Integrand size = 27, antiderivative size = 99 \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {13 x}{8 a^3}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\cos (c+d x)}{a^3 d}-\frac {\cos ^3(c+d x)}{a^3 d}-\frac {13 \cos (c+d x) \sin (c+d x)}{8 a^3 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a^3 d} \] Output:
-13/8*x/a^3-arctanh(cos(d*x+c))/a^3/d+cos(d*x+c)/a^3/d-cos(d*x+c)^3/a^3/d- 13/8*cos(d*x+c)*sin(d*x+c)/a^3/d+1/4*cos(d*x+c)^3*sin(d*x+c)/a^3/d
Time = 1.63 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.81 \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-52 c-52 d x+8 \cos (c+d x)-8 \cos (3 (c+d x))-32 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+32 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-24 \sin (2 (c+d x))+\sin (4 (c+d x))}{32 a^3 d} \] Input:
Integrate[(Cos[c + d*x]^7*Cot[c + d*x])/(a + a*Sin[c + d*x])^3,x]
Output:
(-52*c - 52*d*x + 8*Cos[c + d*x] - 8*Cos[3*(c + d*x)] - 32*Log[Cos[(c + d* x)/2]] + 32*Log[Sin[(c + d*x)/2]] - 24*Sin[2*(c + d*x)] + Sin[4*(c + d*x)] )/(32*a^3*d)
Time = 0.50 (sec) , antiderivative size = 103, 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, 3354, 3042, 3352, 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.
| \(\displaystyle \int \frac {\cos ^7(c+d x) \cot (c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^8}{\sin (c+d x) (a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \cos (c+d x) \cot (c+d x) (a-a \sin (c+d x))^3dx}{a^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\cos (c+d x)^2 (a-a \sin (c+d x))^3}{\sin (c+d x)}dx}{a^6}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (-3 \cos ^2(c+d x) a^3-\cos ^2(c+d x) \sin ^2(c+d x) a^3+\cos (c+d x) \cot (c+d x) a^3+3 \cos ^2(c+d x) \sin (c+d x) a^3\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}-\frac {a^3 \cos ^3(c+d x)}{d}+\frac {a^3 \cos (c+d x)}{d}+\frac {a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac {13 a^3 \sin (c+d x) \cos (c+d x)}{8 d}-\frac {13 a^3 x}{8}}{a^6}\) |
Input:
Int[(Cos[c + d*x]^7*Cot[c + d*x])/(a + a*Sin[c + d*x])^3,x]
Output:
((-13*a^3*x)/8 - (a^3*ArcTanh[Cos[c + d*x]])/d + (a^3*Cos[c + d*x])/d - (a ^3*Cos[c + d*x]^3)/d - (13*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^3*Cos [c + d*x]^3*Sin[c + d*x])/(4*d))/a^6
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a/g)^(2* m) Int[(g*Cos[e + f*x])^(2*m + p)*((d*Sin[e + f*x])^n/(a - b*Sin[e + f*x] )^m), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]
Time = 3.01 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.26
| method | result | size |
| derivativedivides | \(\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2 \left (-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\frac {19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8}+\frac {19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {13 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d \,a^{3}}\) | \(125\) |
| default | \(\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2 \left (-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\frac {19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8}+\frac {19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {13 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d \,a^{3}}\) | \(125\) |
| risch | \(-\frac {13 x}{8 a^{3}}+\frac {{\mathrm e}^{i \left (d x +c \right )}}{8 d \,a^{3}}+\frac {{\mathrm e}^{-i \left (d x +c \right )}}{8 d \,a^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{3}}+\frac {\sin \left (4 d x +4 c \right )}{32 d \,a^{3}}-\frac {\cos \left (3 d x +3 c \right )}{4 a^{3} d}-\frac {3 \sin \left (2 d x +2 c \right )}{4 d \,a^{3}}\) | \(132\) |
Input:
int(cos(d*x+c)^7*cot(d*x+c)/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d/a^3*(ln(tan(1/2*d*x+1/2*c))-2*(-11/8*tan(1/2*d*x+1/2*c)^7+2*tan(1/2*d* x+1/2*c)^6-19/8*tan(1/2*d*x+1/2*c)^5+19/8*tan(1/2*d*x+1/2*c)^3-2*tan(1/2*d *x+1/2*c)^2+11/8*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^4-13/4*arcta n(tan(1/2*d*x+1/2*c)))
Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.85 \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {8 \, \cos \left (d x + c\right )^{3} + 13 \, d x - {\left (2 \, \cos \left (d x + c\right )^{3} - 13 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 8 \, \cos \left (d x + c\right ) + 4 \, \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 4 \, \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{8 \, a^{3} d} \] Input:
integrate(cos(d*x+c)^7*cot(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
-1/8*(8*cos(d*x + c)^3 + 13*d*x - (2*cos(d*x + c)^3 - 13*cos(d*x + c))*sin (d*x + c) - 8*cos(d*x + c) + 4*log(1/2*cos(d*x + c) + 1/2) - 4*log(-1/2*co s(d*x + c) + 1/2))/(a^3*d)
Timed out. \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**7*cot(d*x+c)/(a+a*sin(d*x+c))**3,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (93) = 186\).
Time = 0.12 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.72 \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {\frac {11 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {16 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {19 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {19 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {16 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {11 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{3} + \frac {4 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} + \frac {13 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {4 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{4 \, d} \] Input:
integrate(cos(d*x+c)^7*cot(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
-1/4*((11*sin(d*x + c)/(cos(d*x + c) + 1) - 16*sin(d*x + c)^2/(cos(d*x + c ) + 1)^2 + 19*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 19*sin(d*x + c)^5/(cos (d*x + c) + 1)^5 + 16*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 11*sin(d*x + c )^7/(cos(d*x + c) + 1)^7)/(a^3 + 4*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*a^3*sin(d*x + c)^6/(cos(d *x + c) + 1)^6 + a^3*sin(d*x + c)^8/(cos(d*x + c) + 1)^8) + 13*arctan(sin( d*x + c)/(cos(d*x + c) + 1))/a^3 - 4*log(sin(d*x + c)/(cos(d*x + c) + 1))/ a^3)/d
Time = 0.17 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.30 \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {13 \, {\left (d x + c\right )}}{a^{3}} - \frac {8 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {2 \, {\left (11 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 16 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 19 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 19 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 16 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 11 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a^{3}}}{8 \, d} \] Input:
integrate(cos(d*x+c)^7*cot(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
-1/8*(13*(d*x + c)/a^3 - 8*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - 2*(11*tan( 1/2*d*x + 1/2*c)^7 - 16*tan(1/2*d*x + 1/2*c)^6 + 19*tan(1/2*d*x + 1/2*c)^5 - 19*tan(1/2*d*x + 1/2*c)^3 + 16*tan(1/2*d*x + 1/2*c)^2 - 11*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^4*a^3))/d
Time = 33.58 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.24 \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {13\,\mathrm {atan}\left (\frac {169}{16\,\left (\frac {169\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}+\frac {13}{2}\right )}-\frac {13\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (\frac {169\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}+\frac {13}{2}\right )}\right )}{4\,a^3\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {19\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+\frac {19\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {11\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )} \] Input:
int((cos(c + d*x)^7*cot(c + d*x))/(a + a*sin(c + d*x))^3,x)
Output:
(13*atan(169/(16*((169*tan(c/2 + (d*x)/2))/16 + 13/2)) - (13*tan(c/2 + (d* x)/2))/(2*((169*tan(c/2 + (d*x)/2))/16 + 13/2))))/(4*a^3*d) + log(tan(c/2 + (d*x)/2))/(a^3*d) - ((11*tan(c/2 + (d*x)/2))/4 - 4*tan(c/2 + (d*x)/2)^2 + (19*tan(c/2 + (d*x)/2)^3)/4 - (19*tan(c/2 + (d*x)/2)^5)/4 + 4*tan(c/2 + (d*x)/2)^6 - (11*tan(c/2 + (d*x)/2)^7)/4)/(d*(4*a^3*tan(c/2 + (d*x)/2)^2 + 6*a^3*tan(c/2 + (d*x)/2)^4 + 4*a^3*tan(c/2 + (d*x)/2)^6 + a^3*tan(c/2 + ( d*x)/2)^8 + a^3))
\[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\int \frac {\cos \left (d x +c \right )^{7} \cot \left (d x +c \right )}{\left (\sin \left (d x +c \right ) a +a \right )^{3}}d x \] Input:
int(cos(d*x+c)^7*cot(d*x+c)/(a+a*sin(d*x+c))^3,x)
Output:
int(cos(d*x+c)^7*cot(d*x+c)/(a+a*sin(d*x+c))^3,x)