Integrand size = 29, antiderivative size = 116 \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {9 x}{8 a^2}+\frac {2 \text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {2 \cos (c+d x)}{a^2 d}-\frac {2 \cos ^3(c+d x)}{3 a^2 d}-\frac {\cot (c+d x)}{a^2 d}+\frac {\cos (c+d x) \sin (c+d x)}{8 a^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a^2 d} \] Output:
-9/8*x/a^2+2*arctanh(cos(d*x+c))/a^2/d-2*cos(d*x+c)/a^2/d-2/3*cos(d*x+c)^3 /a^2/d-cot(d*x+c)/a^2/d+1/8*cos(d*x+c)*sin(d*x+c)/a^2/d-1/4*cos(d*x+c)*sin (d*x+c)^3/a^2/d
Time = 3.58 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.10 \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 \left (-108 (c+d x)-240 \cos (c+d x)-16 \cos (3 (c+d x))-48 \cot \left (\frac {1}{2} (c+d x)\right )+192 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-192 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+3 \sin (4 (c+d x))+48 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{96 d (a+a \sin (c+d x))^2} \] Input:
Integrate[(Cos[c + d*x]^6*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]
Output:
((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(-108*(c + d*x) - 240*Cos[c + d*x ] - 16*Cos[3*(c + d*x)] - 48*Cot[(c + d*x)/2] + 192*Log[Cos[(c + d*x)/2]] - 192*Log[Sin[(c + d*x)/2]] + 3*Sin[4*(c + d*x)] + 48*Tan[(c + d*x)/2]))/( 96*d*(a + a*Sin[c + d*x])^2)
Time = 0.61 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03, 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.
| \(\displaystyle \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^8}{\sin (c+d x)^2 (a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \cos ^2(c+d x) \cot ^2(c+d x) (a-a \sin (c+d x))^2dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\cos (c+d x)^4 (a-a \sin (c+d x))^2}{\sin (c+d x)^2}dx}{a^4}\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle \frac {\int \left (\sin ^4(c+d x) a^6-2 \sin ^3(c+d x) a^6+\csc ^2(c+d x) a^6-\sin ^2(c+d x) a^6-2 \csc (c+d x) a^6+4 \sin (c+d x) a^6-a^6\right )dx}{a^8}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2 a^6 \text {arctanh}(\cos (c+d x))}{d}-\frac {2 a^6 \cos ^3(c+d x)}{3 d}-\frac {2 a^6 \cos (c+d x)}{d}-\frac {a^6 \cot (c+d x)}{d}-\frac {a^6 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac {a^6 \sin (c+d x) \cos (c+d x)}{8 d}-\frac {9 a^6 x}{8}}{a^8}\) |
Input:
Int[(Cos[c + d*x]^6*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]
Output:
((-9*a^6*x)/8 + (2*a^6*ArcTanh[Cos[c + d*x]])/d - (2*a^6*Cos[c + d*x])/d - (2*a^6*Cos[c + d*x]^3)/(3*d) - (a^6*Cot[c + d*x])/d + (a^6*Cos[c + d*x]*S in[c + d*x])/(8*d) - (a^6*Cos[c + d*x]*Sin[c + d*x]^3)/(4*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]
Result contains complex when optimal does not.
Time = 5.06 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.19
| method | result | size |
| risch | \(-\frac {9 x}{8 a^{2}}-\frac {5 \,{\mathrm e}^{i \left (d x +c \right )}}{4 d \,a^{2}}-\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )}}{4 d \,a^{2}}-\frac {2 i}{a^{2} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{2}}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{2}}+\frac {\sin \left (4 d x +4 c \right )}{32 d \,a^{2}}-\frac {\cos \left (3 d x +3 c \right )}{6 d \,a^{2}}\) | \(138\) |
| derivativedivides | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {4 \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8}+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8}+8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8}+\frac {20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}+\frac {8}{3}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {9 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}}{2 d \,a^{2}}\) | \(164\) |
| default | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {4 \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8}+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8}+8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8}+\frac {20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}+\frac {8}{3}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {9 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}}{2 d \,a^{2}}\) | \(164\) |
Input:
int(cos(d*x+c)^6*cot(d*x+c)^2/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-9/8*x/a^2-5/4/d/a^2*exp(I*(d*x+c))-5/4/d/a^2*exp(-I*(d*x+c))-2*I/a^2/d/(e xp(2*I*(d*x+c))-1)-2/d/a^2*ln(exp(I*(d*x+c))-1)+2/d/a^2*ln(exp(I*(d*x+c))+ 1)+1/32/d/a^2*sin(4*d*x+4*c)-1/6/d/a^2*cos(3*d*x+3*c)
Time = 0.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.97 \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {6 \, \cos \left (d x + c\right )^{5} - 9 \, \cos \left (d x + c\right )^{3} + {\left (16 \, \cos \left (d x + c\right )^{3} + 27 \, d x + 48 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 24 \, \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 24 \, \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 27 \, \cos \left (d x + c\right )}{24 \, a^{2} d \sin \left (d x + c\right )} \] Input:
integrate(cos(d*x+c)^6*cot(d*x+c)^2/(a+a*sin(d*x+c))^2,x, algorithm="frica s")
Output:
-1/24*(6*cos(d*x + c)^5 - 9*cos(d*x + c)^3 + (16*cos(d*x + c)^3 + 27*d*x + 48*cos(d*x + c))*sin(d*x + c) - 24*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 24*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 27*cos(d*x + c))/(a^2* d*sin(d*x + c))
\[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\int \frac {\cos ^{6}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \] Input:
integrate(cos(d*x+c)**6*cot(d*x+c)**2/(a+a*sin(d*x+c))**2,x)
Output:
Integral(cos(c + d*x)**6*cot(c + d*x)**2/(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. 348 vs. \(2 (108) = 216\).
Time = 0.12 (sec) , antiderivative size = 348, normalized size of antiderivative = 3.00 \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {\frac {64 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {21 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {160 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {57 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {192 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {96 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {9 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 6}{\frac {a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {4 \, a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {4 \, a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {a^{2} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}} + \frac {27 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {24 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac {6 \, \sin \left (d x + c\right )}{a^{2} {\left (\cos \left (d x + c\right ) + 1\right )}}}{12 \, d} \] Input:
integrate(cos(d*x+c)^6*cot(d*x+c)^2/(a+a*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
-1/12*((64*sin(d*x + c)/(cos(d*x + c) + 1) + 21*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 160*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 57*sin(d*x + c)^4/(c os(d*x + c) + 1)^4 + 192*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 96*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 9*si n(d*x + c)^8/(cos(d*x + c) + 1)^8 + 6)/(a^2*sin(d*x + c)/(cos(d*x + c) + 1 ) + 4*a^2*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 6*a^2*sin(d*x + c)^5/(cos( d*x + c) + 1)^5 + 4*a^2*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + a^2*sin(d*x + c)^9/(cos(d*x + c) + 1)^9) + 27*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/ a^2 + 24*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 - 6*sin(d*x + c)/(a^2*(c os(d*x + c) + 1)))/d
Time = 0.20 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.60 \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {27 \, {\left (d x + c\right )}}{a^{2}} + \frac {48 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{2}} - \frac {12 \, {\left (4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 96 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 192 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 160 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 64\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a^{2}}}{24 \, d} \] Input:
integrate(cos(d*x+c)^6*cot(d*x+c)^2/(a+a*sin(d*x+c))^2,x, algorithm="giac" )
Output:
-1/24*(27*(d*x + c)/a^2 + 48*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - 12*tan(1 /2*d*x + 1/2*c)/a^2 - 12*(4*tan(1/2*d*x + 1/2*c) - 1)/(a^2*tan(1/2*d*x + 1 /2*c)) + 2*(3*tan(1/2*d*x + 1/2*c)^7 + 96*tan(1/2*d*x + 1/2*c)^6 - 21*tan( 1/2*d*x + 1/2*c)^5 + 192*tan(1/2*d*x + 1/2*c)^4 + 21*tan(1/2*d*x + 1/2*c)^ 3 + 160*tan(1/2*d*x + 1/2*c)^2 - 3*tan(1/2*d*x + 1/2*c) + 64)/((tan(1/2*d* x + 1/2*c)^2 + 1)^4*a^2))/d
Time = 31.07 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.41 \[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {9\,\mathrm {atan}\left (\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\frac {81\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}-9}+\frac {81}{16\,\left (\frac {81\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}-9\right )}\right )}{4\,a^2\,d}-\frac {2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}-\frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2}+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2}+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {19\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+\frac {80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+\frac {32\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+1}{d\,\left (2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+8\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+12\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^2\,d} \] Input:
int((cos(c + d*x)^6*cot(c + d*x)^2)/(a + a*sin(c + d*x))^2,x)
Output:
(9*atan((9*tan(c/2 + (d*x)/2))/((81*tan(c/2 + (d*x)/2))/16 - 9) + 81/(16*( (81*tan(c/2 + (d*x)/2))/16 - 9))))/(4*a^2*d) - (2*log(tan(c/2 + (d*x)/2))) /(a^2*d) - ((32*tan(c/2 + (d*x)/2))/3 + (7*tan(c/2 + (d*x)/2)^2)/2 + (80*t an(c/2 + (d*x)/2)^3)/3 + (19*tan(c/2 + (d*x)/2)^4)/2 + 32*tan(c/2 + (d*x)/ 2)^5 + tan(c/2 + (d*x)/2)^6/2 + 16*tan(c/2 + (d*x)/2)^7 + (3*tan(c/2 + (d* x)/2)^8)/2 + 1)/(d*(8*a^2*tan(c/2 + (d*x)/2)^3 + 12*a^2*tan(c/2 + (d*x)/2) ^5 + 8*a^2*tan(c/2 + (d*x)/2)^7 + 2*a^2*tan(c/2 + (d*x)/2)^9 + 2*a^2*tan(c /2 + (d*x)/2))) + tan(c/2 + (d*x)/2)/(2*a^2*d)
\[ \int \frac {\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\int \frac {\cos \left (d x +c \right )^{6} \cot \left (d x +c \right )^{2}}{\left (\sin \left (d x +c \right ) a +a \right )^{2}}d x \] Input:
int(cos(d*x+c)^6*cot(d*x+c)^2/(a+a*sin(d*x+c))^2,x)
Output:
int(cos(d*x+c)^6*cot(d*x+c)^2/(a+a*sin(d*x+c))^2,x)