Integrand size = 21, antiderivative size = 58 \[ \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^8} \, dx=-\frac {\cos ^7(c+d x)}{9 d (a+a \sin (c+d x))^8}-\frac {\cos ^7(c+d x)}{63 a d (a+a \sin (c+d x))^7} \] Output:
-1/9*cos(d*x+c)^7/d/(a+a*sin(d*x+c))^8-1/63*cos(d*x+c)^7/a/d/(a+a*sin(d*x+ c))^7
Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.62 \[ \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^8} \, dx=-\frac {\cos ^7(c+d x) (8+\sin (c+d x))}{63 a^8 d (1+\sin (c+d x))^8} \] Input:
Integrate[Cos[c + d*x]^6/(a + a*Sin[c + d*x])^8,x]
Output:
-1/63*(Cos[c + d*x]^7*(8 + Sin[c + d*x]))/(a^8*d*(1 + Sin[c + d*x])^8)
Time = 0.32 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3151, 3042, 3150}
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.
| \(\displaystyle \int \frac {\cos ^6(c+d x)}{(a \sin (c+d x)+a)^8} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6}{(a \sin (c+d x)+a)^8}dx\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {\int \frac {\cos ^6(c+d x)}{(\sin (c+d x) a+a)^7}dx}{9 a}-\frac {\cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\cos (c+d x)^6}{(\sin (c+d x) a+a)^7}dx}{9 a}-\frac {\cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^8}\) |
| \(\Big \downarrow \) 3150 |
| \(\displaystyle -\frac {\cos ^7(c+d x)}{63 a d (a \sin (c+d x)+a)^7}-\frac {\cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^8}\) |
Input:
Int[Cos[c + d*x]^6/(a + a*Sin[c + d*x])^8,x]
Output:
-1/9*Cos[c + d*x]^7/(d*(a + a*Sin[c + d*x])^8) - Cos[c + d*x]^7/(63*a*d*(a + a*Sin[c + d*x])^7)
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^m/(a*f*g*m)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[Simplify[m + p + 1], 0] && !ILtQ[p, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Simp[Simplify[m + p + 1]/(a*Simpl ify[2*m + p + 1]) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x] , x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simpli fy[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && !IGtQ[m, 0]
Result contains complex when optimal does not.
Time = 1.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.84
| method | result | size |
| risch | \(\frac {2 i \left (-105 i {\mathrm e}^{6 i \left (d x +c \right )}+63 \,{\mathrm e}^{7 i \left (d x +c \right )}+189 i {\mathrm e}^{4 i \left (d x +c \right )}-315 \,{\mathrm e}^{5 i \left (d x +c \right )}-27 i {\mathrm e}^{2 i \left (d x +c \right )}+189 \,{\mathrm e}^{3 i \left (d x +c \right )}-i-9 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{63 d \,a^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{9}}\) | \(107\) |
| parallelrisch | \(\frac {-\frac {16}{63}-\frac {46 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3}-22 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{7}-\frac {50 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{7}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d \,a^{8} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}\) | \(126\) |
| derivativedivides | \(\frac {-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {1856}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {152}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {256}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {14}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {272}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {992}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {172}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}}{d \,a^{8}}\) | \(145\) |
| default | \(\frac {-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {1856}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {152}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {256}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {14}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {272}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {992}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {172}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}}{d \,a^{8}}\) | \(145\) |
Input:
int(cos(d*x+c)^6/(a+a*sin(d*x+c))^8,x,method=_RETURNVERBOSE)
Output:
2/63*I*(-105*I*exp(6*I*(d*x+c))+63*exp(7*I*(d*x+c))+189*I*exp(4*I*(d*x+c)) -315*exp(5*I*(d*x+c))-27*I*exp(2*I*(d*x+c))+189*exp(3*I*(d*x+c))-I-9*exp(I *(d*x+c)))/d/a^8/(exp(I*(d*x+c))+I)^9
Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (54) = 108\).
Time = 0.10 (sec) , antiderivative size = 239, normalized size of antiderivative = 4.12 \[ \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {\cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{4} + 19 \, \cos \left (d x + c\right )^{3} + 52 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{3} + 24 \, \cos \left (d x + c\right )^{2} - 28 \, \cos \left (d x + c\right ) - 56\right )} \sin \left (d x + c\right ) - 28 \, \cos \left (d x + c\right ) - 56}{63 \, {\left (a^{8} d \cos \left (d x + c\right )^{5} + 5 \, a^{8} d \cos \left (d x + c\right )^{4} - 8 \, a^{8} d \cos \left (d x + c\right )^{3} - 20 \, a^{8} d \cos \left (d x + c\right )^{2} + 8 \, a^{8} d \cos \left (d x + c\right ) + 16 \, a^{8} d + {\left (a^{8} d \cos \left (d x + c\right )^{4} - 4 \, a^{8} d \cos \left (d x + c\right )^{3} - 12 \, a^{8} d \cos \left (d x + c\right )^{2} + 8 \, a^{8} d \cos \left (d x + c\right ) + 16 \, a^{8} d\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(cos(d*x+c)^6/(a+a*sin(d*x+c))^8,x, algorithm="fricas")
Output:
1/63*(cos(d*x + c)^5 - 4*cos(d*x + c)^4 + 19*cos(d*x + c)^3 + 52*cos(d*x + c)^2 - (cos(d*x + c)^4 + 5*cos(d*x + c)^3 + 24*cos(d*x + c)^2 - 28*cos(d* x + c) - 56)*sin(d*x + c) - 28*cos(d*x + c) - 56)/(a^8*d*cos(d*x + c)^5 + 5*a^8*d*cos(d*x + c)^4 - 8*a^8*d*cos(d*x + c)^3 - 20*a^8*d*cos(d*x + c)^2 + 8*a^8*d*cos(d*x + c) + 16*a^8*d + (a^8*d*cos(d*x + c)^4 - 4*a^8*d*cos(d* x + c)^3 - 12*a^8*d*cos(d*x + c)^2 + 8*a^8*d*cos(d*x + c) + 16*a^8*d)*sin( d*x + c))
Timed out. \[ \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**6/(a+a*sin(d*x+c))**8,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (54) = 108\).
Time = 0.04 (sec) , antiderivative size = 375, normalized size of antiderivative = 6.47 \[ \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^8} \, dx=-\frac {2 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {225 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {189 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {693 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {315 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {483 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {63 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {63 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 8\right )}}{63 \, {\left (a^{8} + \frac {9 \, a^{8} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {36 \, a^{8} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {84 \, a^{8} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {126 \, a^{8} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {126 \, a^{8} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {84 \, a^{8} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {36 \, a^{8} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {9 \, a^{8} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a^{8} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}\right )} d} \] Input:
integrate(cos(d*x+c)^6/(a+a*sin(d*x+c))^8,x, algorithm="maxima")
Output:
-2/63*(9*sin(d*x + c)/(cos(d*x + c) + 1) + 225*sin(d*x + c)^2/(cos(d*x + c ) + 1)^2 + 189*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 693*sin(d*x + c)^4/(c os(d*x + c) + 1)^4 + 315*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 483*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 63*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 63 *sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 8)/((a^8 + 9*a^8*sin(d*x + c)/(cos( d*x + c) + 1) + 36*a^8*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 84*a^8*sin(d* x + c)^3/(cos(d*x + c) + 1)^3 + 126*a^8*sin(d*x + c)^4/(cos(d*x + c) + 1)^ 4 + 126*a^8*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 84*a^8*sin(d*x + c)^6/(c os(d*x + c) + 1)^6 + 36*a^8*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 9*a^8*si n(d*x + c)^8/(cos(d*x + c) + 1)^8 + a^8*sin(d*x + c)^9/(cos(d*x + c) + 1)^ 9)*d)
Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (54) = 108\).
Time = 0.19 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.16 \[ \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^8} \, dx=-\frac {2 \, {\left (63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 483 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 693 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 189 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 225 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8\right )}}{63 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{9}} \] Input:
integrate(cos(d*x+c)^6/(a+a*sin(d*x+c))^8,x, algorithm="giac")
Output:
-2/63*(63*tan(1/2*d*x + 1/2*c)^8 + 63*tan(1/2*d*x + 1/2*c)^7 + 483*tan(1/2 *d*x + 1/2*c)^6 + 315*tan(1/2*d*x + 1/2*c)^5 + 693*tan(1/2*d*x + 1/2*c)^4 + 189*tan(1/2*d*x + 1/2*c)^3 + 225*tan(1/2*d*x + 1/2*c)^2 + 9*tan(1/2*d*x + 1/2*c) + 8)/(a^8*d*(tan(1/2*d*x + 1/2*c) + 1)^9)
Time = 27.63 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.03 \[ \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^8} \, dx=-\frac {\sqrt {2}\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {63\,\sin \left (c+d\,x\right )}{2}-\frac {257\,\cos \left (c+d\,x\right )}{8}-\frac {113\,\cos \left (2\,c+2\,d\,x\right )}{4}+\frac {37\,\cos \left (3\,c+3\,d\,x\right )}{8}+\frac {7\,\cos \left (4\,c+4\,d\,x\right )}{16}-\frac {63\,\sin \left (2\,c+2\,d\,x\right )}{8}-\frac {9\,\sin \left (3\,c+3\,d\,x\right )}{2}+\frac {9\,\sin \left (4\,c+4\,d\,x\right )}{16}+\frac {1013}{16}\right )}{1008\,a^8\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^9} \] Input:
int(cos(c + d*x)^6/(a + a*sin(c + d*x))^8,x)
Output:
-(2^(1/2)*cos(c/2 + (d*x)/2)*((63*sin(c + d*x))/2 - (257*cos(c + d*x))/8 - (113*cos(2*c + 2*d*x))/4 + (37*cos(3*c + 3*d*x))/8 + (7*cos(4*c + 4*d*x)) /16 - (63*sin(2*c + 2*d*x))/8 - (9*sin(3*c + 3*d*x))/2 + (9*sin(4*c + 4*d* x))/16 + 1013/16))/(1008*a^8*d*cos(c/2 - pi/4 + (d*x)/2)^9)
Time = 0.22 (sec) , antiderivative size = 244, normalized size of antiderivative = 4.21 \[ \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {5 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+19 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+57 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+\cos \left (d x +c \right ) \sin \left (d x +c \right )+14 \cos \left (d x +c \right )-7 \sin \left (d x +c \right )^{5}-34 \sin \left (d x +c \right )^{4}-34 \sin \left (d x +c \right )^{3}-104 \sin \left (d x +c \right )^{2}+\sin \left (d x +c \right )-14}{63 a^{8} d \left (\cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+4 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+4 \cos \left (d x +c \right ) \sin \left (d x +c \right )+\cos \left (d x +c \right )-\sin \left (d x +c \right )^{5}-5 \sin \left (d x +c \right )^{4}-10 \sin \left (d x +c \right )^{3}-10 \sin \left (d x +c \right )^{2}-5 \sin \left (d x +c \right )-1\right )} \] Input:
int(cos(d*x+c)^6/(a+a*sin(d*x+c))^8,x)
Output:
(5*cos(c + d*x)*sin(c + d*x)**4 + 19*cos(c + d*x)*sin(c + d*x)**3 + 57*cos (c + d*x)*sin(c + d*x)**2 + cos(c + d*x)*sin(c + d*x) + 14*cos(c + d*x) - 7*sin(c + d*x)**5 - 34*sin(c + d*x)**4 - 34*sin(c + d*x)**3 - 104*sin(c + d*x)**2 + sin(c + d*x) - 14)/(63*a**8*d*(cos(c + d*x)*sin(c + d*x)**4 + 4* cos(c + d*x)*sin(c + d*x)**3 + 6*cos(c + d*x)*sin(c + d*x)**2 + 4*cos(c + d*x)*sin(c + d*x) + cos(c + d*x) - sin(c + d*x)**5 - 5*sin(c + d*x)**4 - 1 0*sin(c + d*x)**3 - 10*sin(c + d*x)**2 - 5*sin(c + d*x) - 1))