\(\int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx\) [705]

Optimal result

Integrand size = 29, antiderivative size = 209 \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5 x}{1024 a}-\frac {\cos ^7(c+d x)}{7 a d}+\frac {2 \cos ^9(c+d x)}{9 a d}-\frac {\cos ^{11}(c+d x)}{11 a d}-\frac {5 \cos (c+d x) \sin (c+d x)}{1024 a d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{1536 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{384 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{64 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{24 a d}+\frac {\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d} \]

[Out]

-5/1024*x/a-1/7*cos(d*x+c)^7/a/d+2/9*cos(d*x+c)^9/a/d-1/11*cos(d*x+c)^11/a/d-5/1024*cos(d*x+c)*sin(d*x+c)/a/d-
5/1536*cos(d*x+c)^3*sin(d*x+c)/a/d-1/384*cos(d*x+c)^5*sin(d*x+c)/a/d+1/64*cos(d*x+c)^7*sin(d*x+c)/a/d+1/24*cos
(d*x+c)^7*sin(d*x+c)^3/a/d+1/12*cos(d*x+c)^7*sin(d*x+c)^5/a/d

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2918, 2645, 276, 2648, 2715, 8} \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\cos ^{11}(c+d x)}{11 a d}+\frac {2 \cos ^9(c+d x)}{9 a d}-\frac {\cos ^7(c+d x)}{7 a d}+\frac {\sin ^5(c+d x) \cos ^7(c+d x)}{12 a d}+\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{24 a d}+\frac {\sin (c+d x) \cos ^7(c+d x)}{64 a d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{384 a d}-\frac {5 \sin (c+d x) \cos ^3(c+d x)}{1536 a d}-\frac {5 \sin (c+d x) \cos (c+d x)}{1024 a d}-\frac {5 x}{1024 a} \]

[In]

Int[(Cos[c + d*x]^8*Sin[c + d*x]^5)/(a + a*Sin[c + d*x]),x]

[Out]

(-5*x)/(1024*a) - Cos[c + d*x]^7/(7*a*d) + (2*Cos[c + d*x]^9)/(9*a*d) - Cos[c + d*x]^11/(11*a*d) - (5*Cos[c +
d*x]*Sin[c + d*x])/(1024*a*d) - (5*Cos[c + d*x]^3*Sin[c + d*x])/(1536*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(38
4*a*d) + (Cos[c + d*x]^7*Sin[c + d*x])/(64*a*d) + (Cos[c + d*x]^7*Sin[c + d*x]^3)/(24*a*d) + (Cos[c + d*x]^7*S
in[c + d*x]^5)/(12*a*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2645

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(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])

Rule 2648

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] + Dist[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]

Rule 2715

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] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2918

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \cos ^6(c+d x) \sin ^5(c+d x) \, dx}{a}-\frac {\int \cos ^6(c+d x) \sin ^6(c+d x) \, dx}{a} \\ & = \frac {\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}-\frac {5 \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx}{12 a}-\frac {\text {Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{a d} \\ & = \frac {\cos ^7(c+d x) \sin ^3(c+d x)}{24 a d}+\frac {\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}-\frac {\int \cos ^6(c+d x) \sin ^2(c+d x) \, dx}{8 a}-\frac {\text {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{a d} \\ & = -\frac {\cos ^7(c+d x)}{7 a d}+\frac {2 \cos ^9(c+d x)}{9 a d}-\frac {\cos ^{11}(c+d x)}{11 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{64 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{24 a d}+\frac {\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}-\frac {\int \cos ^6(c+d x) \, dx}{64 a} \\ & = -\frac {\cos ^7(c+d x)}{7 a d}+\frac {2 \cos ^9(c+d x)}{9 a d}-\frac {\cos ^{11}(c+d x)}{11 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{384 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{64 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{24 a d}+\frac {\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}-\frac {5 \int \cos ^4(c+d x) \, dx}{384 a} \\ & = -\frac {\cos ^7(c+d x)}{7 a d}+\frac {2 \cos ^9(c+d x)}{9 a d}-\frac {\cos ^{11}(c+d x)}{11 a d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{1536 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{384 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{64 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{24 a d}+\frac {\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}-\frac {5 \int \cos ^2(c+d x) \, dx}{512 a} \\ & = -\frac {\cos ^7(c+d x)}{7 a d}+\frac {2 \cos ^9(c+d x)}{9 a d}-\frac {\cos ^{11}(c+d x)}{11 a d}-\frac {5 \cos (c+d x) \sin (c+d x)}{1024 a d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{1536 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{384 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{64 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{24 a d}+\frac {\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}-\frac {5 \int 1 \, dx}{1024 a} \\ & = -\frac {5 x}{1024 a}-\frac {\cos ^7(c+d x)}{7 a d}+\frac {2 \cos ^9(c+d x)}{9 a d}-\frac {\cos ^{11}(c+d x)}{11 a d}-\frac {5 \cos (c+d x) \sin (c+d x)}{1024 a d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{1536 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{384 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{64 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{24 a d}+\frac {\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(518\) vs. \(2(209)=418\).

Time = 9.53 (sec) , antiderivative size = 518, normalized size of antiderivative = 2.48 \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {55440 d x \cos \left (\frac {c}{2}\right )+55440 \cos \left (\frac {c}{2}+d x\right )+55440 \cos \left (\frac {3 c}{2}+d x\right )+18480 \cos \left (\frac {5 c}{2}+3 d x\right )+18480 \cos \left (\frac {7 c}{2}+3 d x\right )-10395 \cos \left (\frac {7 c}{2}+4 d x\right )+10395 \cos \left (\frac {9 c}{2}+4 d x\right )-5544 \cos \left (\frac {9 c}{2}+5 d x\right )-5544 \cos \left (\frac {11 c}{2}+5 d x\right )-3960 \cos \left (\frac {13 c}{2}+7 d x\right )-3960 \cos \left (\frac {15 c}{2}+7 d x\right )+2079 \cos \left (\frac {15 c}{2}+8 d x\right )-2079 \cos \left (\frac {17 c}{2}+8 d x\right )+616 \cos \left (\frac {17 c}{2}+9 d x\right )+616 \cos \left (\frac {19 c}{2}+9 d x\right )+504 \cos \left (\frac {21 c}{2}+11 d x\right )+504 \cos \left (\frac {23 c}{2}+11 d x\right )-231 \cos \left (\frac {23 c}{2}+12 d x\right )+231 \cos \left (\frac {25 c}{2}+12 d x\right )+99792 \sin \left (\frac {c}{2}\right )+55440 d x \sin \left (\frac {c}{2}\right )-55440 \sin \left (\frac {c}{2}+d x\right )+55440 \sin \left (\frac {3 c}{2}+d x\right )-18480 \sin \left (\frac {5 c}{2}+3 d x\right )+18480 \sin \left (\frac {7 c}{2}+3 d x\right )-10395 \sin \left (\frac {7 c}{2}+4 d x\right )-10395 \sin \left (\frac {9 c}{2}+4 d x\right )+5544 \sin \left (\frac {9 c}{2}+5 d x\right )-5544 \sin \left (\frac {11 c}{2}+5 d x\right )+3960 \sin \left (\frac {13 c}{2}+7 d x\right )-3960 \sin \left (\frac {15 c}{2}+7 d x\right )+2079 \sin \left (\frac {15 c}{2}+8 d x\right )+2079 \sin \left (\frac {17 c}{2}+8 d x\right )-616 \sin \left (\frac {17 c}{2}+9 d x\right )+616 \sin \left (\frac {19 c}{2}+9 d x\right )-504 \sin \left (\frac {21 c}{2}+11 d x\right )+504 \sin \left (\frac {23 c}{2}+11 d x\right )-231 \sin \left (\frac {23 c}{2}+12 d x\right )-231 \sin \left (\frac {25 c}{2}+12 d x\right )}{11354112 a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]

[In]

Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^5)/(a + a*Sin[c + d*x]),x]

[Out]

-1/11354112*(55440*d*x*Cos[c/2] + 55440*Cos[c/2 + d*x] + 55440*Cos[(3*c)/2 + d*x] + 18480*Cos[(5*c)/2 + 3*d*x]
+ 18480*Cos[(7*c)/2 + 3*d*x] - 10395*Cos[(7*c)/2 + 4*d*x] + 10395*Cos[(9*c)/2 + 4*d*x] - 5544*Cos[(9*c)/2 + 5
*d*x] - 5544*Cos[(11*c)/2 + 5*d*x] - 3960*Cos[(13*c)/2 + 7*d*x] - 3960*Cos[(15*c)/2 + 7*d*x] + 2079*Cos[(15*c)
/2 + 8*d*x] - 2079*Cos[(17*c)/2 + 8*d*x] + 616*Cos[(17*c)/2 + 9*d*x] + 616*Cos[(19*c)/2 + 9*d*x] + 504*Cos[(21
*c)/2 + 11*d*x] + 504*Cos[(23*c)/2 + 11*d*x] - 231*Cos[(23*c)/2 + 12*d*x] + 231*Cos[(25*c)/2 + 12*d*x] + 99792
*Sin[c/2] + 55440*d*x*Sin[c/2] - 55440*Sin[c/2 + d*x] + 55440*Sin[(3*c)/2 + d*x] - 18480*Sin[(5*c)/2 + 3*d*x]
+ 18480*Sin[(7*c)/2 + 3*d*x] - 10395*Sin[(7*c)/2 + 4*d*x] - 10395*Sin[(9*c)/2 + 4*d*x] + 5544*Sin[(9*c)/2 + 5*
d*x] - 5544*Sin[(11*c)/2 + 5*d*x] + 3960*Sin[(13*c)/2 + 7*d*x] - 3960*Sin[(15*c)/2 + 7*d*x] + 2079*Sin[(15*c)/
2 + 8*d*x] + 2079*Sin[(17*c)/2 + 8*d*x] - 616*Sin[(17*c)/2 + 9*d*x] + 616*Sin[(19*c)/2 + 9*d*x] - 504*Sin[(21*
c)/2 + 11*d*x] + 504*Sin[(23*c)/2 + 11*d*x] - 231*Sin[(23*c)/2 + 12*d*x] - 231*Sin[(25*c)/2 + 12*d*x])/(a*d*(C
os[c/2] + Sin[c/2]))

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.53

method	result	size
parallelrisch	\(\frac {-27720 d x +3960 \cos \left (7 d x +7 c \right )+5544 \cos \left (5 d x +5 c \right )-18480 \cos \left (3 d x +3 c \right )-55440 \cos \left (d x +c \right )+231 \sin \left (12 d x +12 c \right )-504 \cos \left (11 d x +11 c \right )-616 \cos \left (9 d x +9 c \right )-2079 \sin \left (8 d x +8 c \right )+10395 \sin \left (4 d x +4 c \right )-65536}{5677056 d a}\)	\(111\)
risch	\(-\frac {5 x}{1024 a}-\frac {5 \cos \left (d x +c \right )}{512 a d}+\frac {\sin \left (12 d x +12 c \right )}{24576 d a}-\frac {\cos \left (11 d x +11 c \right )}{11264 a d}-\frac {\cos \left (9 d x +9 c \right )}{9216 a d}-\frac {3 \sin \left (8 d x +8 c \right )}{8192 d a}+\frac {5 \cos \left (7 d x +7 c \right )}{7168 a d}+\frac {\cos \left (5 d x +5 c \right )}{1024 a d}+\frac {15 \sin \left (4 d x +4 c \right )}{8192 d a}-\frac {5 \cos \left (3 d x +3 c \right )}{1536 a d}\)	\(158\)
derivativedivides	\(\frac {\frac {64 \left (-\frac {1}{2772}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32768}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{231}+\frac {175 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{98304}-\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{42}+\frac {311 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32768}+\frac {11 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{126}-\frac {8361 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32768}-\frac {3 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {42259 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{49152}+\frac {3 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{14}-\frac {25295 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16384}-\frac {\left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}+\frac {25295 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16384}-\frac {\left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {42259 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{49152}+\frac {\left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {8361 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32768}-\frac {\left (\tan ^{18}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {311 \left (\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32768}-\frac {175 \left (\tan ^{21}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{98304}-\frac {5 \left (\tan ^{23}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32768}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{12}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}}{d a}\)	\(311\)
default	\(\frac {\frac {64 \left (-\frac {1}{2772}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32768}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{231}+\frac {175 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{98304}-\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{42}+\frac {311 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32768}+\frac {11 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{126}-\frac {8361 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32768}-\frac {3 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {42259 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{49152}+\frac {3 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{14}-\frac {25295 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16384}-\frac {\left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}+\frac {25295 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16384}-\frac {\left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {42259 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{49152}+\frac {\left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {8361 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32768}-\frac {\left (\tan ^{18}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {311 \left (\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32768}-\frac {175 \left (\tan ^{21}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{98304}-\frac {5 \left (\tan ^{23}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32768}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{12}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}}{d a}\)	\(311\)

[In]

int(cos(d*x+c)^8*sin(d*x+c)^5/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/5677056*(-27720*d*x+3960*cos(7*d*x+7*c)+5544*cos(5*d*x+5*c)-18480*cos(3*d*x+3*c)-55440*cos(d*x+c)+231*sin(12
*d*x+12*c)-504*cos(11*d*x+11*c)-616*cos(9*d*x+9*c)-2079*sin(8*d*x+8*c)+10395*sin(4*d*x+4*c)-65536)/d/a

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.53 \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {64512 \, \cos \left (d x + c\right )^{11} - 157696 \, \cos \left (d x + c\right )^{9} + 101376 \, \cos \left (d x + c\right )^{7} + 3465 \, d x - 231 \, {\left (256 \, \cos \left (d x + c\right )^{11} - 640 \, \cos \left (d x + c\right )^{9} + 432 \, \cos \left (d x + c\right )^{7} - 8 \, \cos \left (d x + c\right )^{5} - 10 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{709632 \, a d} \]

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/709632*(64512*cos(d*x + c)^11 - 157696*cos(d*x + c)^9 + 101376*cos(d*x + c)^7 + 3465*d*x - 231*(256*cos(d*x
+ c)^11 - 640*cos(d*x + c)^9 + 432*cos(d*x + c)^7 - 8*cos(d*x + c)^5 - 10*cos(d*x + c)^3 - 15*cos(d*x + c))*s
in(d*x + c))/(a*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**8*sin(d*x+c)**5/(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 705 vs. \(2 (189) = 378\).

Time = 0.31 (sec) , antiderivative size = 705, normalized size of antiderivative = 3.37 \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/354816*((3465*sin(d*x + c)/(cos(d*x + c) + 1) - 98304*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 40425*sin(d*x +
c)^3/(cos(d*x + c) + 1)^3 - 540672*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 215523*sin(d*x + c)^5/(cos(d*x + c) +
1)^5 + 1982464*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 5794173*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 9732096*si
n(d*x + c)^8/(cos(d*x + c) + 1)^8 + 19523658*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 4866048*sin(d*x + c)^10/(co
s(d*x + c) + 1)^10 - 35058870*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 3784704*sin(d*x + c)^12/(cos(d*x + c) +
1)^12 + 35058870*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 11354112*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 - 1952
3658*sin(d*x + c)^15/(cos(d*x + c) + 1)^15 + 5677056*sin(d*x + c)^16/(cos(d*x + c) + 1)^16 + 5794173*sin(d*x +
c)^17/(cos(d*x + c) + 1)^17 - 3784704*sin(d*x + c)^18/(cos(d*x + c) + 1)^18 - 215523*sin(d*x + c)^19/(cos(d*x
+ c) + 1)^19 - 40425*sin(d*x + c)^21/(cos(d*x + c) + 1)^21 - 3465*sin(d*x + c)^23/(cos(d*x + c) + 1)^23 - 819
2)/(a + 12*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 66*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 220*a*sin(d*x +
c)^6/(cos(d*x + c) + 1)^6 + 495*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 792*a*sin(d*x + c)^10/(cos(d*x + c) +
1)^10 + 924*a*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 792*a*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + 495*a*sin(
d*x + c)^16/(cos(d*x + c) + 1)^16 + 220*a*sin(d*x + c)^18/(cos(d*x + c) + 1)^18 + 66*a*sin(d*x + c)^20/(cos(d*
x + c) + 1)^20 + 12*a*sin(d*x + c)^22/(cos(d*x + c) + 1)^22 + a*sin(d*x + c)^24/(cos(d*x + c) + 1)^24) - 3465*
arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.48 \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {3465 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (3465 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{23} + 40425 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{21} + 215523 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{19} + 3784704 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{18} - 5794173 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} - 5677056 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} + 19523658 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 11354112 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 35058870 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 3784704 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 35058870 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 4866048 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 19523658 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 9732096 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 5794173 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1982464 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 215523 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 540672 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 40425 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 98304 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3465 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8192\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{12} a}}{709632 \, d} \]

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/709632*(3465*(d*x + c)/a + 2*(3465*tan(1/2*d*x + 1/2*c)^23 + 40425*tan(1/2*d*x + 1/2*c)^21 + 215523*tan(1/2
*d*x + 1/2*c)^19 + 3784704*tan(1/2*d*x + 1/2*c)^18 - 5794173*tan(1/2*d*x + 1/2*c)^17 - 5677056*tan(1/2*d*x + 1
/2*c)^16 + 19523658*tan(1/2*d*x + 1/2*c)^15 + 11354112*tan(1/2*d*x + 1/2*c)^14 - 35058870*tan(1/2*d*x + 1/2*c)
^13 + 3784704*tan(1/2*d*x + 1/2*c)^12 + 35058870*tan(1/2*d*x + 1/2*c)^11 - 4866048*tan(1/2*d*x + 1/2*c)^10 - 1
9523658*tan(1/2*d*x + 1/2*c)^9 + 9732096*tan(1/2*d*x + 1/2*c)^8 + 5794173*tan(1/2*d*x + 1/2*c)^7 - 1982464*tan
(1/2*d*x + 1/2*c)^6 - 215523*tan(1/2*d*x + 1/2*c)^5 + 540672*tan(1/2*d*x + 1/2*c)^4 - 40425*tan(1/2*d*x + 1/2*
c)^3 + 98304*tan(1/2*d*x + 1/2*c)^2 - 3465*tan(1/2*d*x + 1/2*c) + 8192)/((tan(1/2*d*x + 1/2*c)^2 + 1)^12*a))/d

Mupad [B] (verification not implemented)

Time = 13.23 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.45 \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5\,x}{1024\,a}-\frac {\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{23}}{512}+\frac {175\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{1536}+\frac {311\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{512}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}}{3}-\frac {8361\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{512}-16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+\frac {42259\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{768}+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-\frac {25295\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{256}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}+\frac {25295\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{256}-\frac {96\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{7}-\frac {42259\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{768}+\frac {192\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{7}+\frac {8361\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{512}-\frac {352\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{63}-\frac {311\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{512}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{21}-\frac {175\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{1536}+\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{231}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{512}+\frac {16}{693}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{12}} \]

[In]

int((cos(c + d*x)^8*sin(c + d*x)^5)/(a + a*sin(c + d*x)),x)

[Out]

- (5*x)/(1024*a) - ((64*tan(c/2 + (d*x)/2)^2)/231 - (5*tan(c/2 + (d*x)/2))/512 - (175*tan(c/2 + (d*x)/2)^3)/15
36 + (32*tan(c/2 + (d*x)/2)^4)/21 - (311*tan(c/2 + (d*x)/2)^5)/512 - (352*tan(c/2 + (d*x)/2)^6)/63 + (8361*tan
(c/2 + (d*x)/2)^7)/512 + (192*tan(c/2 + (d*x)/2)^8)/7 - (42259*tan(c/2 + (d*x)/2)^9)/768 - (96*tan(c/2 + (d*x)
/2)^10)/7 + (25295*tan(c/2 + (d*x)/2)^11)/256 + (32*tan(c/2 + (d*x)/2)^12)/3 - (25295*tan(c/2 + (d*x)/2)^13)/2
56 + 32*tan(c/2 + (d*x)/2)^14 + (42259*tan(c/2 + (d*x)/2)^15)/768 - 16*tan(c/2 + (d*x)/2)^16 - (8361*tan(c/2 +
(d*x)/2)^17)/512 + (32*tan(c/2 + (d*x)/2)^18)/3 + (311*tan(c/2 + (d*x)/2)^19)/512 + (175*tan(c/2 + (d*x)/2)^2
1)/1536 + (5*tan(c/2 + (d*x)/2)^23)/512 + 16/693)/(a*d*(tan(c/2 + (d*x)/2)^2 + 1)^12)