Integrand size = 29, antiderivative size = 92 \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {x}{2 a^3}+\frac {3 \text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {3 \cos (c+d x)}{a^3 d}+\frac {\cos ^3(c+d x)}{3 a^3 d}-\frac {\cot (c+d x)}{a^3 d}+\frac {3 \cos (c+d x) \sin (c+d x)}{2 a^3 d} \] Output:
1/2*x/a^3+3*arctanh(cos(d*x+c))/a^3/d-3*cos(d*x+c)/a^3/d+1/3*cos(d*x+c)^3/ a^3/d-cot(d*x+c)/a^3/d+3/2*cos(d*x+c)*sin(d*x+c)/a^3/d
Time = 2.56 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.37 \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6 \left (6 (c+d x)-33 \cos (c+d x)+\cos (3 (c+d x))-6 \cot \left (\frac {1}{2} (c+d x)\right )+36 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-36 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+9 \sin (2 (c+d x))+6 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{12 d (a+a \sin (c+d x))^3} \] Input:
Integrate[(Cos[c + d*x]^6*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]
Output:
((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(6*(c + d*x) - 33*Cos[c + d*x] + Cos[3*(c + d*x)] - 6*Cot[(c + d*x)/2] + 36*Log[Cos[(c + d*x)/2]] - 36*Log[ Sin[(c + d*x)/2]] + 9*Sin[2*(c + d*x)] + 6*Tan[(c + d*x)/2]))/(12*d*(a + a *Sin[c + d*x])^3)
Time = 0.51 (sec) , antiderivative size = 96, 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, 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.
| \(\displaystyle \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^8}{\sin (c+d x)^2 (a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \cot ^2(c+d x) (a-a \sin (c+d x))^3dx}{a^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a-a \sin (c+d x))^3}{\tan (c+d x)^2}dx}{a^6}\) |
| \(\Big \downarrow \) 3188 |
| \(\displaystyle \frac {\int \left (\sin ^3(c+d x) a^5+\csc ^2(c+d x) a^5-3 \sin ^2(c+d x) a^5-3 \csc (c+d x) a^5+2 \sin (c+d x) a^5+2 a^5\right )dx}{a^8}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3 a^5 \text {arctanh}(\cos (c+d x))}{d}+\frac {a^5 \cos ^3(c+d x)}{3 d}-\frac {3 a^5 \cos (c+d x)}{d}-\frac {a^5 \cot (c+d x)}{d}+\frac {3 a^5 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {a^5 x}{2}}{a^8}\) |
Input:
Int[(Cos[c + d*x]^6*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]
Output:
((a^5*x)/2 + (3*a^5*ArcTanh[Cos[c + d*x]])/d - (3*a^5*Cos[c + d*x])/d + (a ^5*Cos[c + d*x]^3)/(3*d) - (a^5*Cot[c + d*x])/d + (3*a^5*Cos[c + d*x]*Sin[ c + d*x])/(2*d))/a^8
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])
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 = 3.67 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.36
| method | result | size |
| derivativedivides | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {32}{3}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{3}}\) | \(125\) |
| default | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {32}{3}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{3}}\) | \(125\) |
| risch | \(\frac {x}{2 a^{3}}-\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )}}{8 d \,a^{3}}-\frac {11 \,{\mathrm e}^{i \left (d x +c \right )}}{8 d \,a^{3}}-\frac {11 \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d \,a^{3}}+\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d \,a^{3}}-\frac {2 i}{a^{3} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{3}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{3}}+\frac {\cos \left (3 d x +3 c \right )}{12 a^{3} d}\) | \(157\) |
Input:
int(cos(d*x+c)^6*cot(d*x+c)^2/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/2/d/a^3*(tan(1/2*d*x+1/2*c)+8*(-3/4*tan(1/2*d*x+1/2*c)^5-tan(1/2*d*x+1/2 *c)^4-3*tan(1/2*d*x+1/2*c)^2+3/4*tan(1/2*d*x+1/2*c)-4/3)/(1+tan(1/2*d*x+1/ 2*c)^2)^3+2*arctan(tan(1/2*d*x+1/2*c))-1/tan(1/2*d*x+1/2*c)-6*ln(tan(1/2*d *x+1/2*c)))
Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.13 \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {9 \, \cos \left (d x + c\right )^{3} - {\left (2 \, \cos \left (d x + c\right )^{3} + 3 \, d x - 18 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 9 \, \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 9 \, \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 3 \, \cos \left (d x + c\right )}{6 \, a^{3} d \sin \left (d x + c\right )} \] Input:
integrate(cos(d*x+c)^6*cot(d*x+c)^2/(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
-1/6*(9*cos(d*x + c)^3 - (2*cos(d*x + c)^3 + 3*d*x - 18*cos(d*x + c))*sin( d*x + c) - 9*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 9*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 3*cos(d*x + c))/(a^3*d*sin(d*x + c))
\[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\cos ^{6}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(cos(d*x+c)**6*cot(d*x+c)**2/(a+a*sin(d*x+c))**3,x)
Output:
Integral(cos(c + d*x)**6*cot(c + d*x)**2/(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. 285 vs. \(2 (86) = 172\).
Time = 0.11 (sec) , antiderivative size = 285, normalized size of antiderivative = 3.10 \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {\frac {32 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {9 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {72 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {9 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {24 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {21 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 3}{\frac {a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}} - \frac {6 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac {18 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {3 \, \sin \left (d x + c\right )}{a^{3} {\left (\cos \left (d x + c\right ) + 1\right )}}}{6 \, d} \] Input:
integrate(cos(d*x+c)^6*cot(d*x+c)^2/(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
-1/6*((32*sin(d*x + c)/(cos(d*x + c) + 1) - 9*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 72*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 9*sin(d*x + c)^4/(cos(d *x + c) + 1)^4 + 24*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 21*sin(d*x + c)^ 6/(cos(d*x + c) + 1)^6 + 3)/(a^3*sin(d*x + c)/(cos(d*x + c) + 1) + 3*a^3*s in(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*a^3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + a^3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7) - 6*arctan(sin(d*x + c)/(c os(d*x + c) + 1))/a^3 + 18*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^3 - 3*si n(d*x + c)/(a^3*(cos(d*x + c) + 1)))/d
Time = 0.19 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.60 \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {3 \, {\left (d x + c\right )}}{a^{3}} - \frac {18 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac {3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{3}} + \frac {3 \, {\left (6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - \frac {2 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 36 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 16\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{3}}}{6 \, d} \] Input:
integrate(cos(d*x+c)^6*cot(d*x+c)^2/(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/6*(3*(d*x + c)/a^3 - 18*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 + 3*tan(1/2*d *x + 1/2*c)/a^3 + 3*(6*tan(1/2*d*x + 1/2*c) - 1)/(a^3*tan(1/2*d*x + 1/2*c) ) - 2*(9*tan(1/2*d*x + 1/2*c)^5 + 12*tan(1/2*d*x + 1/2*c)^4 + 36*tan(1/2*d *x + 1/2*c)^2 - 9*tan(1/2*d*x + 1/2*c) + 16)/((tan(1/2*d*x + 1/2*c)^2 + 1) ^3*a^3))/d
Time = 31.37 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.51 \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d}-\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {32\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+1}{d\,\left (2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+6\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+6\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {\mathrm {atan}\left (\frac {1}{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+6}-\frac {6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+6}\right )}{a^3\,d} \] Input:
int((cos(c + d*x)^6*cot(c + d*x)^2)/(a + a*sin(c + d*x))^3,x)
Output:
tan(c/2 + (d*x)/2)/(2*a^3*d) - ((32*tan(c/2 + (d*x)/2))/3 - 3*tan(c/2 + (d *x)/2)^2 + 24*tan(c/2 + (d*x)/2)^3 + 3*tan(c/2 + (d*x)/2)^4 + 8*tan(c/2 + (d*x)/2)^5 + 7*tan(c/2 + (d*x)/2)^6 + 1)/(d*(6*a^3*tan(c/2 + (d*x)/2)^3 + 6*a^3*tan(c/2 + (d*x)/2)^5 + 2*a^3*tan(c/2 + (d*x)/2)^7 + 2*a^3*tan(c/2 + (d*x)/2))) - (3*log(tan(c/2 + (d*x)/2)))/(a^3*d) - atan(1/(tan(c/2 + (d*x) /2) + 6) - (6*tan(c/2 + (d*x)/2))/(tan(c/2 + (d*x)/2) + 6))/(a^3*d)
Time = 0.17 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.16 \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-2 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+9 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-16 \cos \left (d x +c \right ) \sin \left (d x +c \right )-6 \cos \left (d x +c \right )-18 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )+3 \sin \left (d x +c \right ) d x +16 \sin \left (d x +c \right )}{6 \sin \left (d x +c \right ) a^{3} d} \] Input:
int(cos(d*x+c)^6*cot(d*x+c)^2/(a+a*sin(d*x+c))^3,x)
Output:
( - 2*cos(c + d*x)*sin(c + d*x)**3 + 9*cos(c + d*x)*sin(c + d*x)**2 - 16*c os(c + d*x)*sin(c + d*x) - 6*cos(c + d*x) - 18*log(tan((c + d*x)/2))*sin(c + d*x) + 3*sin(c + d*x)*d*x + 16*sin(c + d*x))/(6*sin(c + d*x)*a**3*d)