Integrand size = 27, antiderivative size = 93 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {15 \text {arctanh}(\cos (c+d x))}{8 a^3 d}+\frac {4 \cot (c+d x)}{a^3 d}+\frac {\cot ^3(c+d x)}{a^3 d}-\frac {15 \cot (c+d x) \csc (c+d x)}{8 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d} \] Output:
-15/8*arctanh(cos(d*x+c))/a^3/d+4*cot(d*x+c)/a^3/d+cot(d*x+c)^3/a^3/d-15/8 *cot(d*x+c)*csc(d*x+c)/a^3/d-1/4*cot(d*x+c)*csc(d*x+c)^3/a^3/d
Time = 3.58 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.34 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\csc ^4(c+d x) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6 \left (46 \cos (c+d x)+120 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin ^4(c+d x)+6 \cos (3 (c+d x)) (-5+8 \sin (c+d x))-56 \sin (2 (c+d x))\right )}{64 a^3 d (1+\sin (c+d x))^3} \] Input:
Integrate[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + a*Sin[c + d*x])^3,x]
Output:
-1/64*(Csc[c + d*x]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(46*Cos[c + d*x] + 120*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^4 + 6*Cos[3*(c + d*x)]*(-5 + 8*Sin[c + d*x]) - 56*Sin[2*(c + d*x)]))/(a^3*d* (1 + Sin[c + d*x])^3)
Time = 0.48 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, 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.
| \(\displaystyle \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6}{\sin (c+d x)^5 (a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3348 |
| \(\displaystyle \frac {\int \csc ^5(c+d x) (a-a \sin (c+d x))^3dx}{a^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a-a \sin (c+d x))^3}{\sin (c+d x)^5}dx}{a^6}\) |
| \(\Big \downarrow \) 3236 |
| \(\displaystyle \frac {\int \left (a^3 \csc ^5(c+d x)-3 a^3 \csc ^4(c+d x)+3 a^3 \csc ^3(c+d x)-a^3 \csc ^2(c+d x)\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {15 a^3 \text {arctanh}(\cos (c+d x))}{8 d}+\frac {a^3 \cot ^3(c+d x)}{d}+\frac {4 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {15 a^3 \cot (c+d x) \csc (c+d x)}{8 d}}{a^6}\) |
Input:
Int[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + a*Sin[c + d*x])^3,x]
Output:
((-15*a^3*ArcTanh[Cos[c + d*x]])/(8*d) + (4*a^3*Cot[c + d*x])/d + (a^3*Cot [c + d*x]^3)/d - (15*a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^3*Cot[c + d *x]*Csc[c + d*x]^3)/(4*d))/a^6
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]
Time = 4.63 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.33
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-26 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {26}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+30 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{3}}\) | \(124\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-26 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {26}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+30 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{3}}\) | \(124\) |
| risch | \(\frac {15 \,{\mathrm e}^{7 i \left (d x +c \right )}-23 \,{\mathrm e}^{5 i \left (d x +c \right )}+8 i {\mathrm e}^{6 i \left (d x +c \right )}-23 \,{\mathrm e}^{3 i \left (d x +c \right )}-72 i {\mathrm e}^{4 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )}+88 i {\mathrm e}^{2 i \left (d x +c \right )}-24 i}{4 d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d \,a^{3}}-\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d \,a^{3}}\) | \(146\) |
Input:
int(cos(d*x+c)*cot(d*x+c)^5/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/16/d/a^3*(1/4*tan(1/2*d*x+1/2*c)^4-2*tan(1/2*d*x+1/2*c)^3+8*tan(1/2*d*x+ 1/2*c)^2-26*tan(1/2*d*x+1/2*c)-1/4/tan(1/2*d*x+1/2*c)^4+2/tan(1/2*d*x+1/2* c)^3-8/tan(1/2*d*x+1/2*c)^2+26/tan(1/2*d*x+1/2*c)+30*ln(tan(1/2*d*x+1/2*c) ))
Time = 0.09 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.60 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {30 \, \cos \left (d x + c\right )^{3} - 15 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 16 \, {\left (3 \, \cos \left (d x + c\right )^{3} - 4 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 34 \, \cos \left (d x + c\right )}{16 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )}} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
1/16*(30*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) + 15*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*log(-1 /2*cos(d*x + c) + 1/2) - 16*(3*cos(d*x + c)^3 - 4*cos(d*x + c))*sin(d*x + c) - 34*cos(d*x + c))/(a^3*d*cos(d*x + c)^4 - 2*a^3*d*cos(d*x + c)^2 + a^3 *d)
\[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\cos {\left (c + d x \right )} \cot ^{5}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)**5/(a+a*sin(d*x+c))**3,x)
Output:
Integral(cos(c + d*x)*cot(c + d*x)**5/(sin(c + d*x)**3 + 3*sin(c + d*x)**2 + 3*sin(c + d*x) + 1), x)/a**3
Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (87) = 174\).
Time = 0.05 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.10 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {\frac {104 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {32 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {8 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a^{3}} - \frac {120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {{\left (\frac {8 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {32 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {104 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{a^{3} \sin \left (d x + c\right )^{4}}}{64 \, d} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
-1/64*((104*sin(d*x + c)/(cos(d*x + c) + 1) - 32*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 8*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - sin(d*x + c)^4/(cos(d *x + c) + 1)^4)/a^3 - 120*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^3 - (8*si n(d*x + c)/(cos(d*x + c) + 1) - 32*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1 04*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 1)*(cos(d*x + c) + 1)^4/(a^3*sin( d*x + c)^4))/d
Time = 0.20 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.68 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {250 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 104 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 32 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} + \frac {a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 32 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 104 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{12}}}{64 \, d} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/64*(120*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - (250*tan(1/2*d*x + 1/2*c)^4 - 104*tan(1/2*d*x + 1/2*c)^3 + 32*tan(1/2*d*x + 1/2*c)^2 - 8*tan(1/2*d*x + 1/2*c) + 1)/(a^3*tan(1/2*d*x + 1/2*c)^4) + (a^9*tan(1/2*d*x + 1/2*c)^4 - 8*a^9*tan(1/2*d*x + 1/2*c)^3 + 32*a^9*tan(1/2*d*x + 1/2*c)^2 - 104*a^9*ta n(1/2*d*x + 1/2*c))/a^12)/d
Time = 33.52 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.62 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,a^3\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,a^3\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a^3\,d}+\frac {15\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,a^3\,d}-\frac {13\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a^3\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (26\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{4}\right )}{16\,a^3\,d} \] Input:
int((cos(c + d*x)*cot(c + d*x)^5)/(a + a*sin(c + d*x))^3,x)
Output:
tan(c/2 + (d*x)/2)^2/(2*a^3*d) - tan(c/2 + (d*x)/2)^3/(8*a^3*d) + tan(c/2 + (d*x)/2)^4/(64*a^3*d) + (15*log(tan(c/2 + (d*x)/2)))/(8*a^3*d) - (13*tan (c/2 + (d*x)/2))/(8*a^3*d) + (cot(c/2 + (d*x)/2)^4*(2*tan(c/2 + (d*x)/2) - 8*tan(c/2 + (d*x)/2)^2 + 26*tan(c/2 + (d*x)/2)^3 - 1/4))/(16*a^3*d)
Time = 0.16 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.98 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {24 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-15 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+8 \cos \left (d x +c \right ) \sin \left (d x +c \right )-2 \cos \left (d x +c \right )+15 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4}}{8 \sin \left (d x +c \right )^{4} a^{3} d} \] Input:
int(cos(d*x+c)*cot(d*x+c)^5/(a+a*sin(d*x+c))^3,x)
Output:
(24*cos(c + d*x)*sin(c + d*x)**3 - 15*cos(c + d*x)*sin(c + d*x)**2 + 8*cos (c + d*x)*sin(c + d*x) - 2*cos(c + d*x) + 15*log(tan((c + d*x)/2))*sin(c + d*x)**4)/(8*sin(c + d*x)**4*a**3*d)