)/2)^3/(24*a^3*d) - (10 *log(tan(c/2 + (d*x)/2)))/(a^3*d) + (20*log(tan(c/2 + (d*x)/2) + 1))/(a^3* d) - (25*tan(c/2 + (d*x)/2))/(8*a^3*d) - (15*tan(c/2 + (d*x)/2)^2 - (5*tan (c/2 + (d*x)/2))/3 + (250*tan(c/2 + (d*x)/2)^3)/3 + (175*tan(c/2 + (d*x)/2 )^4)/3 - 47*tan(c/2 + (d*x)/2)^5 - 55*tan(c/2 + (d*x)/2)^6 + 1/3)/(d*(8*a^ 3*tan(c/2 + (d*x)/2)^3 + 32*a^3*tan(c/2 + (d*x)/2)^4 + 48*a^3*tan(c/2 + (d *x)/2)^5 + 32*a^3*tan(c/2 + (d*x)/2)^6 + 8*a^3*tan(c/2 + (d*x)/2)^7))
Time = 0.17 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.89 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-8 \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{3}-4 \csc \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2}+240 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{4}+480 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{3}+240 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2}-120 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4}-240 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{3}-120 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2}+75 \sin \left (d x +c \right )^{4}+30 \sin \left (d x +c \right )^{3}-105 \sin \left (d x +c \right )^{2}-40 \sin \left (d x +c \right )+18}{12 \sin \left (d x +c \right )^{2} a^{3} d \left (\sin \left (d x +c \right )^{2}+2 \sin \left (d x +c \right )+1\right )} \] Input:
int(cot(d*x+c)*csc(d*x+c)^3/(a+a*sin(d*x+c))^3,x)
Output:
( - 8*csc(c + d*x)**3*sin(c + d*x)**3 - 4*csc(c + d*x)**3*sin(c + d*x)**2 + 240*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4 + 480*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**3 + 240*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2 - 12 0*log(tan((c + d*x)/2))*sin(c + d*x)**4 - 240*log(tan((c + d*x)/2))*sin(c + d*x)**3 - 120*log(tan((c + d*x)/2))*sin(c + d*x)**2 + 75*sin(c + d*x)**4 + 30*sin(c + d*x)**3 - 105*sin(c + d*x)**2 - 40*sin(c + d*x) + 18)/(12*si n(c + d*x)**2*a**3*d*(sin(c + d*x)**2 + 2*sin(c + d*x) + 1))