Integrand size = 29, antiderivative size = 138 \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {11 \text {arctanh}(\cos (c+d x))}{16 a^2 d}+\frac {2 \cot (c+d x)}{a^2 d}+\frac {4 \cot ^3(c+d x)}{3 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {11 \cot (c+d x) \csc (c+d x)}{16 a^2 d}-\frac {11 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d} \] Output:
-11/16*arctanh(cos(d*x+c))/a^2/d+2*cot(d*x+c)/a^2/d+4/3*cot(d*x+c)^3/a^2/d +2/5*cot(d*x+c)^5/a^2/d-11/16*cot(d*x+c)*csc(d*x+c)/a^2/d-11/24*cot(d*x+c) *csc(d*x+c)^3/a^2/d-1/6*cot(d*x+c)*csc(d*x+c)^5/a^2/d
Time = 1.70 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.66 \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\csc ^6(c+d x) \left (-2820 \cos (c+d x)+1870 \cos (3 (c+d x))-330 \cos (5 (c+d x))-1650 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+2475 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-990 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+165 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+1650 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2475 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+990 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-165 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+3840 \sin (2 (c+d x))-1536 \sin (4 (c+d x))+256 \sin (6 (c+d x))\right )}{7680 a^2 d} \] Input:
Integrate[(Cot[c + d*x]^4*Csc[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]
Output:
(Csc[c + d*x]^6*(-2820*Cos[c + d*x] + 1870*Cos[3*(c + d*x)] - 330*Cos[5*(c + d*x)] - 1650*Log[Cos[(c + d*x)/2]] + 2475*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 990*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 165*Cos[6*(c + d* x)]*Log[Cos[(c + d*x)/2]] + 1650*Log[Sin[(c + d*x)/2]] - 2475*Cos[2*(c + d *x)]*Log[Sin[(c + d*x)/2]] + 990*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 165*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 3840*Sin[2*(c + d*x)] - 1536* Sin[4*(c + d*x)] + 256*Sin[6*(c + d*x)]))/(7680*a^2*d)
Time = 0.53 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3348, 3042, 3236, 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.
| \(\displaystyle \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4}{\sin (c+d x)^7 (a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3348 |
| \(\displaystyle \frac {\int \csc ^7(c+d x) (a-a \sin (c+d x))^2dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a-a \sin (c+d x))^2}{\sin (c+d x)^7}dx}{a^4}\) |
| \(\Big \downarrow \) 3236 |
| \(\displaystyle \frac {\int \left (a^2 \csc ^7(c+d x)-2 a^2 \csc ^6(c+d x)+a^2 \csc ^5(c+d x)\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {11 a^2 \text {arctanh}(\cos (c+d x))}{16 d}+\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {4 a^2 \cot ^3(c+d x)}{3 d}+\frac {2 a^2 \cot (c+d x)}{d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {11 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{16 d}}{a^4}\) |
Input:
Int[(Cot[c + d*x]^4*Csc[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]
Output:
((-11*a^2*ArcTanh[Cos[c + d*x]])/(16*d) + (2*a^2*Cot[c + d*x])/d + (4*a^2* Cot[c + d*x]^3)/(3*d) + (2*a^2*Cot[c + d*x]^5)/(5*d) - (11*a^2*Cot[c + d*x ]*Csc[c + d*x])/(16*d) - (11*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^ 2*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d))/a^4
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)])^(m_.), x_Symbol] :> Int[ExpandTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IGt Q[m, 0] && RationalQ[n]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[a^(2*m) Int[(d* Sin[e + f*x])^n/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[2*m + p, 0]
Result contains complex when optimal does not.
Time = 3.94 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.22
| method | result | size |
| risch | \(\frac {165 \,{\mathrm e}^{11 i \left (d x +c \right )}-935 \,{\mathrm e}^{9 i \left (d x +c \right )}+2560 i {\mathrm e}^{6 i \left (d x +c \right )}+1410 \,{\mathrm e}^{7 i \left (d x +c \right )}-3840 i {\mathrm e}^{4 i \left (d x +c \right )}+1410 \,{\mathrm e}^{5 i \left (d x +c \right )}+1536 i {\mathrm e}^{2 i \left (d x +c \right )}-935 \,{\mathrm e}^{3 i \left (d x +c \right )}-256 i+165 \,{\mathrm e}^{i \left (d x +c \right )}}{120 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {11 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d \,a^{2}}-\frac {11 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d \,a^{2}}\) | \(168\) |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{6}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}-\frac {20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\frac {31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-40 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+44 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {31}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {20}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {4}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {5}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {40}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{64 d \,a^{2}}\) | \(176\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{6}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}-\frac {20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\frac {31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-40 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+44 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {31}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {20}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {4}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {5}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {40}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{64 d \,a^{2}}\) | \(176\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)^3/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/120*(165*exp(11*I*(d*x+c))-935*exp(9*I*(d*x+c))+2560*I*exp(6*I*(d*x+c))+ 1410*exp(7*I*(d*x+c))-3840*I*exp(4*I*(d*x+c))+1410*exp(5*I*(d*x+c))+1536*I *exp(2*I*(d*x+c))-935*exp(3*I*(d*x+c))-256*I+165*exp(I*(d*x+c)))/d/a^2/(ex p(2*I*(d*x+c))-1)^6+11/16/d/a^2*ln(exp(I*(d*x+c))-1)-11/16/d/a^2*ln(exp(I* (d*x+c))+1)
Time = 0.09 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.48 \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {330 \, \cos \left (d x + c\right )^{5} - 880 \, \cos \left (d x + c\right )^{3} - 165 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 165 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 64 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 20 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 630 \, \cos \left (d x + c\right )}{480 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} - 3 \, a^{2} d \cos \left (d x + c\right )^{4} + 3 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )}} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm="frica s")
Output:
1/480*(330*cos(d*x + c)^5 - 880*cos(d*x + c)^3 - 165*(cos(d*x + c)^6 - 3*c os(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(1/2*cos(d*x + c) + 1/2) + 165*(c os(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c) + 1/2) - 64*(8*cos(d*x + c)^5 - 20*cos(d*x + c)^3 + 15*cos(d*x + c))* sin(d*x + c) + 630*cos(d*x + c))/(a^2*d*cos(d*x + c)^6 - 3*a^2*d*cos(d*x + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^2*d)
\[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\int \frac {\cot ^{4}{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)**3/(a+a*sin(d*x+c))**2,x)
Output:
Integral(cot(c + d*x)**4*csc(c + d*x)**3/(sin(c + d*x)**2 + 2*sin(c + d*x) + 1), x)/a**2
Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (126) = 252\).
Time = 0.04 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.99 \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {\frac {1200 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {465 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {200 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {75 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {24 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{a^{2}} - \frac {1320 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac {{\left (\frac {24 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {75 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {200 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {465 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1200 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{6}}{a^{2} \sin \left (d x + c\right )^{6}}}{1920 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
-1/1920*((1200*sin(d*x + c)/(cos(d*x + c) + 1) - 465*sin(d*x + c)^2/(cos(d *x + c) + 1)^2 + 200*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 75*sin(d*x + c) ^4/(cos(d*x + c) + 1)^4 + 24*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5*sin(d *x + c)^6/(cos(d*x + c) + 1)^6)/a^2 - 1320*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 - (24*sin(d*x + c)/(cos(d*x + c) + 1) - 75*sin(d*x + c)^2/(cos(d *x + c) + 1)^2 + 200*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 465*sin(d*x + c )^4/(cos(d*x + c) + 1)^4 + 1200*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5)*( cos(d*x + c) + 1)^6/(a^2*sin(d*x + c)^6))/d
Time = 0.20 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.56 \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {1320 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {3234 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1200 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 465 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 200 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 75 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}} + \frac {5 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 24 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 75 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 200 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 465 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1200 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{12}}}{1920 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm="giac" )
Output:
1/1920*(1320*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - (3234*tan(1/2*d*x + 1/2* c)^6 - 1200*tan(1/2*d*x + 1/2*c)^5 + 465*tan(1/2*d*x + 1/2*c)^4 - 200*tan( 1/2*d*x + 1/2*c)^3 + 75*tan(1/2*d*x + 1/2*c)^2 - 24*tan(1/2*d*x + 1/2*c) + 5)/(a^2*tan(1/2*d*x + 1/2*c)^6) + (5*a^10*tan(1/2*d*x + 1/2*c)^6 - 24*a^1 0*tan(1/2*d*x + 1/2*c)^5 + 75*a^10*tan(1/2*d*x + 1/2*c)^4 - 200*a^10*tan(1 /2*d*x + 1/2*c)^3 + 465*a^10*tan(1/2*d*x + 1/2*c)^2 - 1200*a^10*tan(1/2*d* x + 1/2*c))/a^12)/d
Time = 18.30 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.46 \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-24\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+75\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-200\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+465\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-1200\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+1200\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-465\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+200\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-75\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1320\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{1920\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \] Input:
int(cot(c + d*x)^4/(sin(c + d*x)^3*(a + a*sin(c + d*x))^2),x)
Output:
(5*sin(c/2 + (d*x)/2)^12 - 5*cos(c/2 + (d*x)/2)^12 - 24*cos(c/2 + (d*x)/2) *sin(c/2 + (d*x)/2)^11 + 24*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2) + 75* cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 - 200*cos(c/2 + (d*x)/2)^3*sin( c/2 + (d*x)/2)^9 + 465*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^8 - 1200*co s(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^7 + 1200*cos(c/2 + (d*x)/2)^7*sin(c/ 2 + (d*x)/2)^5 - 465*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^4 + 200*cos(c /2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^3 - 75*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^2 + 1320*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d* x)/2)^6*sin(c/2 + (d*x)/2)^6)/(1920*a^2*d*cos(c/2 + (d*x)/2)^6*sin(c/2 + ( d*x)/2)^6)
Time = 0.17 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.89 \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {256 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-165 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+128 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-110 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+96 \cos \left (d x +c \right ) \sin \left (d x +c \right )-40 \cos \left (d x +c \right )+165 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6}}{240 \sin \left (d x +c \right )^{6} a^{2} d} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)^3/(a+a*sin(d*x+c))^2,x)
Output:
(256*cos(c + d*x)*sin(c + d*x)**5 - 165*cos(c + d*x)*sin(c + d*x)**4 + 128 *cos(c + d*x)*sin(c + d*x)**3 - 110*cos(c + d*x)*sin(c + d*x)**2 + 96*cos( c + d*x)*sin(c + d*x) - 40*cos(c + d*x) + 165*log(tan((c + d*x)/2))*sin(c + d*x)**6)/(240*sin(c + d*x)**6*a**2*d)