Integrand size = 27, antiderivative size = 121 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5 x}{128 a}-\frac {\cos ^7(c+d x)}{7 a d}-\frac {5 \cos (c+d x) \sin (c+d x)}{128 a d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{192 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{48 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a d} \] Output:
-5/128*x/a-1/7*cos(d*x+c)^7/a/d-5/128*cos(d*x+c)*sin(d*x+c)/a/d-5/192*cos( d*x+c)^3*sin(d*x+c)/a/d-1/48*cos(d*x+c)^5*sin(d*x+c)/a/d+1/8*cos(d*x+c)^7* sin(d*x+c)/a/d
Leaf count is larger than twice the leaf count of optimal. \(481\) vs. \(2(121)=242\).
Time = 12.83 (sec) , antiderivative size = 481, normalized size of antiderivative = 3.98 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {-336 (7 c-5 d x) \cos \left (\frac {c}{2}\right )+1680 \cos \left (\frac {c}{2}+d x\right )+1680 \cos \left (\frac {3 c}{2}+d x\right )+336 \cos \left (\frac {3 c}{2}+2 d x\right )-336 \cos \left (\frac {5 c}{2}+2 d x\right )+1008 \cos \left (\frac {5 c}{2}+3 d x\right )+1008 \cos \left (\frac {7 c}{2}+3 d x\right )-168 \cos \left (\frac {7 c}{2}+4 d x\right )+168 \cos \left (\frac {9 c}{2}+4 d x\right )+336 \cos \left (\frac {9 c}{2}+5 d x\right )+336 \cos \left (\frac {11 c}{2}+5 d x\right )-112 \cos \left (\frac {11 c}{2}+6 d x\right )+112 \cos \left (\frac {13 c}{2}+6 d x\right )+48 \cos \left (\frac {13 c}{2}+7 d x\right )+48 \cos \left (\frac {15 c}{2}+7 d x\right )-21 \cos \left (\frac {15 c}{2}+8 d x\right )+21 \cos \left (\frac {17 c}{2}+8 d x\right )+4704 \sin \left (\frac {c}{2}\right )-2352 c \sin \left (\frac {c}{2}\right )+1680 d x \sin \left (\frac {c}{2}\right )-1680 \sin \left (\frac {c}{2}+d x\right )+1680 \sin \left (\frac {3 c}{2}+d x\right )+336 \sin \left (\frac {3 c}{2}+2 d x\right )+336 \sin \left (\frac {5 c}{2}+2 d x\right )-1008 \sin \left (\frac {5 c}{2}+3 d x\right )+1008 \sin \left (\frac {7 c}{2}+3 d x\right )-168 \sin \left (\frac {7 c}{2}+4 d x\right )-168 \sin \left (\frac {9 c}{2}+4 d x\right )-336 \sin \left (\frac {9 c}{2}+5 d x\right )+336 \sin \left (\frac {11 c}{2}+5 d x\right )-112 \sin \left (\frac {11 c}{2}+6 d x\right )-112 \sin \left (\frac {13 c}{2}+6 d x\right )-48 \sin \left (\frac {13 c}{2}+7 d x\right )+48 \sin \left (\frac {15 c}{2}+7 d x\right )-21 \sin \left (\frac {15 c}{2}+8 d x\right )-21 \sin \left (\frac {17 c}{2}+8 d x\right )}{43008 a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \] Input:
Integrate[(Cos[c + d*x]^8*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]
Output:
-1/43008*(-336*(7*c - 5*d*x)*Cos[c/2] + 1680*Cos[c/2 + d*x] + 1680*Cos[(3* c)/2 + d*x] + 336*Cos[(3*c)/2 + 2*d*x] - 336*Cos[(5*c)/2 + 2*d*x] + 1008*C os[(5*c)/2 + 3*d*x] + 1008*Cos[(7*c)/2 + 3*d*x] - 168*Cos[(7*c)/2 + 4*d*x] + 168*Cos[(9*c)/2 + 4*d*x] + 336*Cos[(9*c)/2 + 5*d*x] + 336*Cos[(11*c)/2 + 5*d*x] - 112*Cos[(11*c)/2 + 6*d*x] + 112*Cos[(13*c)/2 + 6*d*x] + 48*Cos[ (13*c)/2 + 7*d*x] + 48*Cos[(15*c)/2 + 7*d*x] - 21*Cos[(15*c)/2 + 8*d*x] + 21*Cos[(17*c)/2 + 8*d*x] + 4704*Sin[c/2] - 2352*c*Sin[c/2] + 1680*d*x*Sin[ c/2] - 1680*Sin[c/2 + d*x] + 1680*Sin[(3*c)/2 + d*x] + 336*Sin[(3*c)/2 + 2 *d*x] + 336*Sin[(5*c)/2 + 2*d*x] - 1008*Sin[(5*c)/2 + 3*d*x] + 1008*Sin[(7 *c)/2 + 3*d*x] - 168*Sin[(7*c)/2 + 4*d*x] - 168*Sin[(9*c)/2 + 4*d*x] - 336 *Sin[(9*c)/2 + 5*d*x] + 336*Sin[(11*c)/2 + 5*d*x] - 112*Sin[(11*c)/2 + 6*d *x] - 112*Sin[(13*c)/2 + 6*d*x] - 48*Sin[(13*c)/2 + 7*d*x] + 48*Sin[(15*c) /2 + 7*d*x] - 21*Sin[(15*c)/2 + 8*d*x] - 21*Sin[(17*c)/2 + 8*d*x])/(a*d*(C os[c/2] + Sin[c/2]))
Time = 0.66 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {3042, 3318, 3042, 3045, 15, 3048, 3042, 3115, 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.
\(\displaystyle \int \frac {\sin (c+d x) \cos ^8(c+d x)}{a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (c+d x) \cos (c+d x)^8}{a \sin (c+d x)+a}dx\) |
\(\Big \downarrow \) 3318 |
\(\displaystyle \frac {\int \cos ^6(c+d x) \sin (c+d x)dx}{a}-\frac {\int \cos ^6(c+d x) \sin ^2(c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \cos (c+d x)^6 \sin (c+d x)dx}{a}-\frac {\int \cos (c+d x)^6 \sin (c+d x)^2dx}{a}\) |
\(\Big \downarrow \) 3045 |
\(\displaystyle -\frac {\int \cos ^6(c+d x)d\cos (c+d x)}{a d}-\frac {\int \cos (c+d x)^6 \sin (c+d x)^2dx}{a}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\int \cos (c+d x)^6 \sin (c+d x)^2dx}{a}-\frac {\cos ^7(c+d x)}{7 a d}\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle -\frac {\frac {1}{8} \int \cos ^6(c+d x)dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}-\frac {\cos ^7(c+d x)}{7 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{8} \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}-\frac {\cos ^7(c+d x)}{7 a d}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}-\frac {\cos ^7(c+d x)}{7 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}-\frac {\cos ^7(c+d x)}{7 a d}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}-\frac {\cos ^7(c+d x)}{7 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}-\frac {\cos ^7(c+d x)}{7 a d}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle -\frac {\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}-\frac {\cos ^7(c+d x)}{7 a d}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {\cos ^7(c+d x)}{7 a d}-\frac {\frac {1}{8} \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}\) |
Input:
Int[(Cos[c + d*x]^8*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]
Output:
-1/7*Cos[c + d*x]^7/(a*d) - (-1/8*(Cos[c + d*x]^7*Sin[c + d*x])/d + ((Cos[ c + d*x]^5*Sin[c + d*x])/(6*d) + (5*((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[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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 = 1.64 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.83
method | result | size |
parallelrisch | \(\frac {-840 d x +112 \sin \left (6 d x +6 c \right )+168 \sin \left (4 d x +4 c \right )-336 \sin \left (2 d x +2 c \right )+21 \sin \left (8 d x +8 c \right )-48 \cos \left (7 d x +7 c \right )-336 \cos \left (5 d x +5 c \right )-1008 \cos \left (3 d x +3 c \right )-1680 \cos \left (d x +c \right )-3072}{21504 d a}\) | \(100\) |
risch | \(-\frac {5 x}{128 a}-\frac {5 \cos \left (d x +c \right )}{64 a d}+\frac {\sin \left (8 d x +8 c \right )}{1024 d a}-\frac {\cos \left (7 d x +7 c \right )}{448 a d}+\frac {\sin \left (6 d x +6 c \right )}{192 d a}-\frac {\cos \left (5 d x +5 c \right )}{64 a d}+\frac {\sin \left (4 d x +4 c \right )}{128 d a}-\frac {3 \cos \left (3 d x +3 c \right )}{64 a d}-\frac {\sin \left (2 d x +2 c \right )}{64 d a}\) | \(141\) |
derivativedivides | \(\frac {\frac {4 \left (-\frac {1}{14}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{256}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{14}-\frac {397 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{768}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}+\frac {895 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{768}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2}-\frac {1765 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{768}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}+\frac {1765 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{768}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}-\frac {895 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{768}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{2}+\frac {397 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{768}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{2}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{256}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d a}\) | \(233\) |
default | \(\frac {\frac {4 \left (-\frac {1}{14}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{256}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{14}-\frac {397 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{768}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}+\frac {895 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{768}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2}-\frac {1765 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{768}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}+\frac {1765 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{768}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}-\frac {895 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{768}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{2}+\frac {397 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{768}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{2}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{256}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d a}\) | \(233\) |
Input:
int(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/21504*(-840*d*x+112*sin(6*d*x+6*c)+168*sin(4*d*x+4*c)-336*sin(2*d*x+2*c) +21*sin(8*d*x+8*c)-48*cos(7*d*x+7*c)-336*cos(5*d*x+5*c)-1008*cos(3*d*x+3*c )-1680*cos(d*x+c)-3072)/d/a
Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.58 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {384 \, \cos \left (d x + c\right )^{7} + 105 \, d x - 7 \, {\left (48 \, \cos \left (d x + c\right )^{7} - 8 \, \cos \left (d x + c\right )^{5} - 10 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2688 \, a d} \] Input:
integrate(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="fricas")
Output:
-1/2688*(384*cos(d*x + c)^7 + 105*d*x - 7*(48*cos(d*x + c)^7 - 8*cos(d*x + c)^5 - 10*cos(d*x + c)^3 - 15*cos(d*x + c))*sin(d*x + c))/(a*d)
Leaf count of result is larger than twice the leaf count of optimal. 3888 vs. \(2 (102) = 204\).
Time = 52.32 (sec) , antiderivative size = 3888, normalized size of antiderivative = 32.13 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)**8*sin(d*x+c)/(a+a*sin(d*x+c)),x)
Output:
Piecewise((-105*d*x*tan(c/2 + d*x/2)**16/(2688*a*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a *d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan( c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 + d*x/ 2)**2 + 2688*a*d) - 840*d*x*tan(c/2 + d*x/2)**14/(2688*a*d*tan(c/2 + d*x/2 )**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528* a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/ 2 + d*x/2)**2 + 2688*a*d) - 2940*d*x*tan(c/2 + d*x/2)**12/(2688*a*d*tan(c/ 2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/ 2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a *d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 5880*d*x*tan(c/2 + d*x/2)**10/(2688*a *d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c /2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d *x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 7350*d*x*tan(c/2 + d*x/2)**8 /(2688*a*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a *d*tan(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan (c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 +...
Leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (109) = 218\).
Time = 0.15 (sec) , antiderivative size = 501, normalized size of antiderivative = 4.14 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="maxima")
Output:
1/1344*((105*sin(d*x + c)/(cos(d*x + c) + 1) - 384*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 2779*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 8064*sin(d*x + c )^4/(cos(d*x + c) + 1)^4 + 6265*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 8064 *sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 12355*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 13440*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 12355*sin(d*x + c)^9/ (cos(d*x + c) + 1)^9 - 13440*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 6265* sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 2688*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 2779*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 2688*sin(d*x + c)^ 14/(cos(d*x + c) + 1)^14 - 105*sin(d*x + c)^15/(cos(d*x + c) + 1)^15 - 384 )/(a + 8*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 28*a*sin(d*x + c)^4/(cos( d*x + c) + 1)^4 + 56*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 70*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 56*a*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 28*a*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 8*a*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + a*sin(d*x + c)^16/(cos(d*x + c) + 1)^16) - 105*arctan(sin( d*x + c)/(cos(d*x + c) + 1))/a)/d
Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (109) = 218\).
Time = 0.14 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.91 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {105 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 2688 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 2779 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2688 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 6265 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 13440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 12355 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 13440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 12355 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 8064 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 6265 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 8064 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2779 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 384 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 384\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{8} a}}{2688 \, d} \] Input:
integrate(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-1/2688*(105*(d*x + c)/a + 2*(105*tan(1/2*d*x + 1/2*c)^15 + 2688*tan(1/2*d *x + 1/2*c)^14 - 2779*tan(1/2*d*x + 1/2*c)^13 + 2688*tan(1/2*d*x + 1/2*c)^ 12 + 6265*tan(1/2*d*x + 1/2*c)^11 + 13440*tan(1/2*d*x + 1/2*c)^10 - 12355* tan(1/2*d*x + 1/2*c)^9 + 13440*tan(1/2*d*x + 1/2*c)^8 + 12355*tan(1/2*d*x + 1/2*c)^7 + 8064*tan(1/2*d*x + 1/2*c)^6 - 6265*tan(1/2*d*x + 1/2*c)^5 + 8 064*tan(1/2*d*x + 1/2*c)^4 + 2779*tan(1/2*d*x + 1/2*c)^3 + 384*tan(1/2*d*x + 1/2*c)^2 - 105*tan(1/2*d*x + 1/2*c) + 384)/((tan(1/2*d*x + 1/2*c)^2 + 1 )^8*a))/d
Time = 34.74 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.86 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5\,x}{128\,a}-\frac {\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-\frac {397\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+\frac {895\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{192}+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {1765\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {1765\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{192}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {895\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {397\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{7}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {2}{7}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8} \] Input:
int((cos(c + d*x)^8*sin(c + d*x))/(a + a*sin(c + d*x)),x)
Output:
- (5*x)/(128*a) - ((2*tan(c/2 + (d*x)/2)^2)/7 - (5*tan(c/2 + (d*x)/2))/64 + (397*tan(c/2 + (d*x)/2)^3)/192 + 6*tan(c/2 + (d*x)/2)^4 - (895*tan(c/2 + (d*x)/2)^5)/192 + 6*tan(c/2 + (d*x)/2)^6 + (1765*tan(c/2 + (d*x)/2)^7)/19 2 + 10*tan(c/2 + (d*x)/2)^8 - (1765*tan(c/2 + (d*x)/2)^9)/192 + 10*tan(c/2 + (d*x)/2)^10 + (895*tan(c/2 + (d*x)/2)^11)/192 + 2*tan(c/2 + (d*x)/2)^12 - (397*tan(c/2 + (d*x)/2)^13)/192 + 2*tan(c/2 + (d*x)/2)^14 + (5*tan(c/2 + (d*x)/2)^15)/64 + 2/7)/(a*d*(tan(c/2 + (d*x)/2)^2 + 1)^8)
Time = 0.17 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {-336 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}+384 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+952 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-1152 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-826 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+1152 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+105 \cos \left (d x +c \right ) \sin \left (d x +c \right )-384 \cos \left (d x +c \right )-105 d x +384}{2688 a d} \] Input:
int(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c)),x)
Output:
( - 336*cos(c + d*x)*sin(c + d*x)**7 + 384*cos(c + d*x)*sin(c + d*x)**6 + 952*cos(c + d*x)*sin(c + d*x)**5 - 1152*cos(c + d*x)*sin(c + d*x)**4 - 826 *cos(c + d*x)*sin(c + d*x)**3 + 1152*cos(c + d*x)*sin(c + d*x)**2 + 105*co s(c + d*x)*sin(c + d*x) - 384*cos(c + d*x) - 105*d*x + 384)/(2688*a*d)