Integrand size = 29, antiderivative size = 159 \[ \int \frac {\cos ^6(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 x}{128 a}+\frac {\cos ^5(c+d x)}{5 a d}-\frac {2 \cos ^7(c+d x)}{7 a d}+\frac {\cos ^9(c+d x)}{9 a d}+\frac {3 \cos (c+d x) \sin (c+d x)}{128 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{64 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{16 a d}-\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{8 a d} \] Output:
3/128*x/a+1/5*cos(d*x+c)^5/a/d-2/7*cos(d*x+c)^7/a/d+1/9*cos(d*x+c)^9/a/d+3 /128*cos(d*x+c)*sin(d*x+c)/a/d+1/64*cos(d*x+c)^3*sin(d*x+c)/a/d-1/16*cos(d *x+c)^5*sin(d*x+c)/a/d-1/8*cos(d*x+c)^5*sin(d*x+c)^3/a/d
Leaf count is larger than twice the leaf count of optimal. \(429\) vs. \(2(159)=318\).
Time = 10.25 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.70 \[ \int \frac {\cos ^6(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2520 (5 c+6 d x) \cos \left (\frac {c}{2}\right )+7560 \cos \left (\frac {c}{2}+d x\right )+7560 \cos \left (\frac {3 c}{2}+d x\right )+1680 \cos \left (\frac {5 c}{2}+3 d x\right )+1680 \cos \left (\frac {7 c}{2}+3 d x\right )-2520 \cos \left (\frac {7 c}{2}+4 d x\right )+2520 \cos \left (\frac {9 c}{2}+4 d x\right )-1008 \cos \left (\frac {9 c}{2}+5 d x\right )-1008 \cos \left (\frac {11 c}{2}+5 d x\right )-180 \cos \left (\frac {13 c}{2}+7 d x\right )-180 \cos \left (\frac {15 c}{2}+7 d x\right )+315 \cos \left (\frac {15 c}{2}+8 d x\right )-315 \cos \left (\frac {17 c}{2}+8 d x\right )+140 \cos \left (\frac {17 c}{2}+9 d x\right )+140 \cos \left (\frac {19 c}{2}+9 d x\right )+12600 \sin \left (\frac {c}{2}\right )+12600 c \sin \left (\frac {c}{2}\right )+15120 d x \sin \left (\frac {c}{2}\right )-7560 \sin \left (\frac {c}{2}+d x\right )+7560 \sin \left (\frac {3 c}{2}+d x\right )-1680 \sin \left (\frac {5 c}{2}+3 d x\right )+1680 \sin \left (\frac {7 c}{2}+3 d x\right )-2520 \sin \left (\frac {7 c}{2}+4 d x\right )-2520 \sin \left (\frac {9 c}{2}+4 d x\right )+1008 \sin \left (\frac {9 c}{2}+5 d x\right )-1008 \sin \left (\frac {11 c}{2}+5 d x\right )+180 \sin \left (\frac {13 c}{2}+7 d x\right )-180 \sin \left (\frac {15 c}{2}+7 d x\right )+315 \sin \left (\frac {15 c}{2}+8 d x\right )+315 \sin \left (\frac {17 c}{2}+8 d x\right )-140 \sin \left (\frac {17 c}{2}+9 d x\right )+140 \sin \left (\frac {19 c}{2}+9 d x\right )}{645120 a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \] Input:
Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]
Output:
(2520*(5*c + 6*d*x)*Cos[c/2] + 7560*Cos[c/2 + d*x] + 7560*Cos[(3*c)/2 + d* x] + 1680*Cos[(5*c)/2 + 3*d*x] + 1680*Cos[(7*c)/2 + 3*d*x] - 2520*Cos[(7*c )/2 + 4*d*x] + 2520*Cos[(9*c)/2 + 4*d*x] - 1008*Cos[(9*c)/2 + 5*d*x] - 100 8*Cos[(11*c)/2 + 5*d*x] - 180*Cos[(13*c)/2 + 7*d*x] - 180*Cos[(15*c)/2 + 7 *d*x] + 315*Cos[(15*c)/2 + 8*d*x] - 315*Cos[(17*c)/2 + 8*d*x] + 140*Cos[(1 7*c)/2 + 9*d*x] + 140*Cos[(19*c)/2 + 9*d*x] + 12600*Sin[c/2] + 12600*c*Sin [c/2] + 15120*d*x*Sin[c/2] - 7560*Sin[c/2 + d*x] + 7560*Sin[(3*c)/2 + d*x] - 1680*Sin[(5*c)/2 + 3*d*x] + 1680*Sin[(7*c)/2 + 3*d*x] - 2520*Sin[(7*c)/ 2 + 4*d*x] - 2520*Sin[(9*c)/2 + 4*d*x] + 1008*Sin[(9*c)/2 + 5*d*x] - 1008* Sin[(11*c)/2 + 5*d*x] + 180*Sin[(13*c)/2 + 7*d*x] - 180*Sin[(15*c)/2 + 7*d *x] + 315*Sin[(15*c)/2 + 8*d*x] + 315*Sin[(17*c)/2 + 8*d*x] - 140*Sin[(17* c)/2 + 9*d*x] + 140*Sin[(19*c)/2 + 9*d*x])/(645120*a*d*(Cos[c/2] + Sin[c/2 ]))
Time = 0.77 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.97, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {3042, 3318, 3042, 3045, 244, 2009, 3048, 3042, 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.
| \(\displaystyle \int \frac {\sin ^4(c+d x) \cos ^6(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^4 \cos (c+d x)^6}{a \sin (c+d x)+a}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \cos ^4(c+d x) \sin ^4(c+d x)dx}{a}-\frac {\int \cos ^4(c+d x) \sin ^5(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \cos (c+d x)^4 \sin (c+d x)^4dx}{a}-\frac {\int \cos (c+d x)^4 \sin (c+d x)^5dx}{a}\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle \frac {\int \cos ^4(c+d x) \left (1-\cos ^2(c+d x)\right )^2d\cos (c+d x)}{a d}+\frac {\int \cos (c+d x)^4 \sin (c+d x)^4dx}{a}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {\int \left (\cos ^8(c+d x)-2 \cos ^6(c+d x)+\cos ^4(c+d x)\right )d\cos (c+d x)}{a d}+\frac {\int \cos (c+d x)^4 \sin (c+d x)^4dx}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\int \cos (c+d x)^4 \sin (c+d x)^4dx}{a}+\frac {\frac {1}{9} \cos ^9(c+d x)-\frac {2}{7} \cos ^7(c+d x)+\frac {1}{5} \cos ^5(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {\frac {3}{8} \int \cos ^4(c+d x) \sin ^2(c+d x)dx-\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 d}}{a}+\frac {\frac {1}{9} \cos ^9(c+d x)-\frac {2}{7} \cos ^7(c+d x)+\frac {1}{5} \cos ^5(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{8} \int \cos (c+d x)^4 \sin (c+d x)^2dx-\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 d}}{a}+\frac {\frac {1}{9} \cos ^9(c+d x)-\frac {2}{7} \cos ^7(c+d x)+\frac {1}{5} \cos ^5(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {\frac {3}{8} \left (\frac {1}{6} \int \cos ^4(c+d x)dx-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 d}}{a}+\frac {\frac {1}{9} \cos ^9(c+d x)-\frac {2}{7} \cos ^7(c+d x)+\frac {1}{5} \cos ^5(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{8} \left (\frac {1}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 d}}{a}+\frac {\frac {1}{9} \cos ^9(c+d x)-\frac {2}{7} \cos ^7(c+d x)+\frac {1}{5} \cos ^5(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {3}{8} \left (\frac {1}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 d}}{a}+\frac {\frac {1}{9} \cos ^9(c+d x)-\frac {2}{7} \cos ^7(c+d x)+\frac {1}{5} \cos ^5(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{8} \left (\frac {1}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 d}}{a}+\frac {\frac {1}{9} \cos ^9(c+d x)-\frac {2}{7} \cos ^7(c+d x)+\frac {1}{5} \cos ^5(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {3}{8} \left (\frac {1}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 d}}{a}+\frac {\frac {1}{9} \cos ^9(c+d x)-\frac {2}{7} \cos ^7(c+d x)+\frac {1}{5} \cos ^5(c+d x)}{a d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{9} \cos ^9(c+d x)-\frac {2}{7} \cos ^7(c+d x)+\frac {1}{5} \cos ^5(c+d x)}{a d}+\frac {\frac {3}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 d}}{a}\) |
Input:
Int[(Cos[c + d*x]^6*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]
Output:
(Cos[c + d*x]^5/5 - (2*Cos[c + d*x]^7)/7 + Cos[c + d*x]^9/9)/(a*d) + (-1/8 *(Cos[c + d*x]^5*Sin[c + d*x]^3)/d + (3*(-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))/8)/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 = 1.19 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.56
| method | result | size |
| parallelrisch | \(\frac {7560 d x -1008 \cos \left (5 d x +5 c \right )+1680 \cos \left (3 d x +3 c \right )+7560 \cos \left (d x +c \right )+140 \cos \left (9 d x +9 c \right )+315 \sin \left (8 d x +8 c \right )-180 \cos \left (7 d x +7 c \right )-2520 \sin \left (4 d x +4 c \right )+8192}{322560 d a}\) | \(89\) |
| risch | \(\frac {3 x}{128 a}+\frac {3 \cos \left (d x +c \right )}{128 a d}+\frac {\cos \left (9 d x +9 c \right )}{2304 a d}+\frac {\sin \left (8 d x +8 c \right )}{1024 d a}-\frac {\cos \left (7 d x +7 c \right )}{1792 a d}-\frac {\cos \left (5 d x +5 c \right )}{320 a d}-\frac {\sin \left (4 d x +4 c \right )}{128 d a}+\frac {\cos \left (3 d x +3 c \right )}{192 a d}\) | \(124\) |
| derivativedivides | \(\frac {\frac {32 \left (\frac {1}{630}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2048}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{70}-\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{1024}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{35}+\frac {155 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{1024}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{5}-\frac {169 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{1024}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{10}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}+\frac {169 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{1024}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{3}-\frac {155 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{1024}+\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{1024}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}}{2048}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{9}}+\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d a}\) | \(220\) |
| default | \(\frac {\frac {32 \left (\frac {1}{630}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2048}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{70}-\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{1024}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{35}+\frac {155 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{1024}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{5}-\frac {169 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{1024}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{10}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}+\frac {169 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{1024}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{3}-\frac {155 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{1024}+\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{1024}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}}{2048}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{9}}+\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d a}\) | \(220\) |
Input:
int(cos(d*x+c)^6*sin(d*x+c)^4/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/322560*(7560*d*x-1008*cos(5*d*x+5*c)+1680*cos(3*d*x+3*c)+7560*cos(d*x+c) +140*cos(9*d*x+9*c)+315*sin(8*d*x+8*c)-180*cos(7*d*x+7*c)-2520*sin(4*d*x+4 *c)+8192)/d/a
Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.57 \[ \int \frac {\cos ^6(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {4480 \, \cos \left (d x + c\right )^{9} - 11520 \, \cos \left (d x + c\right )^{7} + 8064 \, \cos \left (d x + c\right )^{5} + 945 \, d x + 315 \, {\left (16 \, \cos \left (d x + c\right )^{7} - 24 \, \cos \left (d x + c\right )^{5} + 2 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{40320 \, a d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/40320*(4480*cos(d*x + c)^9 - 11520*cos(d*x + c)^7 + 8064*cos(d*x + c)^5 + 945*d*x + 315*(16*cos(d*x + c)^7 - 24*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. 4318 vs. \(2 (131) = 262\).
Time = 82.68 (sec) , antiderivative size = 4318, normalized size of antiderivative = 27.16 \[ \int \frac {\cos ^6(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)**6*sin(d*x+c)**4/(a+a*sin(d*x+c)),x)
Output:
Piecewise((945*d*x*tan(c/2 + d*x/2)**18/(40320*a*d*tan(c/2 + d*x/2)**18 + 362880*a*d*tan(c/2 + d*x/2)**16 + 1451520*a*d*tan(c/2 + d*x/2)**14 + 33868 80*a*d*tan(c/2 + d*x/2)**12 + 5080320*a*d*tan(c/2 + d*x/2)**10 + 5080320*a *d*tan(c/2 + d*x/2)**8 + 3386880*a*d*tan(c/2 + d*x/2)**6 + 1451520*a*d*tan (c/2 + d*x/2)**4 + 362880*a*d*tan(c/2 + d*x/2)**2 + 40320*a*d) + 8505*d*x* tan(c/2 + d*x/2)**16/(40320*a*d*tan(c/2 + d*x/2)**18 + 362880*a*d*tan(c/2 + d*x/2)**16 + 1451520*a*d*tan(c/2 + d*x/2)**14 + 3386880*a*d*tan(c/2 + d* x/2)**12 + 5080320*a*d*tan(c/2 + d*x/2)**10 + 5080320*a*d*tan(c/2 + d*x/2) **8 + 3386880*a*d*tan(c/2 + d*x/2)**6 + 1451520*a*d*tan(c/2 + d*x/2)**4 + 362880*a*d*tan(c/2 + d*x/2)**2 + 40320*a*d) + 34020*d*x*tan(c/2 + d*x/2)** 14/(40320*a*d*tan(c/2 + d*x/2)**18 + 362880*a*d*tan(c/2 + d*x/2)**16 + 145 1520*a*d*tan(c/2 + d*x/2)**14 + 3386880*a*d*tan(c/2 + d*x/2)**12 + 5080320 *a*d*tan(c/2 + d*x/2)**10 + 5080320*a*d*tan(c/2 + d*x/2)**8 + 3386880*a*d* tan(c/2 + d*x/2)**6 + 1451520*a*d*tan(c/2 + d*x/2)**4 + 362880*a*d*tan(c/2 + d*x/2)**2 + 40320*a*d) + 79380*d*x*tan(c/2 + d*x/2)**12/(40320*a*d*tan( c/2 + d*x/2)**18 + 362880*a*d*tan(c/2 + d*x/2)**16 + 1451520*a*d*tan(c/2 + d*x/2)**14 + 3386880*a*d*tan(c/2 + d*x/2)**12 + 5080320*a*d*tan(c/2 + d*x /2)**10 + 5080320*a*d*tan(c/2 + d*x/2)**8 + 3386880*a*d*tan(c/2 + d*x/2)** 6 + 1451520*a*d*tan(c/2 + d*x/2)**4 + 362880*a*d*tan(c/2 + d*x/2)**2 + 403 20*a*d) + 119070*d*x*tan(c/2 + d*x/2)**10/(40320*a*d*tan(c/2 + d*x/2)**...
Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (143) = 286\).
Time = 0.13 (sec) , antiderivative size = 502, normalized size of antiderivative = 3.16 \[ \int \frac {\cos ^6(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/20160*((945*sin(d*x + c)/(cos(d*x + c) + 1) - 9216*sin(d*x + c)^2/(cos( d*x + c) + 1)^2 + 8190*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 36864*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 97650*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 129024*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 106470*sin(d*x + c)^7/(cos(d *x + c) + 1)^7 - 451584*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 322560*sin(d *x + c)^10/(cos(d*x + c) + 1)^10 - 106470*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 215040*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 97650*sin(d*x + c)^ 13/(cos(d*x + c) + 1)^13 - 8190*sin(d*x + c)^15/(cos(d*x + c) + 1)^15 - 94 5*sin(d*x + c)^17/(cos(d*x + c) + 1)^17 - 1024)/(a + 9*a*sin(d*x + c)^2/(c os(d*x + c) + 1)^2 + 36*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 84*a*sin(d *x + c)^6/(cos(d*x + c) + 1)^6 + 126*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 126*a*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 84*a*sin(d*x + c)^12/(cos (d*x + c) + 1)^12 + 36*a*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + 9*a*sin(d *x + c)^16/(cos(d*x + c) + 1)^16 + a*sin(d*x + c)^18/(cos(d*x + c) + 1)^18 ) - 945*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d
Time = 0.14 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.37 \[ \int \frac {\cos ^6(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {945 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (945 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 8190 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} - 97650 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 215040 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 106470 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 322560 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 451584 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 106470 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 129024 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 97650 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36864 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8190 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9216 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 945 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1024\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{9} a}}{40320 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/40320*(945*(d*x + c)/a + 2*(945*tan(1/2*d*x + 1/2*c)^17 + 8190*tan(1/2*d *x + 1/2*c)^15 - 97650*tan(1/2*d*x + 1/2*c)^13 + 215040*tan(1/2*d*x + 1/2* c)^12 + 106470*tan(1/2*d*x + 1/2*c)^11 - 322560*tan(1/2*d*x + 1/2*c)^10 + 451584*tan(1/2*d*x + 1/2*c)^8 - 106470*tan(1/2*d*x + 1/2*c)^7 - 129024*tan (1/2*d*x + 1/2*c)^6 + 97650*tan(1/2*d*x + 1/2*c)^5 + 36864*tan(1/2*d*x + 1 /2*c)^4 - 8190*tan(1/2*d*x + 1/2*c)^3 + 9216*tan(1/2*d*x + 1/2*c)^2 - 945* tan(1/2*d*x + 1/2*c) + 1024)/((tan(1/2*d*x + 1/2*c)^2 + 1)^9*a))/d
Time = 36.86 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.33 \[ \int \frac {\cos ^6(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3\,x}{128\,a}+\frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{64}+\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{32}-\frac {155\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}+\frac {169\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{32}-16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {112\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{5}-\frac {169\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}-\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{5}+\frac {155\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{32}+\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{35}-\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32}+\frac {16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{35}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {16}{315}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^9} \] Input:
int((cos(c + d*x)^6*sin(c + d*x)^4)/(a + a*sin(c + d*x)),x)
Output:
(3*x)/(128*a) + ((16*tan(c/2 + (d*x)/2)^2)/35 - (3*tan(c/2 + (d*x)/2))/64 - (13*tan(c/2 + (d*x)/2)^3)/32 + (64*tan(c/2 + (d*x)/2)^4)/35 + (155*tan(c /2 + (d*x)/2)^5)/32 - (32*tan(c/2 + (d*x)/2)^6)/5 - (169*tan(c/2 + (d*x)/2 )^7)/32 + (112*tan(c/2 + (d*x)/2)^8)/5 - 16*tan(c/2 + (d*x)/2)^10 + (169*t an(c/2 + (d*x)/2)^11)/32 + (32*tan(c/2 + (d*x)/2)^12)/3 - (155*tan(c/2 + ( d*x)/2)^13)/32 + (13*tan(c/2 + (d*x)/2)^15)/32 + (3*tan(c/2 + (d*x)/2)^17) /64 + 16/315)/(a*d*(tan(c/2 + (d*x)/2)^2 + 1)^9)
Time = 0.20 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.93 \[ \int \frac {\cos ^6(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {4480 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}-5040 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-6400 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+7560 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+384 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-630 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+512 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-945 \cos \left (d x +c \right ) \sin \left (d x +c \right )+1024 \cos \left (d x +c \right )+945 d x -1024}{40320 a d} \] Input:
int(cos(d*x+c)^6*sin(d*x+c)^4/(a+a*sin(d*x+c)),x)
Output:
(4480*cos(c + d*x)*sin(c + d*x)**8 - 5040*cos(c + d*x)*sin(c + d*x)**7 - 6 400*cos(c + d*x)*sin(c + d*x)**6 + 7560*cos(c + d*x)*sin(c + d*x)**5 + 384 *cos(c + d*x)*sin(c + d*x)**4 - 630*cos(c + d*x)*sin(c + d*x)**3 + 512*cos (c + d*x)*sin(c + d*x)**2 - 945*cos(c + d*x)*sin(c + d*x) + 1024*cos(c + d *x) + 945*d*x - 1024)/(40320*a*d)