Integrand size = 25, antiderivative size = 90 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\csc (c+d x)}{a^3 d}-\frac {3 \log (\sin (c+d x))}{a^3 d}+\frac {3 \log (1+\sin (c+d x))}{a^3 d}-\frac {1}{2 a d (a+a \sin (c+d x))^2}-\frac {2}{d \left (a^3+a^3 \sin (c+d x)\right )} \] Output:
-csc(d*x+c)/a^3/d-3*ln(sin(d*x+c))/a^3/d+3*ln(1+sin(d*x+c))/a^3/d-1/2/a/d/ (a+a*sin(d*x+c))^2-2/d/(a^3+a^3*sin(d*x+c))
Time = 0.22 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.68 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {2 \csc (c+d x)+6 \log (\sin (c+d x))-6 \log (1+\sin (c+d x))+\frac {1}{(1+\sin (c+d x))^2}+\frac {4}{1+\sin (c+d x)}}{2 a^3 d} \] Input:
Integrate[(Cot[c + d*x]*Csc[c + d*x])/(a + a*Sin[c + d*x])^3,x]
Output:
-1/2*(2*Csc[c + d*x] + 6*Log[Sin[c + d*x]] - 6*Log[1 + Sin[c + d*x]] + (1 + Sin[c + d*x])^(-2) + 4/(1 + Sin[c + d*x]))/(a^3*d)
Time = 0.31 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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.
\(\displaystyle \int \frac {\cot (c+d x) \csc (c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)}{\sin (c+d x)^2 (a \sin (c+d x)+a)^3}dx\) |
\(\Big \downarrow \) 3312 |
\(\displaystyle \frac {\int \frac {\csc ^2(c+d x)}{(\sin (c+d x) a+a)^3}d(a \sin (c+d x))}{a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a \int \frac {\csc ^2(c+d x)}{a^2 (\sin (c+d x) a+a)^3}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {a \int \left (\frac {\csc ^2(c+d x)}{a^5}-\frac {3 \csc (c+d x)}{a^5}+\frac {3}{a^4 (\sin (c+d x) a+a)}+\frac {2}{a^3 (\sin (c+d x) a+a)^2}+\frac {1}{a^2 (\sin (c+d x) a+a)^3}\right )d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a \left (-\frac {\csc (c+d x)}{a^4}-\frac {3 \log (a \sin (c+d x))}{a^4}+\frac {3 \log (a \sin (c+d x)+a)}{a^4}-\frac {2}{a^3 (a \sin (c+d x)+a)}-\frac {1}{2 a^2 (a \sin (c+d x)+a)^2}\right )}{d}\) |
Input:
Int[(Cot[c + d*x]*Csc[c + d*x])/(a + a*Sin[c + d*x])^3,x]
Output:
(a*(-(Csc[c + d*x]/a^4) - (3*Log[a*Sin[c + d*x]])/a^4 + (3*Log[a + a*Sin[c + d*x]])/a^4 - 1/(2*a^2*(a + a*Sin[c + d*x])^2) - 2/(a^3*(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 = 1.99 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.57
method | result | size |
derivativedivides | \(-\frac {\csc \left (d x +c \right )+\frac {1}{2 \left (1+\csc \left (d x +c \right )\right )^{2}}-\frac {3}{1+\csc \left (d x +c \right )}-3 \ln \left (1+\csc \left (d x +c \right )\right )}{d \,a^{3}}\) | \(51\) |
default | \(-\frac {\csc \left (d x +c \right )+\frac {1}{2 \left (1+\csc \left (d x +c \right )\right )^{2}}-\frac {3}{1+\csc \left (d x +c \right )}-3 \ln \left (1+\csc \left (d x +c \right )\right )}{d \,a^{3}}\) | \(51\) |
risch | \(-\frac {2 i \left (9 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{5 i \left (d x +c \right )}-9 i {\mathrm e}^{2 i \left (d x +c \right )}-10 \,{\mathrm e}^{3 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{4} d \,a^{3}}+\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,a^{3}}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d \,a^{3}}\) | \(137\) |
Input:
int(cot(d*x+c)*csc(d*x+c)/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-1/d/a^3*(csc(d*x+c)+1/2/(1+csc(d*x+c))^2-3/(1+csc(d*x+c))-3*ln(1+csc(d*x+ c)))
Time = 0.09 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.69 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {6 \, \cos \left (d x + c\right )^{2} + 6 \, {\left (2 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) - 2\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 6 \, {\left (2 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) - 2\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \, \sin \left (d x + c\right ) - 8}{2 \, {\left (2 \, a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm="fricas")
Output:
-1/2*(6*cos(d*x + c)^2 + 6*(2*cos(d*x + c)^2 + (cos(d*x + c)^2 - 2)*sin(d* x + c) - 2)*log(1/2*sin(d*x + c)) - 6*(2*cos(d*x + c)^2 + (cos(d*x + c)^2 - 2)*sin(d*x + c) - 2)*log(sin(d*x + c) + 1) - 9*sin(d*x + c) - 8)/(2*a^3* d*cos(d*x + c)^2 - 2*a^3*d + (a^3*d*cos(d*x + c)^2 - 2*a^3*d)*sin(d*x + c) )
\[ \int \frac {\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\cot {\left (c + d x \right )} \csc {\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)/(a+a*sin(d*x+c))**3,x)
Output:
Integral(cot(c + d*x)*csc(c + d*x)/(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 = 91, normalized size of antiderivative = 1.01 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {6 \, \sin \left (d x + c\right )^{2} + 9 \, \sin \left (d x + c\right ) + 2}{a^{3} \sin \left (d x + c\right )^{3} + 2 \, a^{3} \sin \left (d x + c\right )^{2} + a^{3} \sin \left (d x + c\right )} - \frac {6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {6 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm="maxima")
Output:
-1/2*((6*sin(d*x + c)^2 + 9*sin(d*x + c) + 2)/(a^3*sin(d*x + c)^3 + 2*a^3* sin(d*x + c)^2 + a^3*sin(d*x + c)) - 6*log(sin(d*x + c) + 1)/a^3 + 6*log(s in(d*x + c))/a^3)/d
Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.90 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {3 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} d} - \frac {3 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3} d} - \frac {6 \, \sin \left (d x + c\right )^{2} + 9 \, \sin \left (d x + c\right ) + 2}{2 \, a^{3} d {\left (\sin \left (d x + c\right ) + 1\right )}^{2} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
3*log(abs(sin(d*x + c) + 1))/(a^3*d) - 3*log(abs(sin(d*x + c)))/(a^3*d) - 1/2*(6*sin(d*x + c)^2 + 9*sin(d*x + c) + 2)/(a^3*d*(sin(d*x + c) + 1)^2*si n(d*x + c))
Time = 18.16 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.14 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1}{d\,\left (2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+12\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+8\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}+\frac {6\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^3\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d} \] Input:
int(cot(c + d*x)/(sin(c + d*x)*(a + a*sin(c + d*x))^3),x)
Output:
(6*tan(c/2 + (d*x)/2)^2 - 4*tan(c/2 + (d*x)/2) + 16*tan(c/2 + (d*x)/2)^3 + 11*tan(c/2 + (d*x)/2)^4 - 1)/(d*(8*a^3*tan(c/2 + (d*x)/2)^2 + 12*a^3*tan( c/2 + (d*x)/2)^3 + 8*a^3*tan(c/2 + (d*x)/2)^4 + 2*a^3*tan(c/2 + (d*x)/2)^5 + 2*a^3*tan(c/2 + (d*x)/2))) - (3*log(tan(c/2 + (d*x)/2)))/(a^3*d) + (6*l og(tan(c/2 + (d*x)/2) + 1))/(a^3*d) - tan(c/2 + (d*x)/2)/(2*a^3*d)
Time = 0.25 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.87 \[ \int \frac {\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-4 \csc \left (d x +c \right ) \sin \left (d x +c \right )-2 \csc \left (d x +c \right )+12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2}+24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )+12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2}-12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )-6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \sin \left (d x +c \right )^{2}-2}{2 a^{3} d \left (\sin \left (d x +c \right )^{2}+2 \sin \left (d x +c \right )+1\right )} \] Input:
int(cot(d*x+c)*csc(d*x+c)/(a+a*sin(d*x+c))^3,x)
Output:
( - 4*csc(c + d*x)*sin(c + d*x) - 2*csc(c + d*x) + 12*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2 + 24*log(tan((c + d*x)/2) + 1)*sin(c + d*x) + 12*log (tan((c + d*x)/2) + 1) - 6*log(tan((c + d*x)/2))*sin(c + d*x)**2 - 12*log( tan((c + d*x)/2))*sin(c + d*x) - 6*log(tan((c + d*x)/2)) + 3*sin(c + d*x)* *2 - 2)/(2*a**3*d*(sin(c + d*x)**2 + 2*sin(c + d*x) + 1))