Integrand size = 21, antiderivative size = 100 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\text {arctanh}(\cos (c+d x))}{4 a^2 d}-\frac {2 \cot ^3(c+d x)}{3 a^2 d}-\frac {\cot ^5(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{4 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d} \] Output:
-1/4*arctanh(cos(d*x+c))/a^2/d-2/3*cot(d*x+c)^3/a^2/d-1/5*cot(d*x+c)^5/a^2 /d-1/4*cot(d*x+c)*csc(d*x+c)/a^2/d+1/2*cot(d*x+c)*csc(d*x+c)^3/a^2/d
Time = 1.52 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.89 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\csc ^5(c+d x) \left (200 \cos (c+d x)+20 \cos (3 (c+d x))-28 \cos (5 (c+d x))+150 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-150 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-180 \sin (2 (c+d x))-75 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+75 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-30 \sin (4 (c+d x))+15 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-15 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))\right )}{960 a^2 d} \] Input:
Integrate[Cot[c + d*x]^6/(a + a*Sin[c + d*x])^2,x]
Output:
-1/960*(Csc[c + d*x]^5*(200*Cos[c + d*x] + 20*Cos[3*(c + d*x)] - 28*Cos[5* (c + d*x)] + 150*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 150*Log[Sin[(c + d*x )/2]]*Sin[c + d*x] - 180*Sin[2*(c + d*x)] - 75*Log[Cos[(c + d*x)/2]]*Sin[3 *(c + d*x)] + 75*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 30*Sin[4*(c + d* x)] + 15*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 15*Log[Sin[(c + d*x)/2]] *Sin[5*(c + d*x)]))/(a^2*d)
Time = 0.37 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.04, 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.
| \(\displaystyle \int \frac {\cot ^6(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^6 (a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3188 |
| \(\displaystyle \frac {\int \left (a^4 \csc ^6(c+d x)-2 a^4 \csc ^5(c+d x)+2 a^4 \csc ^3(c+d x)-a^4 \csc ^2(c+d x)\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {a^4 \text {arctanh}(\cos (c+d x))}{4 d}-\frac {a^4 \cot ^5(c+d x)}{5 d}-\frac {2 a^4 \cot ^3(c+d x)}{3 d}+\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{2 d}-\frac {a^4 \cot (c+d x) \csc (c+d x)}{4 d}}{a^6}\) |
Input:
Int[Cot[c + d*x]^6/(a + a*Sin[c + d*x])^2,x]
Output:
(-1/4*(a^4*ArcTanh[Cos[c + d*x]])/d - (2*a^4*Cot[c + d*x]^3)/(3*d) - (a^4* Cot[c + d*x]^5)/(5*d) - (a^4*Cot[c + d*x]*Csc[c + d*x])/(4*d) + (a^4*Cot[c + d*x]*Csc[c + d*x]^3)/(2*d))/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 = 3.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.22
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+8 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{32 d \,a^{2}}\) | \(122\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+8 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{32 d \,a^{2}}\) | \(122\) |
| risch | \(\frac {60 i {\mathrm e}^{8 i \left (d x +c \right )}+15 \,{\mathrm e}^{9 i \left (d x +c \right )}-240 i {\mathrm e}^{6 i \left (d x +c \right )}+90 \,{\mathrm e}^{7 i \left (d x +c \right )}+40 i {\mathrm e}^{4 i \left (d x +c \right )}-80 i {\mathrm e}^{2 i \left (d x +c \right )}-90 \,{\mathrm e}^{3 i \left (d x +c \right )}+28 i-15 \,{\mathrm e}^{i \left (d x +c \right )}}{30 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{4 d \,a^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{4 d \,a^{2}}\) | \(158\) |
Input:
int(cot(d*x+c)^6/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/32/d/a^2*(1/5*tan(1/2*d*x+1/2*c)^5-tan(1/2*d*x+1/2*c)^4+5/3*tan(1/2*d*x+ 1/2*c)^3-6*tan(1/2*d*x+1/2*c)+6/tan(1/2*d*x+1/2*c)-1/5/tan(1/2*d*x+1/2*c)^ 5+1/tan(1/2*d*x+1/2*c)^4+8*ln(tan(1/2*d*x+1/2*c))-5/3/tan(1/2*d*x+1/2*c)^3 )
Time = 0.09 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.67 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {56 \, \cos \left (d x + c\right )^{5} - 80 \, \cos \left (d x + c\right )^{3} - 15 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 15 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 30 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^6/(a+a*sin(d*x+c))^2,x, algorithm="fricas")
Output:
1/120*(56*cos(d*x + c)^5 - 80*cos(d*x + c)^3 - 15*(cos(d*x + c)^4 - 2*cos( d*x + c)^2 + 1)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 15*(cos(d*x + c )^4 - 2*cos(d*x + c)^2 + 1)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 30 *(cos(d*x + c)^3 + cos(d*x + c))*sin(d*x + c))/((a^2*d*cos(d*x + c)^4 - 2* a^2*d*cos(d*x + c)^2 + a^2*d)*sin(d*x + c))
\[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\int \frac {\cot ^{6}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \] Input:
integrate(cot(d*x+c)**6/(a+a*sin(d*x+c))**2,x)
Output:
Integral(cot(c + d*x)**6/(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. 195 vs. \(2 (90) = 180\).
Time = 0.05 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.95 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {\frac {90 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {25 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{2}} - \frac {120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac {{\left (\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {25 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {90 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a^{2} \sin \left (d x + c\right )^{5}}}{480 \, d} \] Input:
integrate(cot(d*x+c)^6/(a+a*sin(d*x+c))^2,x, algorithm="maxima")
Output:
-1/480*((90*sin(d*x + c)/(cos(d*x + c) + 1) - 25*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 15*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 3*sin(d*x + c)^5/(co s(d*x + c) + 1)^5)/a^2 - 120*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 - (1 5*sin(d*x + c)/(cos(d*x + c) + 1) - 25*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 90*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 3)*(cos(d*x + c) + 1)^5/(a^2*s in(d*x + c)^5))/d
Time = 0.19 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.57 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {274 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 90 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 25 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} + \frac {3 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 25 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 90 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{10}}}{480 \, d} \] Input:
integrate(cot(d*x+c)^6/(a+a*sin(d*x+c))^2,x, algorithm="giac")
Output:
1/480*(120*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - (274*tan(1/2*d*x + 1/2*c)^ 5 - 90*tan(1/2*d*x + 1/2*c)^4 + 25*tan(1/2*d*x + 1/2*c)^2 - 15*tan(1/2*d*x + 1/2*c) + 3)/(a^2*tan(1/2*d*x + 1/2*c)^5) + (3*a^8*tan(1/2*d*x + 1/2*c)^ 5 - 15*a^8*tan(1/2*d*x + 1/2*c)^4 + 25*a^8*tan(1/2*d*x + 1/2*c)^3 - 90*a^8 *tan(1/2*d*x + 1/2*c))/a^10)/d
Time = 32.67 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.49 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,a^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,a^2\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,a^2\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,a^2\,d}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,a^2\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{5}\right )}{32\,a^2\,d} \] Input:
int(cot(c + d*x)^6/(a + a*sin(c + d*x))^2,x)
Output:
(5*tan(c/2 + (d*x)/2)^3)/(96*a^2*d) - tan(c/2 + (d*x)/2)^4/(32*a^2*d) + ta n(c/2 + (d*x)/2)^5/(160*a^2*d) + log(tan(c/2 + (d*x)/2))/(4*a^2*d) - (3*ta n(c/2 + (d*x)/2))/(16*a^2*d) + (cot(c/2 + (d*x)/2)^5*(tan(c/2 + (d*x)/2) - (5*tan(c/2 + (d*x)/2)^2)/3 + 6*tan(c/2 + (d*x)/2)^4 - 1/5))/(32*a^2*d)
Time = 0.17 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.07 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {28 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-15 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-16 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+30 \cos \left (d x +c \right ) \sin \left (d x +c \right )-12 \cos \left (d x +c \right )+15 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{5}}{60 \sin \left (d x +c \right )^{5} a^{2} d} \] Input:
int(cot(d*x+c)^6/(a+a*sin(d*x+c))^2,x)
Output:
(28*cos(c + d*x)*sin(c + d*x)**4 - 15*cos(c + d*x)*sin(c + d*x)**3 - 16*co s(c + d*x)*sin(c + d*x)**2 + 30*cos(c + d*x)*sin(c + d*x) - 12*cos(c + d*x ) + 15*log(tan((c + d*x)/2))*sin(c + d*x)**5)/(60*sin(c + d*x)**5*a**2*d)