\(\int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [722]

Optimal result

Integrand size = 29, antiderivative size = 203 \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {3 x}{128 a^2}-\frac {2 \cos ^5(c+d x)}{5 a^2 d}+\frac {5 \cos ^7(c+d x)}{7 a^2 d}-\frac {4 \cos ^9(c+d x)}{9 a^2 d}+\frac {\cos ^{11}(c+d x)}{11 a^2 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{128 a^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{64 a^2 d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{16 a^2 d}+\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{8 a^2 d}+\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{5 a^2 d} \]

[Out]

-3/128*x/a^2-2/5*cos(d*x+c)^5/a^2/d+5/7*cos(d*x+c)^7/a^2/d-4/9*cos(d*x+c)^9/a^2/d+1/11*cos(d*x+c)^11/a^2/d-3/1
28*cos(d*x+c)*sin(d*x+c)/a^2/d-1/64*cos(d*x+c)^3*sin(d*x+c)/a^2/d+1/16*cos(d*x+c)^5*sin(d*x+c)/a^2/d+1/8*cos(d
*x+c)^5*sin(d*x+c)^3/a^2/d+1/5*cos(d*x+c)^5*sin(d*x+c)^5/a^2/d

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2954, 2952, 2645, 276, 2648, 2715, 8} \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\cos ^{11}(c+d x)}{11 a^2 d}-\frac {4 \cos ^9(c+d x)}{9 a^2 d}+\frac {5 \cos ^7(c+d x)}{7 a^2 d}-\frac {2 \cos ^5(c+d x)}{5 a^2 d}+\frac {\sin ^5(c+d x) \cos ^5(c+d x)}{5 a^2 d}+\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 a^2 d}+\frac {\sin (c+d x) \cos ^5(c+d x)}{16 a^2 d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{64 a^2 d}-\frac {3 \sin (c+d x) \cos (c+d x)}{128 a^2 d}-\frac {3 x}{128 a^2} \]

[In]

Int[(Cos[c + d*x]^8*Sin[c + d*x]^5)/(a + a*Sin[c + d*x])^2,x]

[Out]

(-3*x)/(128*a^2) - (2*Cos[c + d*x]^5)/(5*a^2*d) + (5*Cos[c + d*x]^7)/(7*a^2*d) - (4*Cos[c + d*x]^9)/(9*a^2*d)
+ Cos[c + d*x]^11/(11*a^2*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(128*a^2*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(64*a
^2*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(16*a^2*d) + (Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*a^2*d) + (Cos[c + d*x]^5
*Sin[c + d*x]^5)/(5*a^2*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 2952

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2954

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Dist[(a/g)^(2*m), Int[(g*Cos[e + f*x])^(2*m + p)*((d*Sin[e + f*x])^n/(a - b*Sin[e +
f*x])^m), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \cos ^4(c+d x) \sin ^5(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4} \\ & = \frac {\int \left (a^2 \cos ^4(c+d x) \sin ^5(c+d x)-2 a^2 \cos ^4(c+d x) \sin ^6(c+d x)+a^2 \cos ^4(c+d x) \sin ^7(c+d x)\right ) \, dx}{a^4} \\ & = \frac {\int \cos ^4(c+d x) \sin ^5(c+d x) \, dx}{a^2}+\frac {\int \cos ^4(c+d x) \sin ^7(c+d x) \, dx}{a^2}-\frac {2 \int \cos ^4(c+d x) \sin ^6(c+d x) \, dx}{a^2} \\ & = \frac {\cos ^5(c+d x) \sin ^5(c+d x)}{5 a^2 d}-\frac {\int \cos ^4(c+d x) \sin ^4(c+d x) \, dx}{a^2}-\frac {\text {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac {\text {Subst}\left (\int x^4 \left (1-x^2\right )^3 \, dx,x,\cos (c+d x)\right )}{a^2 d} \\ & = \frac {\cos ^5(c+d x) \sin ^3(c+d x)}{8 a^2 d}+\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{5 a^2 d}-\frac {3 \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx}{8 a^2}-\frac {\text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac {\text {Subst}\left (\int \left (x^4-3 x^6+3 x^8-x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d} \\ & = -\frac {2 \cos ^5(c+d x)}{5 a^2 d}+\frac {5 \cos ^7(c+d x)}{7 a^2 d}-\frac {4 \cos ^9(c+d x)}{9 a^2 d}+\frac {\cos ^{11}(c+d x)}{11 a^2 d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{16 a^2 d}+\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{8 a^2 d}+\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{5 a^2 d}-\frac {\int \cos ^4(c+d x) \, dx}{16 a^2} \\ & = -\frac {2 \cos ^5(c+d x)}{5 a^2 d}+\frac {5 \cos ^7(c+d x)}{7 a^2 d}-\frac {4 \cos ^9(c+d x)}{9 a^2 d}+\frac {\cos ^{11}(c+d x)}{11 a^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{64 a^2 d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{16 a^2 d}+\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{8 a^2 d}+\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{5 a^2 d}-\frac {3 \int \cos ^2(c+d x) \, dx}{64 a^2} \\ & = -\frac {2 \cos ^5(c+d x)}{5 a^2 d}+\frac {5 \cos ^7(c+d x)}{7 a^2 d}-\frac {4 \cos ^9(c+d x)}{9 a^2 d}+\frac {\cos ^{11}(c+d x)}{11 a^2 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{128 a^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{64 a^2 d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{16 a^2 d}+\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{8 a^2 d}+\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{5 a^2 d}-\frac {3 \int 1 \, dx}{128 a^2} \\ & = -\frac {3 x}{128 a^2}-\frac {2 \cos ^5(c+d x)}{5 a^2 d}+\frac {5 \cos ^7(c+d x)}{7 a^2 d}-\frac {4 \cos ^9(c+d x)}{9 a^2 d}+\frac {\cos ^{11}(c+d x)}{11 a^2 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{128 a^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{64 a^2 d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{16 a^2 d}+\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{8 a^2 d}+\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{5 a^2 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(638\) vs. \(2(203)=406\).

Time = 8.09 (sec) , antiderivative size = 638, normalized size of antiderivative = 3.14 \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {4620 (-5+222 c-36 d x) \cos \left (\frac {c}{2}\right )-131670 \cos \left (\frac {c}{2}+d x\right )-131670 \cos \left (\frac {3 c}{2}+d x\right )+13860 \cos \left (\frac {3 c}{2}+2 d x\right )-13860 \cos \left (\frac {5 c}{2}+2 d x\right )-25410 \cos \left (\frac {5 c}{2}+3 d x\right )-25410 \cos \left (\frac {7 c}{2}+3 d x\right )+27720 \cos \left (\frac {7 c}{2}+4 d x\right )-27720 \cos \left (\frac {9 c}{2}+4 d x\right )+18711 \cos \left (\frac {9 c}{2}+5 d x\right )+18711 \cos \left (\frac {11 c}{2}+5 d x\right )-6930 \cos \left (\frac {11 c}{2}+6 d x\right )+6930 \cos \left (\frac {13 c}{2}+6 d x\right )+1485 \cos \left (\frac {13 c}{2}+7 d x\right )+1485 \cos \left (\frac {15 c}{2}+7 d x\right )-3465 \cos \left (\frac {15 c}{2}+8 d x\right )+3465 \cos \left (\frac {17 c}{2}+8 d x\right )-2695 \cos \left (\frac {17 c}{2}+9 d x\right )-2695 \cos \left (\frac {19 c}{2}+9 d x\right )+1386 \cos \left (\frac {19 c}{2}+10 d x\right )-1386 \cos \left (\frac {21 c}{2}+10 d x\right )+315 \cos \left (\frac {21 c}{2}+11 d x\right )+315 \cos \left (\frac {23 c}{2}+11 d x\right )-646800 \sin \left (\frac {c}{2}\right )+1025640 c \sin \left (\frac {c}{2}\right )-166320 d x \sin \left (\frac {c}{2}\right )+131670 \sin \left (\frac {c}{2}+d x\right )-131670 \sin \left (\frac {3 c}{2}+d x\right )+13860 \sin \left (\frac {3 c}{2}+2 d x\right )+13860 \sin \left (\frac {5 c}{2}+2 d x\right )+25410 \sin \left (\frac {5 c}{2}+3 d x\right )-25410 \sin \left (\frac {7 c}{2}+3 d x\right )+27720 \sin \left (\frac {7 c}{2}+4 d x\right )+27720 \sin \left (\frac {9 c}{2}+4 d x\right )-18711 \sin \left (\frac {9 c}{2}+5 d x\right )+18711 \sin \left (\frac {11 c}{2}+5 d x\right )-6930 \sin \left (\frac {11 c}{2}+6 d x\right )-6930 \sin \left (\frac {13 c}{2}+6 d x\right )-1485 \sin \left (\frac {13 c}{2}+7 d x\right )+1485 \sin \left (\frac {15 c}{2}+7 d x\right )-3465 \sin \left (\frac {15 c}{2}+8 d x\right )-3465 \sin \left (\frac {17 c}{2}+8 d x\right )+2695 \sin \left (\frac {17 c}{2}+9 d x\right )-2695 \sin \left (\frac {19 c}{2}+9 d x\right )+1386 \sin \left (\frac {19 c}{2}+10 d x\right )+1386 \sin \left (\frac {21 c}{2}+10 d x\right )-315 \sin \left (\frac {21 c}{2}+11 d x\right )+315 \sin \left (\frac {23 c}{2}+11 d x\right )}{7096320 a^2 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])^2,x]

[Out]

(4620*(-5 + 222*c - 36*d*x)*Cos[c/2] - 131670*Cos[c/2 + d*x] - 131670*Cos[(3*c)/2 + d*x] + 13860*Cos[(3*c)/2 +
2*d*x] - 13860*Cos[(5*c)/2 + 2*d*x] - 25410*Cos[(5*c)/2 + 3*d*x] - 25410*Cos[(7*c)/2 + 3*d*x] + 27720*Cos[(7*
c)/2 + 4*d*x] - 27720*Cos[(9*c)/2 + 4*d*x] + 18711*Cos[(9*c)/2 + 5*d*x] + 18711*Cos[(11*c)/2 + 5*d*x] - 6930*C
os[(11*c)/2 + 6*d*x] + 6930*Cos[(13*c)/2 + 6*d*x] + 1485*Cos[(13*c)/2 + 7*d*x] + 1485*Cos[(15*c)/2 + 7*d*x] -
3465*Cos[(15*c)/2 + 8*d*x] + 3465*Cos[(17*c)/2 + 8*d*x] - 2695*Cos[(17*c)/2 + 9*d*x] - 2695*Cos[(19*c)/2 + 9*d
*x] + 1386*Cos[(19*c)/2 + 10*d*x] - 1386*Cos[(21*c)/2 + 10*d*x] + 315*Cos[(21*c)/2 + 11*d*x] + 315*Cos[(23*c)/
2 + 11*d*x] - 646800*Sin[c/2] + 1025640*c*Sin[c/2] - 166320*d*x*Sin[c/2] + 131670*Sin[c/2 + d*x] - 131670*Sin[
(3*c)/2 + d*x] + 13860*Sin[(3*c)/2 + 2*d*x] + 13860*Sin[(5*c)/2 + 2*d*x] + 25410*Sin[(5*c)/2 + 3*d*x] - 25410*
Sin[(7*c)/2 + 3*d*x] + 27720*Sin[(7*c)/2 + 4*d*x] + 27720*Sin[(9*c)/2 + 4*d*x] - 18711*Sin[(9*c)/2 + 5*d*x] +
18711*Sin[(11*c)/2 + 5*d*x] - 6930*Sin[(11*c)/2 + 6*d*x] - 6930*Sin[(13*c)/2 + 6*d*x] - 1485*Sin[(13*c)/2 + 7*
d*x] + 1485*Sin[(15*c)/2 + 7*d*x] - 3465*Sin[(15*c)/2 + 8*d*x] - 3465*Sin[(17*c)/2 + 8*d*x] + 2695*Sin[(17*c)/
2 + 9*d*x] - 2695*Sin[(19*c)/2 + 9*d*x] + 1386*Sin[(19*c)/2 + 10*d*x] + 1386*Sin[(21*c)/2 + 10*d*x] - 315*Sin[
(21*c)/2 + 11*d*x] + 315*Sin[(23*c)/2 + 11*d*x])/(7096320*a^2*d*(Cos[c/2] + Sin[c/2]))

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.66

method	result	size
parallelrisch	\(\frac {-83160 d x +1485 \cos \left (7 d x +7 c \right )+18711 \cos \left (5 d x +5 c \right )-25410 \cos \left (3 d x +3 c \right )-131670 \cos \left (d x +c \right )+315 \cos \left (11 d x +11 c \right )+1386 \sin \left (10 d x +10 c \right )-2695 \cos \left (9 d x +9 c \right )-3465 \sin \left (8 d x +8 c \right )-6930 \sin \left (6 d x +6 c \right )+27720 \sin \left (4 d x +4 c \right )+13860 \sin \left (2 d x +2 c \right )-139264}{3548160 d \,a^{2}}\)	\(133\)
risch	\(-\frac {19 \cos \left (d x +c \right )}{512 a^{2} d}-\frac {3 x}{128 a^{2}}+\frac {\cos \left (11 d x +11 c \right )}{11264 d \,a^{2}}+\frac {\sin \left (10 d x +10 c \right )}{2560 d \,a^{2}}-\frac {7 \cos \left (9 d x +9 c \right )}{9216 d \,a^{2}}-\frac {\sin \left (8 d x +8 c \right )}{1024 d \,a^{2}}+\frac {3 \cos \left (7 d x +7 c \right )}{7168 d \,a^{2}}-\frac {\sin \left (6 d x +6 c \right )}{512 d \,a^{2}}+\frac {27 \cos \left (5 d x +5 c \right )}{5120 d \,a^{2}}+\frac {\sin \left (4 d x +4 c \right )}{128 d \,a^{2}}-\frac {11 \cos \left (3 d x +3 c \right )}{1536 d \,a^{2}}+\frac {\sin \left (2 d x +2 c \right )}{256 d \,a^{2}}\)	\(192\)
derivativedivides	\(\frac {\frac {64 \left (-\frac {17}{13860}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4096}-\frac {17 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1260}+\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}-\frac {17 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{252}+\frac {773 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20480}-\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{28}-\frac {37 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80}+\frac {5 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{28}+\frac {1207 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}-\frac {29 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20}+\frac {53 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60}-\frac {1207 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}-\frac {7 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12}+\frac {37 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80}-\frac {\left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {773 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20480}-\frac {\left (\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}-\frac {3 \left (\tan ^{21}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4096}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d \,a^{2}}\)	\(272\)
default	\(\frac {\frac {64 \left (-\frac {17}{13860}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4096}-\frac {17 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1260}+\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}-\frac {17 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{252}+\frac {773 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20480}-\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{28}-\frac {37 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80}+\frac {5 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{28}+\frac {1207 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}-\frac {29 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20}+\frac {53 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60}-\frac {1207 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}-\frac {7 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12}+\frac {37 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80}-\frac {\left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {773 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20480}-\frac {\left (\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}-\frac {3 \left (\tan ^{21}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4096}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d \,a^{2}}\)	\(272\)

[In]

int(cos(d*x+c)^8*sin(d*x+c)^5/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/3548160*(-83160*d*x+1485*cos(7*d*x+7*c)+18711*cos(5*d*x+5*c)-25410*cos(3*d*x+3*c)-131670*cos(d*x+c)+315*cos(
11*d*x+11*c)+1386*sin(10*d*x+10*c)-2695*cos(9*d*x+9*c)-3465*sin(8*d*x+8*c)-6930*sin(6*d*x+6*c)+27720*sin(4*d*x
+4*c)+13860*sin(2*d*x+2*c)-139264)/d/a^2

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.54 \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {40320 \, \cos \left (d x + c\right )^{11} - 197120 \, \cos \left (d x + c\right )^{9} + 316800 \, \cos \left (d x + c\right )^{7} - 177408 \, \cos \left (d x + c\right )^{5} - 10395 \, d x + 693 \, {\left (128 \, \cos \left (d x + c\right )^{9} - 336 \, \cos \left (d x + c\right )^{7} + 248 \, \cos \left (d x + c\right )^{5} - 10 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{443520 \, a^{2} d} \]

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^5/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/443520*(40320*cos(d*x + c)^11 - 197120*cos(d*x + c)^9 + 316800*cos(d*x + c)^7 - 177408*cos(d*x + c)^5 - 1039
5*d*x + 693*(128*cos(d*x + c)^9 - 336*cos(d*x + c)^7 + 248*cos(d*x + c)^5 - 10*cos(d*x + c)^3 - 15*cos(d*x + c
))*sin(d*x + c))/(a^2*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**8*sin(d*x+c)**5/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 648 vs. \(2 (183) = 366\).

Time = 0.31 (sec) , antiderivative size = 648, normalized size of antiderivative = 3.19 \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {\frac {10395 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {191488 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {110880 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {957440 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {535689 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {506880 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {6564096 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {2534400 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {8364510 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {20579328 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {12536832 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {8364510 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {8279040 \, \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {6564096 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} - \frac {2365440 \, \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} - \frac {535689 \, \sin \left (d x + c\right )^{17}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{17}} - \frac {110880 \, \sin \left (d x + c\right )^{19}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{19}} - \frac {10395 \, \sin \left (d x + c\right )^{21}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{21}} - 17408}{a^{2} + \frac {11 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {55 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {165 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {330 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {462 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {462 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {330 \, a^{2} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {165 \, a^{2} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} + \frac {55 \, a^{2} \sin \left (d x + c\right )^{18}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{18}} + \frac {11 \, a^{2} \sin \left (d x + c\right )^{20}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{20}} + \frac {a^{2} \sin \left (d x + c\right )^{22}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{22}}} - \frac {10395 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{221760 \, d} \]

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^5/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/221760*((10395*sin(d*x + c)/(cos(d*x + c) + 1) - 191488*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 110880*sin(d*x
+ c)^3/(cos(d*x + c) + 1)^3 - 957440*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 535689*sin(d*x + c)^5/(cos(d*x + c
) + 1)^5 - 506880*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 6564096*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 2534400*
sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 8364510*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 20579328*sin(d*x + c)^10/(
cos(d*x + c) + 1)^10 + 12536832*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 - 8364510*sin(d*x + c)^13/(cos(d*x + c)
+ 1)^13 - 8279040*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + 6564096*sin(d*x + c)^15/(cos(d*x + c) + 1)^15 - 2365
440*sin(d*x + c)^16/(cos(d*x + c) + 1)^16 - 535689*sin(d*x + c)^17/(cos(d*x + c) + 1)^17 - 110880*sin(d*x + c)
^19/(cos(d*x + c) + 1)^19 - 10395*sin(d*x + c)^21/(cos(d*x + c) + 1)^21 - 17408)/(a^2 + 11*a^2*sin(d*x + c)^2/
(cos(d*x + c) + 1)^2 + 55*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 165*a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^
6 + 330*a^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 462*a^2*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 462*a^2*sin(
d*x + c)^12/(cos(d*x + c) + 1)^12 + 330*a^2*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + 165*a^2*sin(d*x + c)^16/(c
os(d*x + c) + 1)^16 + 55*a^2*sin(d*x + c)^18/(cos(d*x + c) + 1)^18 + 11*a^2*sin(d*x + c)^20/(cos(d*x + c) + 1)
^20 + a^2*sin(d*x + c)^22/(cos(d*x + c) + 1)^22) - 10395*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.33 \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {10395 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (10395 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{21} + 110880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{19} + 535689 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 2365440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} - 6564096 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 8279040 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} + 8364510 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 12536832 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 20579328 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 8364510 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 2534400 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 6564096 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 506880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 535689 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 957440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 110880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 191488 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10395 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17408\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{11} a^{2}}}{443520 \, d} \]

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^5/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/443520*(10395*(d*x + c)/a^2 + 2*(10395*tan(1/2*d*x + 1/2*c)^21 + 110880*tan(1/2*d*x + 1/2*c)^19 + 535689*ta
n(1/2*d*x + 1/2*c)^17 + 2365440*tan(1/2*d*x + 1/2*c)^16 - 6564096*tan(1/2*d*x + 1/2*c)^15 + 8279040*tan(1/2*d*
x + 1/2*c)^14 + 8364510*tan(1/2*d*x + 1/2*c)^13 - 12536832*tan(1/2*d*x + 1/2*c)^12 + 20579328*tan(1/2*d*x + 1/
2*c)^10 - 8364510*tan(1/2*d*x + 1/2*c)^9 - 2534400*tan(1/2*d*x + 1/2*c)^8 + 6564096*tan(1/2*d*x + 1/2*c)^7 + 5
06880*tan(1/2*d*x + 1/2*c)^6 - 535689*tan(1/2*d*x + 1/2*c)^5 + 957440*tan(1/2*d*x + 1/2*c)^4 - 110880*tan(1/2*
d*x + 1/2*c)^3 + 191488*tan(1/2*d*x + 1/2*c)^2 - 10395*tan(1/2*d*x + 1/2*c) + 17408)/((tan(1/2*d*x + 1/2*c)^2
+ 1)^11*a^2))/d

Mupad [B] (verification not implemented)

Time = 12.38 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.30 \[ \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {3\,x}{128\,a^2}-\frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{64}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{2}+\frac {773\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{320}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{3}-\frac {148\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{5}+\frac {112\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{3}+\frac {1207\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}-\frac {848\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{15}+\frac {464\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{5}-\frac {1207\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{32}-\frac {80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{7}+\frac {148\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}+\frac {16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{7}-\frac {773\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320}+\frac {272\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{63}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {272\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{315}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {272}{3465}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{11}} \]

[In]

int((cos(c + d*x)^8*sin(c + d*x)^5)/(a + a*sin(c + d*x))^2,x)

[Out]

- (3*x)/(128*a^2) - ((272*tan(c/2 + (d*x)/2)^2)/315 - (3*tan(c/2 + (d*x)/2))/64 - tan(c/2 + (d*x)/2)^3/2 + (27
2*tan(c/2 + (d*x)/2)^4)/63 - (773*tan(c/2 + (d*x)/2)^5)/320 + (16*tan(c/2 + (d*x)/2)^6)/7 + (148*tan(c/2 + (d*
x)/2)^7)/5 - (80*tan(c/2 + (d*x)/2)^8)/7 - (1207*tan(c/2 + (d*x)/2)^9)/32 + (464*tan(c/2 + (d*x)/2)^10)/5 - (8
48*tan(c/2 + (d*x)/2)^12)/15 + (1207*tan(c/2 + (d*x)/2)^13)/32 + (112*tan(c/2 + (d*x)/2)^14)/3 - (148*tan(c/2
+ (d*x)/2)^15)/5 + (32*tan(c/2 + (d*x)/2)^16)/3 + (773*tan(c/2 + (d*x)/2)^17)/320 + tan(c/2 + (d*x)/2)^19/2 +
(3*tan(c/2 + (d*x)/2)^21)/64 + 272/3465)/(a^2*d*(tan(c/2 + (d*x)/2)^2 + 1)^11)