Integrand size = 27, antiderivative size = 101 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {3 \csc (c+d x)}{a^2 d}+\frac {\csc ^2(c+d x)}{a^2 d}-\frac {\csc ^3(c+d x)}{3 a^2 d}-\frac {4 \log (\sin (c+d x))}{a^2 d}+\frac {4 \log (1+\sin (c+d x))}{a^2 d}-\frac {1}{d \left (a^2+a^2 \sin (c+d x)\right )} \] Output:
-3*csc(d*x+c)/a^2/d+csc(d*x+c)^2/a^2/d-1/3*csc(d*x+c)^3/a^2/d-4*ln(sin(d*x +c))/a^2/d+4*ln(1+sin(d*x+c))/a^2/d-1/d/(a^2+a^2*sin(d*x+c))
Time = 0.68 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.97 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {3 \csc (c+d x)}{a^2 d}+\frac {\csc ^2(c+d x)}{a^2 d}-\frac {\csc ^3(c+d x)}{3 a^2 d}-\frac {4 \log (\sin (c+d x))}{a^2 d}+\frac {4 \log (1+\sin (c+d x))}{a^2 d}-\frac {1}{a^2 d (1+\sin (c+d x))} \] Input:
Integrate[(Cot[c + d*x]*Csc[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]
Output:
(-3*Csc[c + d*x])/(a^2*d) + Csc[c + d*x]^2/(a^2*d) - Csc[c + d*x]^3/(3*a^2 *d) - (4*Log[Sin[c + d*x]])/(a^2*d) + (4*Log[1 + Sin[c + d*x]])/(a^2*d) - 1/(a^2*d*(1 + Sin[c + d*x]))
Time = 0.32 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, 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.
| \(\displaystyle \int \frac {\cot (c+d x) \csc ^3(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)}{\sin (c+d x)^4 (a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3312 |
| \(\displaystyle \frac {\int \frac {\csc ^4(c+d x)}{(\sin (c+d x) a+a)^2}d(a \sin (c+d x))}{a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^3 \int \frac {\csc ^4(c+d x)}{a^4 (\sin (c+d x) a+a)^2}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {a^3 \int \left (\frac {\csc ^4(c+d x)}{a^6}-\frac {2 \csc ^3(c+d x)}{a^6}+\frac {3 \csc ^2(c+d x)}{a^6}-\frac {4 \csc (c+d x)}{a^6}+\frac {4}{a^5 (\sin (c+d x) a+a)}+\frac {1}{a^4 (\sin (c+d x) a+a)^2}\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 \left (-\frac {\csc ^3(c+d x)}{3 a^5}+\frac {\csc ^2(c+d x)}{a^5}-\frac {3 \csc (c+d x)}{a^5}-\frac {4 \log (a \sin (c+d x))}{a^5}+\frac {4 \log (a \sin (c+d x)+a)}{a^5}-\frac {1}{a^4 (a \sin (c+d x)+a)}\right )}{d}\) |
Input:
Int[(Cot[c + d*x]*Csc[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]
Output:
(a^3*((-3*Csc[c + d*x])/a^5 + Csc[c + d*x]^2/a^5 - Csc[c + d*x]^3/(3*a^5) - (4*Log[a*Sin[c + d*x]])/a^5 + (4*Log[a + a*Sin[c + d*x]])/a^5 - 1/(a^4*( 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 = 1.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.60
| method | result | size |
| derivativedivides | \(-\frac {\frac {\csc \left (d x +c \right )^{3}}{3}-\csc \left (d x +c \right )^{2}+3 \csc \left (d x +c \right )-4 \ln \left (1+\csc \left (d x +c \right )\right )-\frac {1}{1+\csc \left (d x +c \right )}}{d \,a^{2}}\) | \(61\) |
| default | \(-\frac {\frac {\csc \left (d x +c \right )^{3}}{3}-\csc \left (d x +c \right )^{2}+3 \csc \left (d x +c \right )-4 \ln \left (1+\csc \left (d x +c \right )\right )-\frac {1}{1+\csc \left (d x +c \right )}}{d \,a^{2}}\) | \(61\) |
| risch | \(-\frac {8 i \left (3 i {\mathrm e}^{6 i \left (d x +c \right )}+3 \,{\mathrm e}^{7 i \left (d x +c \right )}-8 i {\mathrm e}^{4 i \left (d x +c \right )}-7 \,{\mathrm e}^{5 i \left (d x +c \right )}+3 i {\mathrm e}^{2 i \left (d x +c \right )}+7 \,{\mathrm e}^{3 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{2} d \,a^{2}}+\frac {8 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,a^{2}}-\frac {4 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d \,a^{2}}\) | \(160\) |
Input:
int(cot(d*x+c)*csc(d*x+c)^3/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-1/d/a^2*(1/3*csc(d*x+c)^3-csc(d*x+c)^2+3*csc(d*x+c)-4*ln(1+csc(d*x+c))-1/ (1+csc(d*x+c)))
Time = 0.09 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.93 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {6 \, \cos \left (d x + c\right )^{2} - 12 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 12 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, \cos \left (d x + c\right )^{2} - 5\right )} \sin \left (d x + c\right ) - 7}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d - {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm="fricas" )
Output:
1/3*(6*cos(d*x + c)^2 - 12*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 - (cos(d*x + c)^2 - 1)*sin(d*x + c) + 1)*log(1/2*sin(d*x + c)) + 12*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 - (cos(d*x + c)^2 - 1)*sin(d*x + c) + 1)*log(sin(d*x + c) + 1) + 2*(6*cos(d*x + c)^2 - 5)*sin(d*x + c) - 7)/(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d - (a^2*d*cos(d*x + c)^2 - a^2*d)*sin(d*x + c))
\[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\int \frac {\cot {\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)**3/(a+a*sin(d*x+c))**2,x)
Output:
Integral(cot(c + d*x)*csc(c + d*x)**3/(sin(c + d*x)**2 + 2*sin(c + d*x) + 1), x)/a**2
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.89 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {12 \, \sin \left (d x + c\right )^{3} + 6 \, \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1}{a^{2} \sin \left (d x + c\right )^{4} + a^{2} \sin \left (d x + c\right )^{3}} - \frac {12 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}} + \frac {12 \, \log \left (\sin \left (d x + c\right )\right )}{a^{2}}}{3 \, d} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm="maxima" )
Output:
-1/3*((12*sin(d*x + c)^3 + 6*sin(d*x + c)^2 - 2*sin(d*x + c) + 1)/(a^2*sin (d*x + c)^4 + a^2*sin(d*x + c)^3) - 12*log(sin(d*x + c) + 1)/a^2 + 12*log( sin(d*x + c))/a^2)/d
Time = 0.18 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.90 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {4 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2} d} - \frac {4 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{2} d} - \frac {12 \, \sin \left (d x + c\right )^{3} + 6 \, \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1}{3 \, a^{2} d {\left (\sin \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )^{3}} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm="giac")
Output:
4*log(abs(sin(d*x + c) + 1))/(a^2*d) - 4*log(abs(sin(d*x + c)))/(a^2*d) - 1/3*(12*sin(d*x + c)^3 + 6*sin(d*x + c)^2 - 2*sin(d*x + c) + 1)/(a^2*d*(si n(d*x + c) + 1)*sin(d*x + c)^3)
Time = 18.18 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.00 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,a^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^2\,d}-\frac {-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}-\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {1}{3}}{d\,\left (8\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+16\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}+\frac {8\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^2\,d}-\frac {13\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a^2\,d} \] Input:
int(cot(c + d*x)/(sin(c + d*x)^3*(a + a*sin(c + d*x))^2),x)
Output:
tan(c/2 + (d*x)/2)^2/(4*a^2*d) - tan(c/2 + (d*x)/2)^3/(24*a^2*d) - ((28*ta n(c/2 + (d*x)/2)^2)/3 - (4*tan(c/2 + (d*x)/2))/3 + 24*tan(c/2 + (d*x)/2)^3 - 3*tan(c/2 + (d*x)/2)^4 + 1/3)/(d*(8*a^2*tan(c/2 + (d*x)/2)^3 + 16*a^2*t an(c/2 + (d*x)/2)^4 + 8*a^2*tan(c/2 + (d*x)/2)^5)) - (4*log(tan(c/2 + (d*x )/2)))/(a^2*d) + (8*log(tan(c/2 + (d*x)/2) + 1))/(a^2*d) - (13*tan(c/2 + ( d*x)/2))/(8*a^2*d)
Time = 0.17 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.49 \[ \int \frac {\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {192 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{4}+192 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{3}-96 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4}-96 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{3}+27 \sin \left (d x +c \right )^{4}-69 \sin \left (d x +c \right )^{3}-48 \sin \left (d x +c \right )^{2}+16 \sin \left (d x +c \right )-8}{24 \sin \left (d x +c \right )^{3} a^{2} d \left (\sin \left (d x +c \right )+1\right )} \] Input:
int(cot(d*x+c)*csc(d*x+c)^3/(a+a*sin(d*x+c))^2,x)
Output:
(192*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4 + 192*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**3 - 96*log(tan((c + d*x)/2))*sin(c + d*x)**4 - 96*log(t an((c + d*x)/2))*sin(c + d*x)**3 + 27*sin(c + d*x)**4 - 69*sin(c + d*x)**3 - 48*sin(c + d*x)**2 + 16*sin(c + d*x) - 8)/(24*sin(c + d*x)**3*a**2*d*(s in(c + d*x) + 1))