Integrand size = 29, antiderivative size = 98 \[ \int \frac {\cos ^5(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {5 x}{2 a^3}-\frac {5 \text {arctanh}(\cos (c+d x))}{2 a^3 d}+\frac {3 \cos (c+d x)}{a^3 d}+\frac {3 \cot (c+d x)}{a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {\cos (c+d x) \sin (c+d x)}{2 a^3 d} \] Output:
5/2*x/a^3-5/2*arctanh(cos(d*x+c))/a^3/d+3*cos(d*x+c)/a^3/d+3*cot(d*x+c)/a^ 3/d-1/2*cot(d*x+c)*csc(d*x+c)/a^3/d-1/2*cos(d*x+c)*sin(d*x+c)/a^3/d
Time = 2.33 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.47 \[ \int \frac {\cos ^5(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6 \left (20 (c+d x)+24 \cos (c+d x)+12 \cot \left (\frac {1}{2} (c+d x)\right )-\csc ^2\left (\frac {1}{2} (c+d x)\right )-20 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+20 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\sec ^2\left (\frac {1}{2} (c+d x)\right )-2 \sin (2 (c+d x))-12 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{8 d (a+a \sin (c+d x))^3} \] Input:
Integrate[(Cos[c + d*x]^5*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]
Output:
((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(20*(c + d*x) + 24*Cos[c + d*x] + 12*Cot[(c + d*x)/2] - Csc[(c + d*x)/2]^2 - 20*Log[Cos[(c + d*x)/2]] + 20* Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^2 - 2*Sin[2*(c + d*x)] - 12*Tan[( c + d*x)/2]))/(8*d*(a + a*Sin[c + d*x])^3)
Time = 0.52 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3354, 3042, 3351, 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.
| \(\displaystyle \int \frac {\cos ^5(c+d x) \cot ^3(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^8}{\sin (c+d x)^3 (a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \cot ^2(c+d x) \csc (c+d x) (a-a \sin (c+d x))^3dx}{a^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\cos (c+d x)^2 (a-a \sin (c+d x))^3}{\sin (c+d x)^3}dx}{a^6}\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle \frac {\int \left (\csc ^3(c+d x) a^5-3 \csc ^2(c+d x) a^5+\sin ^2(c+d x) a^5+2 \csc (c+d x) a^5-3 \sin (c+d x) a^5+2 a^5\right )dx}{a^8}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {5 a^5 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {3 a^5 \cos (c+d x)}{d}+\frac {3 a^5 \cot (c+d x)}{d}-\frac {a^5 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {a^5 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {5 a^5 x}{2}}{a^8}\) |
Input:
Int[(Cos[c + d*x]^5*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]
Output:
((5*a^5*x)/2 - (5*a^5*ArcTanh[Cos[c + d*x]])/(2*d) + (3*a^5*Cos[c + d*x])/ d + (3*a^5*Cot[c + d*x])/d - (a^5*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (a^5* Cos[c + d*x]*Sin[c + d*x])/(2*d))/a^8
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/a^p Int[Expan dTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x])^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && In tegersQ[m, n, p/2] && ((GtQ[m, 0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (G tQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 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 = 6.04 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.43
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+24}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+20 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+10 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(140\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+24}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+20 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+10 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(140\) |
| risch | \(\frac {5 x}{2 a^{3}}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 d \,a^{3}}+\frac {3 \,{\mathrm e}^{i \left (d x +c \right )}}{2 d \,a^{3}}+\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d \,a^{3}}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d \,a^{3}}+\frac {{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}+6 i {\mathrm e}^{2 i \left (d x +c \right )}-6 i}{d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \,a^{3}}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \,a^{3}}\) | \(171\) |
Input:
int(cos(d*x+c)^5*cot(d*x+c)^3/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/4/d/a^3*(1/2*tan(1/2*d*x+1/2*c)^2-6*tan(1/2*d*x+1/2*c)+16*(1/4*tan(1/2*d *x+1/2*c)^3+3/2*tan(1/2*d*x+1/2*c)^2-1/4*tan(1/2*d*x+1/2*c)+3/2)/(1+tan(1/ 2*d*x+1/2*c)^2)^2+20*arctan(tan(1/2*d*x+1/2*c))-1/2/tan(1/2*d*x+1/2*c)^2+6 /tan(1/2*d*x+1/2*c)+10*ln(tan(1/2*d*x+1/2*c)))
Time = 0.09 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.33 \[ \int \frac {\cos ^5(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {10 \, d x \cos \left (d x + c\right )^{2} + 12 \, \cos \left (d x + c\right )^{3} - 10 \, d x - 5 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 5 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (\cos \left (d x + c\right )^{3} + 5 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 10 \, \cos \left (d x + c\right )}{4 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )}} \] Input:
integrate(cos(d*x+c)^5*cot(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
1/4*(10*d*x*cos(d*x + c)^2 + 12*cos(d*x + c)^3 - 10*d*x - 5*(cos(d*x + c)^ 2 - 1)*log(1/2*cos(d*x + c) + 1/2) + 5*(cos(d*x + c)^2 - 1)*log(-1/2*cos(d *x + c) + 1/2) - 2*(cos(d*x + c)^3 + 5*cos(d*x + c))*sin(d*x + c) - 10*cos (d*x + c))/(a^3*d*cos(d*x + c)^2 - a^3*d)
Timed out. \[ \int \frac {\cos ^5(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**5*cot(d*x+c)**3/(a+a*sin(d*x+c))**3,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (90) = 180\).
Time = 0.11 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.72 \[ \int \frac {\cos ^5(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {\frac {12 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {46 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {47 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 1}{\frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {\frac {12 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{a^{3}} + \frac {40 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac {20 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{8 \, d} \] Input:
integrate(cos(d*x+c)^5*cot(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
1/8*((12*sin(d*x + c)/(cos(d*x + c) + 1) + 46*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 16*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 47*sin(d*x + c)^4/(cos( d*x + c) + 1)^4 + 20*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1)/(a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 2*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) - (12*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d*x + c)^2/(cos(d*x + c) + 1)^2)/a^3 + 40*arctan(sin(d*x + c)/ (cos(d*x + c) + 1))/a^3 + 20*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d
Time = 0.22 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.76 \[ \int \frac {\cos ^5(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {20 \, {\left (d x + c\right )}}{a^{3}} + \frac {20 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 20 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 16 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{2} a^{3}} + \frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{8 \, d} \] Input:
integrate(cos(d*x+c)^5*cot(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/8*(20*(d*x + c)/a^3 + 20*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - (10*tan(1/ 2*d*x + 1/2*c)^6 - 20*tan(1/2*d*x + 1/2*c)^5 - 27*tan(1/2*d*x + 1/2*c)^4 - 16*tan(1/2*d*x + 1/2*c)^3 - 36*tan(1/2*d*x + 1/2*c)^2 - 12*tan(1/2*d*x + 1/2*c) + 1)/((tan(1/2*d*x + 1/2*c)^3 + tan(1/2*d*x + 1/2*c))^2*a^3) + (a^3 *tan(1/2*d*x + 1/2*c)^2 - 12*a^3*tan(1/2*d*x + 1/2*c))/a^6)/d
Time = 31.53 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.33 \[ \int \frac {\cos ^5(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^3\,d}-\frac {5\,\mathrm {atan}\left (\frac {25\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{25\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-25}+\frac {25}{25\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-25}\right )}{a^3\,d}+\frac {5\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a^3\,d}+\frac {10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {47\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+23\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{2}}{d\,\left (4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+8\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d} \] Input:
int((cos(c + d*x)^5*cot(c + d*x)^3)/(a + a*sin(c + d*x))^3,x)
Output:
tan(c/2 + (d*x)/2)^2/(8*a^3*d) - (5*atan((25*tan(c/2 + (d*x)/2))/(25*tan(c /2 + (d*x)/2) - 25) + 25/(25*tan(c/2 + (d*x)/2) - 25)))/(a^3*d) + (5*log(t an(c/2 + (d*x)/2)))/(2*a^3*d) + (6*tan(c/2 + (d*x)/2) + 23*tan(c/2 + (d*x) /2)^2 + 8*tan(c/2 + (d*x)/2)^3 + (47*tan(c/2 + (d*x)/2)^4)/2 + 10*tan(c/2 + (d*x)/2)^5 - 1/2)/(d*(4*a^3*tan(c/2 + (d*x)/2)^2 + 8*a^3*tan(c/2 + (d*x) /2)^4 + 4*a^3*tan(c/2 + (d*x)/2)^6)) - (3*tan(c/2 + (d*x)/2))/(2*a^3*d)
Time = 0.17 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.15 \[ \int \frac {\cos ^5(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-2 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+12 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+12 \cos \left (d x +c \right ) \sin \left (d x +c \right )-2 \cos \left (d x +c \right )+10 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2}+10 \sin \left (d x +c \right )^{2} d x -11 \sin \left (d x +c \right )^{2}}{4 \sin \left (d x +c \right )^{2} a^{3} d} \] Input:
int(cos(d*x+c)^5*cot(d*x+c)^3/(a+a*sin(d*x+c))^3,x)
Output:
( - 2*cos(c + d*x)*sin(c + d*x)**3 + 12*cos(c + d*x)*sin(c + d*x)**2 + 12* cos(c + d*x)*sin(c + d*x) - 2*cos(c + d*x) + 10*log(tan((c + d*x)/2))*sin( c + d*x)**2 + 10*sin(c + d*x)**2*d*x - 11*sin(c + d*x)**2)/(4*sin(c + d*x) **2*a**3*d)