Integrand size = 21, antiderivative size = 133 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {27 \text {arctanh}(\cos (c+d x))}{2 a^4 d}-\frac {16 \cot (c+d x)}{a^4 d}-\frac {3 \cot ^3(c+d x)}{a^4 d}-\frac {\cot ^5(c+d x)}{5 a^4 d}+\frac {11 \cot (c+d x) \csc (c+d x)}{2 a^4 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{a^4 d}-\frac {8 \cot (c+d x)}{a^4 d (1+\csc (c+d x))} \] Output:
27/2*arctanh(cos(d*x+c))/a^4/d-16*cot(d*x+c)/a^4/d-3*cot(d*x+c)^3/a^4/d-1/ 5*cot(d*x+c)^5/a^4/d+11/2*cot(d*x+c)*csc(d*x+c)/a^4/d+cot(d*x+c)*csc(d*x+c )^3/a^4/d-8*cot(d*x+c)/a^4/d/(1+csc(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(733\) vs. \(2(133)=266\).
Time = 7.11 (sec) , antiderivative size = 733, normalized size of antiderivative = 5.51 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^4} \, dx =\text {Too large to display} \] Input:
Integrate[Cot[c + d*x]^6/(a + a*Sin[c + d*x])^4,x]
Output:
(16*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7)/(d*(a + a*Si n[c + d*x])^4) - (33*Cot[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2] )^8)/(5*d*(a + a*Sin[c + d*x])^4) + (11*Csc[(c + d*x)/2]^2*(Cos[(c + d*x)/ 2] + Sin[(c + d*x)/2])^8)/(8*d*(a + a*Sin[c + d*x])^4) - (53*Cot[(c + d*x) /2]*Csc[(c + d*x)/2]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(160*d*(a + a*Sin[c + d*x])^4) + (Csc[(c + d*x)/2]^4*(Cos[(c + d*x)/2] + Sin[(c + d* x)/2])^8)/(16*d*(a + a*Sin[c + d*x])^4) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/ 2]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(160*d*(a + a*Sin[c + d*x])^ 4) + (27*Log[Cos[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(2 *d*(a + a*Sin[c + d*x])^4) - (27*Log[Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(2*d*(a + a*Sin[c + d*x])^4) - (11*Sec[(c + d*x)/2]^ 2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(8*d*(a + a*Sin[c + d*x])^4) - (Sec[(c + d*x)/2]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)/(16*d*(a + a* Sin[c + d*x])^4) + (33*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8*Tan[(c + d* x)/2])/(5*d*(a + a*Sin[c + d*x])^4) + (53*Sec[(c + d*x)/2]^2*(Cos[(c + d*x )/2] + Sin[(c + d*x)/2])^8*Tan[(c + d*x)/2])/(160*d*(a + a*Sin[c + d*x])^4 ) + (Sec[(c + d*x)/2]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8*Tan[(c + d *x)/2])/(160*d*(a + a*Sin[c + d*x])^4)
Time = 0.47 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3188, 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.
| \(\displaystyle \int \frac {\cot ^6(c+d x)}{(a \sin (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^6 (a \sin (c+d x)+a)^4}dx\) |
| \(\Big \downarrow \) 3188 |
| \(\displaystyle \frac {\int \left (a^2 \csc ^6(c+d x)-4 a^2 \csc ^5(c+d x)+7 a^2 \csc ^4(c+d x)-8 a^2 \csc ^3(c+d x)+8 a^2 \csc ^2(c+d x)-8 a^2 \csc (c+d x)+8 a^2-\frac {8 a^2}{\csc (c+d x)+1}\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {27 a^2 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 \cot ^3(c+d x)}{d}-\frac {16 a^2 \cot (c+d x)}{d}+\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{d}+\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {8 a^2 \cot (c+d x)}{d (\csc (c+d x)+1)}}{a^6}\) |
Input:
Int[Cot[c + d*x]^6/(a + a*Sin[c + d*x])^4,x]
Output:
((27*a^2*ArcTanh[Cos[c + d*x]])/(2*d) - (16*a^2*Cot[c + d*x])/d - (3*a^2*C ot[c + d*x]^3)/d - (a^2*Cot[c + d*x]^5)/(5*d) + (11*a^2*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/d - (8*a^2*Cot[c + d*x] )/(d*(1 + Csc[c + d*x])))/a^6
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_ ), x_Symbol] :> Simp[a^p Int[ExpandIntegrand[Sin[e + f*x]^p*((a + b*Sin[e + f*x])^(m - p/2)/(a - b*Sin[e + f*x])^(p/2)), x], x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p/2] && (LtQ[p, 0] || GtQ[m - p/2, 0])
Time = 40.28 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.24
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-48 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+222 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {11}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {48}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {222}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-432 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {512}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{32 d \,a^{4}}\) | \(165\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-48 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+222 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {11}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {48}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {222}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-432 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {512}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{32 d \,a^{4}}\) | \(165\) |
| risch | \(-\frac {135 i {\mathrm e}^{9 i \left (d x +c \right )}-630 \,{\mathrm e}^{8 i \left (d x +c \right )}+135 \,{\mathrm e}^{10 i \left (d x +c \right )}-610 i {\mathrm e}^{7 i \left (d x +c \right )}+1260 \,{\mathrm e}^{6 i \left (d x +c \right )}+860 i {\mathrm e}^{5 i \left (d x +c \right )}-1510 \,{\mathrm e}^{4 i \left (d x +c \right )}-430 i {\mathrm e}^{3 i \left (d x +c \right )}+925 \,{\mathrm e}^{2 i \left (d x +c \right )}+77 i {\mathrm e}^{i \left (d x +c \right )}-212}{5 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{4} d}-\frac {27 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 a^{4} d}+\frac {27 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 a^{4} d}\) | \(194\) |
Input:
int(cot(d*x+c)^6/(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/32/d/a^4*(1/5*tan(1/2*d*x+1/2*c)^5-2*tan(1/2*d*x+1/2*c)^4+11*tan(1/2*d*x +1/2*c)^3-48*tan(1/2*d*x+1/2*c)^2+222*tan(1/2*d*x+1/2*c)-1/5/tan(1/2*d*x+1 /2*c)^5+2/tan(1/2*d*x+1/2*c)^4-11/tan(1/2*d*x+1/2*c)^3+48/tan(1/2*d*x+1/2* c)^2-222/tan(1/2*d*x+1/2*c)-432*ln(tan(1/2*d*x+1/2*c))-512/(tan(1/2*d*x+1/ 2*c)+1))
Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (127) = 254\).
Time = 0.14 (sec) , antiderivative size = 439, normalized size of antiderivative = 3.30 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {424 \, \cos \left (d x + c\right )^{6} + 154 \, \cos \left (d x + c\right )^{5} - 1060 \, \cos \left (d x + c\right )^{4} - 340 \, \cos \left (d x + c\right )^{3} + 800 \, \cos \left (d x + c\right )^{2} + 135 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 135 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (212 \, \cos \left (d x + c\right )^{5} + 135 \, \cos \left (d x + c\right )^{4} - 395 \, \cos \left (d x + c\right )^{3} - 225 \, \cos \left (d x + c\right )^{2} + 175 \, \cos \left (d x + c\right ) + 80\right )} \sin \left (d x + c\right ) + 190 \, \cos \left (d x + c\right ) - 160}{20 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} - 3 \, a^{4} d \cos \left (d x + c\right )^{4} + 3 \, a^{4} d \cos \left (d x + c\right )^{2} - a^{4} d - {\left (a^{4} d \cos \left (d x + c\right )^{5} + a^{4} d \cos \left (d x + c\right )^{4} - 2 \, a^{4} d \cos \left (d x + c\right )^{3} - 2 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(cot(d*x+c)^6/(a+a*sin(d*x+c))^4,x, algorithm="fricas")
Output:
1/20*(424*cos(d*x + c)^6 + 154*cos(d*x + c)^5 - 1060*cos(d*x + c)^4 - 340* cos(d*x + c)^3 + 800*cos(d*x + c)^2 + 135*(cos(d*x + c)^6 - 3*cos(d*x + c) ^4 + 3*cos(d*x + c)^2 - (cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos(d*x + c)^ 3 - 2*cos(d*x + c)^2 + cos(d*x + c) + 1)*sin(d*x + c) - 1)*log(1/2*cos(d*x + c) + 1/2) - 135*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - (cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 + cos(d*x + c) + 1)*sin(d*x + c) - 1)*log(-1/2*cos(d*x + c) + 1/2) + 2*(212* cos(d*x + c)^5 + 135*cos(d*x + c)^4 - 395*cos(d*x + c)^3 - 225*cos(d*x + c )^2 + 175*cos(d*x + c) + 80)*sin(d*x + c) + 190*cos(d*x + c) - 160)/(a^4*d *cos(d*x + c)^6 - 3*a^4*d*cos(d*x + c)^4 + 3*a^4*d*cos(d*x + c)^2 - a^4*d - (a^4*d*cos(d*x + c)^5 + a^4*d*cos(d*x + c)^4 - 2*a^4*d*cos(d*x + c)^3 - 2*a^4*d*cos(d*x + c)^2 + a^4*d*cos(d*x + c) + a^4*d)*sin(d*x + c))
\[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\int \frac {\cot ^{6}{\left (c + d x \right )}}{\sin ^{4}{\left (c + d x \right )} + 4 \sin ^{3}{\left (c + d x \right )} + 6 \sin ^{2}{\left (c + d x \right )} + 4 \sin {\left (c + d x \right )} + 1}\, dx}{a^{4}} \] Input:
integrate(cot(d*x+c)**6/(a+a*sin(d*x+c))**4,x)
Output:
Integral(cot(c + d*x)**6/(sin(c + d*x)**4 + 4*sin(c + d*x)**3 + 6*sin(c + d*x)**2 + 4*sin(c + d*x) + 1), x)/a**4
Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (127) = 254\).
Time = 0.04 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.10 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {45 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {185 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {870 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3670 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 1}{\frac {a^{4} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {a^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {1110 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {240 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {55 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {10 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{4}} - \frac {2160 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}}{160 \, d} \] Input:
integrate(cot(d*x+c)^6/(a+a*sin(d*x+c))^4,x, algorithm="maxima")
Output:
1/160*((9*sin(d*x + c)/(cos(d*x + c) + 1) - 45*sin(d*x + c)^2/(cos(d*x + c ) + 1)^2 + 185*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 870*sin(d*x + c)^4/(c os(d*x + c) + 1)^4 - 3670*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1)/(a^4*si n(d*x + c)^5/(cos(d*x + c) + 1)^5 + a^4*sin(d*x + c)^6/(cos(d*x + c) + 1)^ 6) + (1110*sin(d*x + c)/(cos(d*x + c) + 1) - 240*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 55*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 10*sin(d*x + c)^4/(c os(d*x + c) + 1)^4 + sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^4 - 2160*log(s in(d*x + c)/(cos(d*x + c) + 1))/a^4)/d
Time = 0.19 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.53 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^4} \, dx=-\frac {\frac {2160 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{4}} + \frac {2560}{a^{4} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}} - \frac {4932 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1110 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 55 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}{a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} - \frac {a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 10 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 55 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 240 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1110 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{20}}}{160 \, d} \] Input:
integrate(cot(d*x+c)^6/(a+a*sin(d*x+c))^4,x, algorithm="giac")
Output:
-1/160*(2160*log(abs(tan(1/2*d*x + 1/2*c)))/a^4 + 2560/(a^4*(tan(1/2*d*x + 1/2*c) + 1)) - (4932*tan(1/2*d*x + 1/2*c)^5 - 1110*tan(1/2*d*x + 1/2*c)^4 + 240*tan(1/2*d*x + 1/2*c)^3 - 55*tan(1/2*d*x + 1/2*c)^2 + 10*tan(1/2*d*x + 1/2*c) - 1)/(a^4*tan(1/2*d*x + 1/2*c)^5) - (a^16*tan(1/2*d*x + 1/2*c)^5 - 10*a^16*tan(1/2*d*x + 1/2*c)^4 + 55*a^16*tan(1/2*d*x + 1/2*c)^3 - 240*a ^16*tan(1/2*d*x + 1/2*c)^2 + 1110*a^16*tan(1/2*d*x + 1/2*c))/a^20)/d
Time = 18.38 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.57 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32\,a^4\,d}-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16\,a^4\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,a^4\,d}-\frac {27\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a^4\,d}+\frac {111\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,a^4\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {367\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{16}+\frac {87\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16}-\frac {37\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32}+\frac {9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32}-\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{160}+\frac {1}{160}\right )}{a^4\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )} \] Input:
int(cot(c + d*x)^6/(a + a*sin(c + d*x))^4,x)
Output:
(11*tan(c/2 + (d*x)/2)^3)/(32*a^4*d) - (3*tan(c/2 + (d*x)/2)^2)/(2*a^4*d) - tan(c/2 + (d*x)/2)^4/(16*a^4*d) + tan(c/2 + (d*x)/2)^5/(160*a^4*d) - (27 *log(tan(c/2 + (d*x)/2)))/(2*a^4*d) + (111*tan(c/2 + (d*x)/2))/(16*a^4*d) - (cot(c/2 + (d*x)/2)^5*((9*tan(c/2 + (d*x)/2)^2)/32 - (9*tan(c/2 + (d*x)/ 2))/160 - (37*tan(c/2 + (d*x)/2)^3)/32 + (87*tan(c/2 + (d*x)/2)^4)/16 + (3 67*tan(c/2 + (d*x)/2)^5)/16 + 1/160))/(a^4*d*(tan(c/2 + (d*x)/2) + 1))
Time = 0.19 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.55 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {-2160 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-2160 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}-9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+45 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}-185 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+870 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+4780 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-870 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+185 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-45 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}{160 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{4} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \] Input:
int(cot(d*x+c)^6/(a+a*sin(d*x+c))^4,x)
Output:
( - 2160*log(tan((c + d*x)/2))*tan((c + d*x)/2)**6 - 2160*log(tan((c + d*x )/2))*tan((c + d*x)/2)**5 + tan((c + d*x)/2)**11 - 9*tan((c + d*x)/2)**10 + 45*tan((c + d*x)/2)**9 - 185*tan((c + d*x)/2)**8 + 870*tan((c + d*x)/2)* *7 + 4780*tan((c + d*x)/2)**6 - 870*tan((c + d*x)/2)**4 + 185*tan((c + d*x )/2)**3 - 45*tan((c + d*x)/2)**2 + 9*tan((c + d*x)/2) - 1)/(160*tan((c + d *x)/2)**5*a**4*d*(tan((c + d*x)/2) + 1))