Integrand size = 29, antiderivative size = 104 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {3 x}{16 a^2}+\frac {\cos ^5(c+d x)}{10 a^2 d}+\frac {3 \cos (c+d x) \sin (c+d x)}{16 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{8 a^2 d}+\frac {\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a^5 d} \] Output:
3/16*x/a^2+1/10*cos(d*x+c)^5/a^2/d+3/16*cos(d*x+c)*sin(d*x+c)/a^2/d+1/8*co s(d*x+c)^3*sin(d*x+c)/a^2/d+1/6*cos(d*x+c)^3*(a-a*sin(d*x+c))^3/a^5/d
Leaf count is larger than twice the leaf count of optimal. \(362\) vs. \(2(104)=208\).
Time = 2.40 (sec) , antiderivative size = 362, normalized size of antiderivative = 3.48 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {360 d x \cos \left (\frac {c}{2}\right )+240 \cos \left (\frac {c}{2}+d x\right )+240 \cos \left (\frac {3 c}{2}+d x\right )-15 \cos \left (\frac {3 c}{2}+2 d x\right )+15 \cos \left (\frac {5 c}{2}+2 d x\right )+40 \cos \left (\frac {5 c}{2}+3 d x\right )+40 \cos \left (\frac {7 c}{2}+3 d x\right )-45 \cos \left (\frac {7 c}{2}+4 d x\right )+45 \cos \left (\frac {9 c}{2}+4 d x\right )-24 \cos \left (\frac {9 c}{2}+5 d x\right )-24 \cos \left (\frac {11 c}{2}+5 d x\right )+5 \cos \left (\frac {11 c}{2}+6 d x\right )-5 \cos \left (\frac {13 c}{2}+6 d x\right )+50 \sin \left (\frac {c}{2}\right )+360 d x \sin \left (\frac {c}{2}\right )-240 \sin \left (\frac {c}{2}+d x\right )+240 \sin \left (\frac {3 c}{2}+d x\right )-15 \sin \left (\frac {3 c}{2}+2 d x\right )-15 \sin \left (\frac {5 c}{2}+2 d x\right )-40 \sin \left (\frac {5 c}{2}+3 d x\right )+40 \sin \left (\frac {7 c}{2}+3 d x\right )-45 \sin \left (\frac {7 c}{2}+4 d x\right )-45 \sin \left (\frac {9 c}{2}+4 d x\right )+24 \sin \left (\frac {9 c}{2}+5 d x\right )-24 \sin \left (\frac {11 c}{2}+5 d x\right )+5 \sin \left (\frac {11 c}{2}+6 d x\right )+5 \sin \left (\frac {13 c}{2}+6 d x\right )}{1920 a^2 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \] Input:
Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]
Output:
(360*d*x*Cos[c/2] + 240*Cos[c/2 + d*x] + 240*Cos[(3*c)/2 + d*x] - 15*Cos[( 3*c)/2 + 2*d*x] + 15*Cos[(5*c)/2 + 2*d*x] + 40*Cos[(5*c)/2 + 3*d*x] + 40*C os[(7*c)/2 + 3*d*x] - 45*Cos[(7*c)/2 + 4*d*x] + 45*Cos[(9*c)/2 + 4*d*x] - 24*Cos[(9*c)/2 + 5*d*x] - 24*Cos[(11*c)/2 + 5*d*x] + 5*Cos[(11*c)/2 + 6*d* x] - 5*Cos[(13*c)/2 + 6*d*x] + 50*Sin[c/2] + 360*d*x*Sin[c/2] - 240*Sin[c/ 2 + d*x] + 240*Sin[(3*c)/2 + d*x] - 15*Sin[(3*c)/2 + 2*d*x] - 15*Sin[(5*c) /2 + 2*d*x] - 40*Sin[(5*c)/2 + 3*d*x] + 40*Sin[(7*c)/2 + 3*d*x] - 45*Sin[( 7*c)/2 + 4*d*x] - 45*Sin[(9*c)/2 + 4*d*x] + 24*Sin[(9*c)/2 + 5*d*x] - 24*S in[(11*c)/2 + 5*d*x] + 5*Sin[(11*c)/2 + 6*d*x] + 5*Sin[(13*c)/2 + 6*d*x])/ (1920*a^2*d*(Cos[c/2] + Sin[c/2]))
Time = 0.73 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {3042, 3354, 3042, 3349, 3042, 3148, 3042, 3115, 3042, 3115, 24}
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.
| \(\displaystyle \int \frac {\sin ^2(c+d x) \cos ^6(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^2 \cos (c+d x)^6}{(a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \cos ^2(c+d x) \sin ^2(c+d x) (a-a \sin (c+d x))^2dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \cos (c+d x)^2 \sin (c+d x)^2 (a-a \sin (c+d x))^2dx}{a^4}\) |
| \(\Big \downarrow \) 3349 |
| \(\displaystyle \frac {\frac {1}{2} a \int \cos ^4(c+d x) (a-a \sin (c+d x))dx+\frac {\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a d}}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} a \int \cos (c+d x)^4 (a-a \sin (c+d x))dx+\frac {\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a d}}{a^4}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {\frac {1}{2} a \left (a \int \cos ^4(c+d x)dx+\frac {a \cos ^5(c+d x)}{5 d}\right )+\frac {\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a d}}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} a \left (a \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {a \cos ^5(c+d x)}{5 d}\right )+\frac {\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a d}}{a^4}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {1}{2} a \left (a \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {a \cos ^5(c+d x)}{5 d}\right )+\frac {\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a d}}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} a \left (a \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {a \cos ^5(c+d x)}{5 d}\right )+\frac {\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a d}}{a^4}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {1}{2} a \left (a \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {a \cos ^5(c+d x)}{5 d}\right )+\frac {\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a d}}{a^4}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a d}+\frac {1}{2} a \left (\frac {a \cos ^5(c+d x)}{5 d}+a \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )}{a^4}\) |
Input:
Int[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]
Output:
((Cos[c + d*x]^3*(a - a*Sin[c + d*x])^3)/(6*a*d) + (a*((a*Cos[c + d*x]^5)/ (5*d) + a*((Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (3*(x/2 + (Cos[c + d*x]*S in[c + d*x])/(2*d)))/4)))/2)/a^4
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-(g*Cos[e + f*x])^( p + 1))*((a + b*Sin[e + f*x])^(m + 1)/(2*b*f*g*(m + 1))), x] + Simp[a/(2*g^ 2) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x], x] /; F reeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[m - p, 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 = 0.85 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.75
| method | result | size |
| parallelrisch | \(\frac {180 d x +5 \sin \left (6 d x +6 c \right )-45 \sin \left (4 d x +4 c \right )-15 \sin \left (2 d x +2 c \right )-24 \cos \left (5 d x +5 c \right )+40 \cos \left (3 d x +3 c \right )+240 \cos \left (d x +c \right )+256}{960 d \,a^{2}}\) | \(78\) |
| risch | \(\frac {3 x}{16 a^{2}}+\frac {\cos \left (d x +c \right )}{4 a^{2} d}+\frac {\sin \left (6 d x +6 c \right )}{192 d \,a^{2}}-\frac {\cos \left (5 d x +5 c \right )}{40 a^{2} d}-\frac {3 \sin \left (4 d x +4 c \right )}{64 d \,a^{2}}+\frac {\cos \left (3 d x +3 c \right )}{24 d \,a^{2}}-\frac {\sin \left (2 d x +2 c \right )}{64 d \,a^{2}}\) | \(107\) |
| derivativedivides | \(\frac {\frac {8 \left (\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{64}-\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{192}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-\frac {25 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{32}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3}+\frac {25 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{32}+\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{192}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{5}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}+\frac {1}{15}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}+\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d \,a^{2}}\) | \(153\) |
| default | \(\frac {\frac {8 \left (\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{64}-\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{192}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-\frac {25 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{32}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3}+\frac {25 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{32}+\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{192}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{5}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}+\frac {1}{15}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}+\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d \,a^{2}}\) | \(153\) |
Input:
int(cos(d*x+c)^6*sin(d*x+c)^2/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/960*(180*d*x+5*sin(6*d*x+6*c)-45*sin(4*d*x+4*c)-15*sin(2*d*x+2*c)-24*cos (5*d*x+5*c)+40*cos(3*d*x+3*c)+240*cos(d*x+c)+256)/d/a^2
Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.67 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {96 \, \cos \left (d x + c\right )^{5} - 160 \, \cos \left (d x + c\right )^{3} - 45 \, d x - 5 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 26 \, \cos \left (d x + c\right )^{3} + 9 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, a^{2} d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^2/(a+a*sin(d*x+c))^2,x, algorithm="frica s")
Output:
-1/240*(96*cos(d*x + c)^5 - 160*cos(d*x + c)^3 - 45*d*x - 5*(8*cos(d*x + c )^5 - 26*cos(d*x + c)^3 + 9*cos(d*x + c))*sin(d*x + c))/(a^2*d)
Leaf count of result is larger than twice the leaf count of optimal. 2271 vs. \(2 (94) = 188\).
Time = 57.89 (sec) , antiderivative size = 2271, normalized size of antiderivative = 21.84 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)**6*sin(d*x+c)**2/(a+a*sin(d*x+c))**2,x)
Output:
Piecewise((45*d*x*tan(c/2 + d*x/2)**12/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800* a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d *tan(c/2 + d*x/2)**2 + 240*a**2*d) + 270*d*x*tan(c/2 + d*x/2)**10/(240*a** 2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d* tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/ 2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 675*d*x*ta n(c/2 + d*x/2)**8/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/ 2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 900*d*x*tan(c/2 + d*x/2)**6/(240*a**2*d*tan(c/2 + d*x/2)** 12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a **2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 675*d*x*tan(c/2 + d*x/2)**4/(240 *a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a** 2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*ta n(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 270*d* x*tan(c/2 + d*x/2)**2/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c /2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x...
Leaf count of result is larger than twice the leaf count of optimal. 353 vs. \(2 (95) = 190\).
Time = 0.12 (sec) , antiderivative size = 353, normalized size of antiderivative = 3.39 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {\frac {45 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {384 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {65 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {750 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {640 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {750 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {960 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {65 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {45 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 64}{a^{2} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac {45 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{120 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^2/(a+a*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
-1/120*((45*sin(d*x + c)/(cos(d*x + c) + 1) - 384*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 65*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 750*sin(d*x + c)^5/ (cos(d*x + c) + 1)^5 - 640*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 750*sin(d *x + c)^7/(cos(d*x + c) + 1)^7 - 960*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 65*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 45*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 64)/(a^2 + 6*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 15*a^2*s in(d*x + c)^4/(cos(d*x + c) + 1)^4 + 20*a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 15*a^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 6*a^2*sin(d*x + c)^10 /(cos(d*x + c) + 1)^10 + a^2*sin(d*x + c)^12/(cos(d*x + c) + 1)^12) - 45*a rctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d
Time = 0.15 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.47 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {45 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 65 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 750 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 640 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 750 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 65 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 384 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 64\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a^{2}}}{240 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^2/(a+a*sin(d*x+c))^2,x, algorithm="giac" )
Output:
1/240*(45*(d*x + c)/a^2 + 2*(45*tan(1/2*d*x + 1/2*c)^11 - 65*tan(1/2*d*x + 1/2*c)^9 + 960*tan(1/2*d*x + 1/2*c)^8 - 750*tan(1/2*d*x + 1/2*c)^7 + 640* tan(1/2*d*x + 1/2*c)^6 + 750*tan(1/2*d*x + 1/2*c)^5 + 65*tan(1/2*d*x + 1/2 *c)^3 + 384*tan(1/2*d*x + 1/2*c)^2 - 45*tan(1/2*d*x + 1/2*c) + 64)/((tan(1 /2*d*x + 1/2*c)^2 + 1)^6*a^2))/d
Time = 35.49 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.40 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {3\,x}{16\,a^2}+\frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}-\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-\frac {25\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {25\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {8}{15}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \] Input:
int((cos(c + d*x)^6*sin(c + d*x)^2)/(a + a*sin(c + d*x))^2,x)
Output:
(3*x)/(16*a^2) + ((16*tan(c/2 + (d*x)/2)^2)/5 - (3*tan(c/2 + (d*x)/2))/8 + (13*tan(c/2 + (d*x)/2)^3)/24 + (25*tan(c/2 + (d*x)/2)^5)/4 + (16*tan(c/2 + (d*x)/2)^6)/3 - (25*tan(c/2 + (d*x)/2)^7)/4 + 8*tan(c/2 + (d*x)/2)^8 - ( 13*tan(c/2 + (d*x)/2)^9)/24 + (3*tan(c/2 + (d*x)/2)^11)/8 + 8/15)/(a^2*d*( tan(c/2 + (d*x)/2)^2 + 1)^6)
Time = 0.16 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.96 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-96 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+50 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+32 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-45 \cos \left (d x +c \right ) \sin \left (d x +c \right )+64 \cos \left (d x +c \right )+45 d x -64}{240 a^{2} d} \] Input:
int(cos(d*x+c)^6*sin(d*x+c)^2/(a+a*sin(d*x+c))^2,x)
Output:
(40*cos(c + d*x)*sin(c + d*x)**5 - 96*cos(c + d*x)*sin(c + d*x)**4 + 50*co s(c + d*x)*sin(c + d*x)**3 + 32*cos(c + d*x)*sin(c + d*x)**2 - 45*cos(c + d*x)*sin(c + d*x) + 64*cos(c + d*x) + 45*d*x - 64)/(240*a**2*d)