Integrand size = 29, antiderivative size = 115 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {x}{16 a}+\frac {\cos ^5(c+d x)}{5 a d}-\frac {\cos ^7(c+d x)}{7 a d}+\frac {\cos (c+d x) \sin (c+d x)}{16 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{24 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d} \] Output:
1/16*x/a+1/5*cos(d*x+c)^5/a/d-1/7*cos(d*x+c)^7/a/d+1/16*cos(d*x+c)*sin(d*x +c)/a/d+1/24*cos(d*x+c)^3*sin(d*x+c)/a/d-1/6*cos(d*x+c)^5*sin(d*x+c)/a/d
Leaf count is larger than twice the leaf count of optimal. \(715\) vs. \(2(115)=230\).
Time = 12.26 (sec) , antiderivative size = 715, normalized size of antiderivative = 6.22 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]
Output:
-1/160*(30*x + (50*Cos[c]*Cos[d*x])/d - (10*Cos[3*c]*Cos[3*d*x])/d + (2*Co s[5*c]*Cos[5*d*x])/d - (20*Cos[2*d*x]*Sin[2*c])/d + (5*Cos[4*d*x]*Sin[4*c] )/d - (50*Sin[c]*Sin[d*x])/d - (20*Cos[2*c]*Sin[2*d*x])/d + (10*Sin[3*c]*S in[3*d*x])/d + (5*Cos[4*c]*Sin[4*d*x])/d - (2*Sin[5*c]*Sin[5*d*x])/d - (10 *Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/ 2])))/a + (x + (Cos[c]*Cos[d*x])/d - (Sin[c]*Sin[d*x])/d - Sin[(d*x)/2]/(d *(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(16*a) - (- 6*x - (9*Cos[c]*Cos[d*x])/d + (Cos[3*c]*Cos[3*d*x])/d + (3*Cos[2*d*x]*Sin[ 2*c])/d + (9*Sin[c]*Sin[d*x])/d + (3*Cos[2*c]*Sin[2*d*x])/d - (Sin[3*c]*Si n[3*d*x])/d + (3*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(48*a) - (-420*x - (735*Cos[c]*Cos[d*x])/d + (175*Co s[3*c]*Cos[3*d*x])/d - (63*Cos[5*c]*Cos[5*d*x])/d + (15*Cos[7*c]*Cos[7*d*x ])/d + (315*Cos[2*d*x]*Sin[2*c])/d - (105*Cos[4*d*x]*Sin[4*c])/d + (35*Cos [6*d*x]*Sin[6*c])/d + (735*Sin[c]*Sin[d*x])/d + (315*Cos[2*c]*Sin[2*d*x])/ d - (175*Sin[3*c]*Sin[3*d*x])/d - (105*Cos[4*c]*Sin[4*d*x])/d + (63*Sin[5* c]*Sin[5*d*x])/d + (35*Cos[6*c]*Sin[6*d*x])/d - (15*Sin[7*c]*Sin[7*d*x])/d + (105*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(6720*a) + (5*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d *x)/2]))/(64*d*(a + a*Sin[c + d*x]))
Time = 0.62 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {3042, 3318, 3042, 3045, 244, 2009, 3048, 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.
| \(\displaystyle \int \frac {\sin ^2(c+d x) \cos ^6(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^2 \cos (c+d x)^6}{a \sin (c+d x)+a}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \cos ^4(c+d x) \sin ^2(c+d x)dx}{a}-\frac {\int \cos ^4(c+d x) \sin ^3(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \cos (c+d x)^4 \sin (c+d x)^2dx}{a}-\frac {\int \cos (c+d x)^4 \sin (c+d x)^3dx}{a}\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle \frac {\int \cos ^4(c+d x) \left (1-\cos ^2(c+d x)\right )d\cos (c+d x)}{a d}+\frac {\int \cos (c+d x)^4 \sin (c+d x)^2dx}{a}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {\int \left (\cos ^4(c+d x)-\cos ^6(c+d x)\right )d\cos (c+d x)}{a d}+\frac {\int \cos (c+d x)^4 \sin (c+d x)^2dx}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\int \cos (c+d x)^4 \sin (c+d x)^2dx}{a}+\frac {\frac {1}{5} \cos ^5(c+d x)-\frac {1}{7} \cos ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {\frac {1}{6} \int \cos ^4(c+d x)dx-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}}{a}+\frac {\frac {1}{5} \cos ^5(c+d x)-\frac {1}{7} \cos ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}}{a}+\frac {\frac {1}{5} \cos ^5(c+d x)-\frac {1}{7} \cos ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}}{a}+\frac {\frac {1}{5} \cos ^5(c+d x)-\frac {1}{7} \cos ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}}{a}+\frac {\frac {1}{5} \cos ^5(c+d x)-\frac {1}{7} \cos ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}}{a}+\frac {\frac {1}{5} \cos ^5(c+d x)-\frac {1}{7} \cos ^7(c+d x)}{a d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{5} \cos ^5(c+d x)-\frac {1}{7} \cos ^7(c+d x)}{a d}+\frac {\frac {1}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}}{a}\) |
Input:
Int[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]
Output:
(Cos[c + d*x]^5/5 - Cos[c + d*x]^7/7)/(a*d) + (-1/6*(Cos[c + d*x]^5*Sin[c + d*x])/d + ((Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (3*(x/2 + (Cos[c + d*x] *Sin[c + d*x])/(2*d)))/4)/6)/a
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> Simp[-(a*f)^(-1) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
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_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d) Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
Time = 1.01 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.77
| method | result | size |
| parallelrisch | \(\frac {420 d x -21 \cos \left (5 d x +5 c \right )+105 \cos \left (3 d x +3 c \right )+315 \cos \left (d x +c \right )-15 \cos \left (7 d x +7 c \right )-35 \sin \left (6 d x +6 c \right )-105 \sin \left (4 d x +4 c \right )+105 \sin \left (2 d x +2 c \right )+384}{6720 d a}\) | \(89\) |
| risch | \(\frac {x}{16 a}+\frac {3 \cos \left (d x +c \right )}{64 a d}-\frac {\cos \left (7 d x +7 c \right )}{448 a d}-\frac {\sin \left (6 d x +6 c \right )}{192 d a}-\frac {\cos \left (5 d x +5 c \right )}{320 a d}-\frac {\sin \left (4 d x +4 c \right )}{64 d a}+\frac {\cos \left (3 d x +3 c \right )}{64 a d}+\frac {\sin \left (2 d x +2 c \right )}{64 d a}\) | \(124\) |
| derivativedivides | \(\frac {\frac {8 \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{64}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{48}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}+\frac {31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{192}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\frac {31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{192}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{48}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{10}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}+\frac {1}{70}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}+\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d a}\) | \(179\) |
| default | \(\frac {\frac {8 \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{64}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{48}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}+\frac {31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{192}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\frac {31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{192}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{48}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{10}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}+\frac {1}{70}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}+\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d a}\) | \(179\) |
| norman | \(\frac {-\frac {3}{280 a d}+\frac {x}{16 a}+\frac {1363 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{840 d a}-\frac {145 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{24 d a}-\frac {451 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{120 d a}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{120 d a}-\frac {311 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{120 d a}-\frac {227 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{120 d a}-\frac {59 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{280 d a}-\frac {83 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d a}-\frac {137 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{24 d a}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{24 d a}+\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a}+\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a}+\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}+\frac {7 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 a}+\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}}{16 a}-\frac {181 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{24 d a}-\frac {29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{24 d a}-\frac {65 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{24 d a}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{8 d a}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}}{8 d a}-\frac {19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{140 d a}+\frac {7 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 a}+\frac {7 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2 a}+\frac {7 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{2 a}+\frac {35 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{8 a}+\frac {7 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2 a}+\frac {7 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{2 a}+\frac {7 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{4 a}+\frac {7 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{4 a}+\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{2 a}+\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{2 a}+\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{16 a}+\frac {35 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8 a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\) | \(634\) |
Input:
int(cos(d*x+c)^6*sin(d*x+c)^2/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/6720*(420*d*x-21*cos(5*d*x+5*c)+105*cos(3*d*x+3*c)+315*cos(d*x+c)-15*cos (7*d*x+7*c)-35*sin(6*d*x+6*c)-105*sin(4*d*x+4*c)+105*sin(2*d*x+2*c)+384)/d /a
Time = 0.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.61 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {240 \, \cos \left (d x + c\right )^{7} - 336 \, \cos \left (d x + c\right )^{5} - 105 \, d x + 35 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, a d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
-1/1680*(240*cos(d*x + c)^7 - 336*cos(d*x + c)^5 - 105*d*x + 35*(8*cos(d*x + c)^5 - 2*cos(d*x + c)^3 - 3*cos(d*x + c))*sin(d*x + c))/(a*d)
Leaf count of result is larger than twice the leaf count of optimal. 2773 vs. \(2 (90) = 180\).
Time = 34.41 (sec) , antiderivative size = 2773, normalized size of antiderivative = 24.11 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)**6*sin(d*x+c)**2/(a+a*sin(d*x+c)),x)
Output:
Piecewise((105*d*x*tan(c/2 + d*x/2)**14/(1680*a*d*tan(c/2 + d*x/2)**14 + 1 1760*a*d*tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/2 + d*x/2)**10 + 58800*a*d *tan(c/2 + d*x/2)**8 + 58800*a*d*tan(c/2 + d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4 + 11760*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) + 735*d*x*tan(c/2 + d*x/2)**12/(1680*a*d*tan(c/2 + d*x/2)**14 + 11760*a*d*tan(c/2 + d*x/2)**1 2 + 35280*a*d*tan(c/2 + d*x/2)**10 + 58800*a*d*tan(c/2 + d*x/2)**8 + 58800 *a*d*tan(c/2 + d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4 + 11760*a*d*tan(c /2 + d*x/2)**2 + 1680*a*d) + 2205*d*x*tan(c/2 + d*x/2)**10/(1680*a*d*tan(c /2 + d*x/2)**14 + 11760*a*d*tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/2 + d*x /2)**10 + 58800*a*d*tan(c/2 + d*x/2)**8 + 58800*a*d*tan(c/2 + d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4 + 11760*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) + 3675*d*x*tan(c/2 + d*x/2)**8/(1680*a*d*tan(c/2 + d*x/2)**14 + 11760*a*d* tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/2 + d*x/2)**10 + 58800*a*d*tan(c/2 + d*x/2)**8 + 58800*a*d*tan(c/2 + d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)** 4 + 11760*a*d*tan(c/2 + d*x/2)**2 + 1680*a*d) + 3675*d*x*tan(c/2 + d*x/2)* *6/(1680*a*d*tan(c/2 + d*x/2)**14 + 11760*a*d*tan(c/2 + d*x/2)**12 + 35280 *a*d*tan(c/2 + d*x/2)**10 + 58800*a*d*tan(c/2 + d*x/2)**8 + 58800*a*d*tan( c/2 + d*x/2)**6 + 35280*a*d*tan(c/2 + d*x/2)**4 + 11760*a*d*tan(c/2 + d*x/ 2)**2 + 1680*a*d) + 2205*d*x*tan(c/2 + d*x/2)**4/(1680*a*d*tan(c/2 + d*x/2 )**14 + 11760*a*d*tan(c/2 + d*x/2)**12 + 35280*a*d*tan(c/2 + d*x/2)**10...
Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (103) = 206\).
Time = 0.14 (sec) , antiderivative size = 400, normalized size of antiderivative = 3.48 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {672 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1540 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1344 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1085 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {6720 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {3360 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {1085 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {3360 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {1540 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {105 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - 96}{a + \frac {7 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {21 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {35 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {35 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {21 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {7 \, a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {a \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}}} - \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{840 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/840*((105*sin(d*x + c)/(cos(d*x + c) + 1) - 672*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 1540*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 1344*sin(d*x + c )^4/(cos(d*x + c) + 1)^4 + 1085*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 6720 *sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 3360*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 1085*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 3360*sin(d*x + c)^10/(c os(d*x + c) + 1)^10 + 1540*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 105*sin (d*x + c)^13/(cos(d*x + c) + 1)^13 - 96)/(a + 7*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 21*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 35*a*sin(d*x + c) ^6/(cos(d*x + c) + 1)^6 + 35*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 21*a* sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 7*a*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + a*sin(d*x + c)^14/(cos(d*x + c) + 1)^14) - 105*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d
Time = 0.17 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.56 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {105 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 1540 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 3360 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 1085 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 3360 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 6720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1085 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1344 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1540 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 672 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7} a}}{1680 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/1680*(105*(d*x + c)/a + 2*(105*tan(1/2*d*x + 1/2*c)^13 - 1540*tan(1/2*d* x + 1/2*c)^11 + 3360*tan(1/2*d*x + 1/2*c)^10 + 1085*tan(1/2*d*x + 1/2*c)^9 - 3360*tan(1/2*d*x + 1/2*c)^8 + 6720*tan(1/2*d*x + 1/2*c)^6 - 1085*tan(1/ 2*d*x + 1/2*c)^5 - 1344*tan(1/2*d*x + 1/2*c)^4 + 1540*tan(1/2*d*x + 1/2*c) ^3 + 672*tan(1/2*d*x + 1/2*c)^2 - 105*tan(1/2*d*x + 1/2*c) + 96)/((tan(1/2 *d*x + 1/2*c)^2 + 1)^7*a))/d
Time = 37.05 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.50 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {x}{16\,a}+\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{8}-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{6}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {31\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {31\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{24}-\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {4}{35}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \] Input:
int((cos(c + d*x)^6*sin(c + d*x)^2)/(a + a*sin(c + d*x)),x)
Output:
x/(16*a) + ((4*tan(c/2 + (d*x)/2)^2)/5 - tan(c/2 + (d*x)/2)/8 + (11*tan(c/ 2 + (d*x)/2)^3)/6 - (8*tan(c/2 + (d*x)/2)^4)/5 - (31*tan(c/2 + (d*x)/2)^5) /24 + 8*tan(c/2 + (d*x)/2)^6 - 4*tan(c/2 + (d*x)/2)^8 + (31*tan(c/2 + (d*x )/2)^9)/24 + 4*tan(c/2 + (d*x)/2)^10 - (11*tan(c/2 + (d*x)/2)^11)/6 + tan( c/2 + (d*x)/2)^13/8 + 4/35)/(a*d*(tan(c/2 + (d*x)/2)^2 + 1)^7)
Time = 0.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.01 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {240 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-280 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-384 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+490 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+48 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-105 \cos \left (d x +c \right ) \sin \left (d x +c \right )+96 \cos \left (d x +c \right )+105 d x -96}{1680 a d} \] Input:
int(cos(d*x+c)^6*sin(d*x+c)^2/(a+a*sin(d*x+c)),x)
Output:
(240*cos(c + d*x)*sin(c + d*x)**6 - 280*cos(c + d*x)*sin(c + d*x)**5 - 384 *cos(c + d*x)*sin(c + d*x)**4 + 490*cos(c + d*x)*sin(c + d*x)**3 + 48*cos( c + d*x)*sin(c + d*x)**2 - 105*cos(c + d*x)*sin(c + d*x) + 96*cos(c + d*x) + 105*d*x - 96)/(1680*a*d)