Integrand size = 27, antiderivative size = 108 \[ \int \frac {\cot (c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {3 \csc (c+d x)}{a^3 d}-\frac {\csc ^2(c+d x)}{2 a^3 d}+\frac {6 \log (\sin (c+d x))}{a^3 d}-\frac {6 \log (1+\sin (c+d x))}{a^3 d}+\frac {1}{2 a d (a+a \sin (c+d x))^2}+\frac {3}{d \left (a^3+a^3 \sin (c+d x)\right )} \]
3*csc(d*x+c)/a^3/d-1/2*csc(d*x+c)^2/a^3/d+6*ln(sin(d*x+c))/a^3/d-6*ln(1+si n(d*x+c))/a^3/d+1/2/a/d/(a+a*sin(d*x+c))^2+3/d/(a^3+a^3*sin(d*x+c))
Time = 0.39 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.66 \[ \int \frac {\cot (c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {6 \csc (c+d x)-\csc ^2(c+d x)+12 \log (\sin (c+d x))-12 \log (1+\sin (c+d x))+\frac {1}{(1+\sin (c+d x))^2}+\frac {6}{1+\sin (c+d x)}}{2 a^3 d} \]
(6*Csc[c + d*x] - Csc[c + d*x]^2 + 12*Log[Sin[c + d*x]] - 12*Log[1 + Sin[c + d*x]] + (1 + Sin[c + d*x])^(-2) + 6/(1 + Sin[c + d*x]))/(2*a^3*d)
Time = 0.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93, 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.
| \(\displaystyle \int \frac {\cot (c+d x) \csc ^2(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)}{\sin (c+d x)^3 (a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3312 |
| \(\displaystyle \frac {\int \frac {\csc ^3(c+d x)}{(\sin (c+d x) a+a)^3}d(a \sin (c+d x))}{a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^2 \int \frac {\csc ^3(c+d x)}{a^3 (\sin (c+d x) a+a)^3}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {a^2 \int \left (\frac {\csc ^3(c+d x)}{a^6}-\frac {3 \csc ^2(c+d x)}{a^6}+\frac {6 \csc (c+d x)}{a^6}-\frac {6}{a^5 (\sin (c+d x) a+a)}-\frac {3}{a^4 (\sin (c+d x) a+a)^2}-\frac {1}{a^3 (\sin (c+d x) a+a)^3}\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 \left (-\frac {\csc ^2(c+d x)}{2 a^5}+\frac {3 \csc (c+d x)}{a^5}+\frac {6 \log (a \sin (c+d x))}{a^5}-\frac {6 \log (a \sin (c+d x)+a)}{a^5}+\frac {3}{a^4 (a \sin (c+d x)+a)}+\frac {1}{2 a^3 (a \sin (c+d x)+a)^2}\right )}{d}\) |
(a^2*((3*Csc[c + d*x])/a^5 - Csc[c + d*x]^2/(2*a^5) + (6*Log[a*Sin[c + d*x ]])/a^5 - (6*Log[a + a*Sin[c + d*x]])/a^5 + 1/(2*a^3*(a + a*Sin[c + d*x])^ 2) + 3/(a^4*(a + a*Sin[c + d*x]))))/d
3.3.47.3.1 Defintions of rubi rules used
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 = 0.38 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.58
| method | result | size |
| derivativedivides | \(-\frac {\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}-3 \csc \left (d x +c \right )+\frac {4}{\csc \left (d x +c \right )+1}-\frac {1}{2 \left (\csc \left (d x +c \right )+1\right )^{2}}+6 \ln \left (\csc \left (d x +c \right )+1\right )}{d \,a^{3}}\) | \(63\) |
| default | \(-\frac {\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}-3 \csc \left (d x +c \right )+\frac {4}{\csc \left (d x +c \right )+1}-\frac {1}{2 \left (\csc \left (d x +c \right )+1\right )^{2}}+6 \ln \left (\csc \left (d x +c \right )+1\right )}{d \,a^{3}}\) | \(63\) |
| risch | \(\frac {4 i \left (9 i {\mathrm e}^{6 i \left (d x +c \right )}+3 \,{\mathrm e}^{7 i \left (d x +c \right )}-16 i {\mathrm e}^{4 i \left (d x +c \right )}-13 \,{\mathrm e}^{5 i \left (d x +c \right )}+9 i {\mathrm e}^{2 i \left (d x +c \right )}+13 \,{\mathrm e}^{3 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{4} d \,a^{3}}-\frac {12 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,a^{3}}+\frac {6 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d \,a^{3}}\) | \(160\) |
| parallelrisch | \(\frac {\left (-48 \cos \left (2 d x +2 c \right )+192 \sin \left (d x +c \right )+144\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (24 \cos \left (2 d x +2 c \right )-96 \sin \left (d x +c \right )-72\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-136 \cos \left (d x +c \right )+34 \cos \left (2 d x +2 c \right )+106\right ) \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-88 \cos \left (d x +c \right )+88\right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (\left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-4\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )-8 \sec \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{3} \left (-3+\cos \left (2 d x +2 c \right )-4 \sin \left (d x +c \right )\right )}\) | \(195\) |
| norman | \(\frac {-\frac {24 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {24 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {1}{8 a d}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {7 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}-\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}-\frac {251 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {251 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}-\frac {12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,a^{3}}\) | \(208\) |
-1/d/a^3*(1/2*csc(d*x+c)^2-3*csc(d*x+c)+4/(csc(d*x+c)+1)-1/2/(csc(d*x+c)+1 )^2+6*ln(csc(d*x+c)+1))
Time = 0.28 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.81 \[ \int \frac {\cot (c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {18 \, \cos \left (d x + c\right )^{2} - 12 \, {\left (\cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) + 2\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 12 \, {\left (\cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) + 2\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 4 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 17}{2 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 2 \, a^{3} d - 2 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
-1/2*(18*cos(d*x + c)^2 - 12*(cos(d*x + c)^4 - 3*cos(d*x + c)^2 - 2*(cos(d *x + c)^2 - 1)*sin(d*x + c) + 2)*log(1/2*sin(d*x + c)) + 12*(cos(d*x + c)^ 4 - 3*cos(d*x + c)^2 - 2*(cos(d*x + c)^2 - 1)*sin(d*x + c) + 2)*log(sin(d* x + c) + 1) + 4*(3*cos(d*x + c)^2 - 4)*sin(d*x + c) - 17)/(a^3*d*cos(d*x + c)^4 - 3*a^3*d*cos(d*x + c)^2 + 2*a^3*d - 2*(a^3*d*cos(d*x + c)^2 - a^3*d )*sin(d*x + c))
\[ \int \frac {\cot (c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\cos {\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Integral(cos(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 = 0.20 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.95 \[ \int \frac {\cot (c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {12 \, \sin \left (d x + c\right )^{3} + 18 \, \sin \left (d x + c\right )^{2} + 4 \, \sin \left (d x + c\right ) - 1}{a^{3} \sin \left (d x + c\right )^{4} + 2 \, a^{3} \sin \left (d x + c\right )^{3} + a^{3} \sin \left (d x + c\right )^{2}} - \frac {12 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {12 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \]
1/2*((12*sin(d*x + c)^3 + 18*sin(d*x + c)^2 + 4*sin(d*x + c) - 1)/(a^3*sin (d*x + c)^4 + 2*a^3*sin(d*x + c)^3 + a^3*sin(d*x + c)^2) - 12*log(sin(d*x + c) + 1)/a^3 + 12*log(sin(d*x + c))/a^3)/d
Time = 0.36 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.80 \[ \int \frac {\cot (c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {12 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3}} - \frac {12 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac {12 \, \sin \left (d x + c\right )^{3} + 18 \, \sin \left (d x + c\right )^{2} + 4 \, \sin \left (d x + c\right ) - 1}{{\left (\sin \left (d x + c\right )^{2} + \sin \left (d x + c\right )\right )}^{2} a^{3}}}{2 \, d} \]
-1/2*(12*log(abs(sin(d*x + c) + 1))/a^3 - 12*log(abs(sin(d*x + c)))/a^3 - (12*sin(d*x + c)^3 + 18*sin(d*x + c)^2 + 4*sin(d*x + c) - 1)/((sin(d*x + c )^2 + sin(d*x + c))^2*a^3))/d
Time = 10.27 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.10 \[ \int \frac {\cot (c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {6\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^3\,d}+\frac {-26\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {65\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{2}}{d\,\left (4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+24\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+16\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {12\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^3\,d}+\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d} \]
(6*log(tan(c/2 + (d*x)/2)))/(a^3*d) - tan(c/2 + (d*x)/2)^2/(8*a^3*d) + (4* tan(c/2 + (d*x)/2) + 21*tan(c/2 + (d*x)/2)^2 + 2*tan(c/2 + (d*x)/2)^3 - (6 5*tan(c/2 + (d*x)/2)^4)/2 - 26*tan(c/2 + (d*x)/2)^5 - 1/2)/(d*(4*a^3*tan(c /2 + (d*x)/2)^2 + 16*a^3*tan(c/2 + (d*x)/2)^3 + 24*a^3*tan(c/2 + (d*x)/2)^ 4 + 16*a^3*tan(c/2 + (d*x)/2)^5 + 4*a^3*tan(c/2 + (d*x)/2)^6)) - (12*log(t an(c/2 + (d*x)/2) + 1))/(a^3*d) + (3*tan(c/2 + (d*x)/2))/(2*a^3*d)