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

Optimal. Leaf size=203 \[ \frac {\cos ^{11}(c+d x)}{11 a^2 d}-\frac {4 \cos ^9(c+d x)}{9 a^2 d}+\frac {5 \cos ^7(c+d x)}{7 a^2 d}-\frac {2 \cos ^5(c+d x)}{5 a^2 d}+\frac {\sin ^5(c+d x) \cos ^5(c+d x)}{5 a^2 d}+\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 a^2 d}+\frac {\sin (c+d x) \cos ^5(c+d x)}{16 a^2 d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{64 a^2 d}-\frac {3 \sin (c+d x) \cos (c+d x)}{128 a^2 d}-\frac {3 x}{128 a^2} \]

[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

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Rubi [A]  time = 0.41, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2875, 2873, 2565, 270, 2568, 2635, 8} \[ \frac {\cos ^{11}(c+d x)}{11 a^2 d}-\frac {4 \cos ^9(c+d x)}{9 a^2 d}+\frac {5 \cos ^7(c+d x)}{7 a^2 d}-\frac {2 \cos ^5(c+d x)}{5 a^2 d}+\frac {\sin ^5(c+d x) \cos ^5(c+d x)}{5 a^2 d}+\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 a^2 d}+\frac {\sin (c+d x) \cos ^5(c+d x)}{16 a^2 d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{64 a^2 d}-\frac {3 \sin (c+d x) \cos (c+d x)}{128 a^2 d}-\frac {3 x}{128 a^2} \]

Antiderivative was successfully verified.

[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 270

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 2565

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 2568

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 2635

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] && Integer
Q[2*n]

Rule 2873

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 2875

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*} \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\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 {\operatorname {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac {\operatorname {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 {\operatorname {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac {\operatorname {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*}

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Mathematica [B]  time = 10.59, size = 1453, normalized size = 7.16 \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*Cos[c + d*x]*(1 + 2*Sin[c + d*x]))/(3072*a^2*d*(1 + Sin[c + d*x])^2) + (27720*(c + d*x) + 41580*Cos[c + d*
x] - 7056*Cos[3*(c + d*x)] + 1764*Cos[5*(c + d*x)] - 360*Cos[7*(c + d*x)] + 28*Cos[9*(c + d*x)] + (42*Sin[(c +
d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - 21/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (15204*Sin[(c
+ d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 15120*Sin[2*(c + d*x)] + 3528*Sin[4*(c + d*x)] - 840*Sin[6
*(c + d*x)] + 126*Sin[8*(c + d*x)])/(86016*a^2*d) + (-360360*(c + d*x) - 566280*Cos[c + d*x] + 108900*Cos[3*(c
+ d*x)] - 33264*Cos[5*(c + d*x)] + 9900*Cos[7*(c + d*x)] - 2200*Cos[9*(c + d*x)] + 180*Cos[11*(c + d*x)] - (3
30*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + 165/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (
166980*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 217800*Sin[2*(c + d*x)] - 59400*Sin[4*(c + d*
x)] + 18480*Sin[6*(c + d*x)] - 4950*Sin[8*(c + d*x)] + 792*Sin[10*(c + d*x)])/(2027520*a^2*d) + (25*(36*d*x*Co
s[(d*x)/2] - 21*Cos[c + (d*x)/2] + 35*Cos[c + (3*d*x)/2] - 12*d*x*Cos[2*c + (3*d*x)/2] - 3*Cos[3*c + (5*d*x)/2
] - 57*Sin[(d*x)/2] + 36*d*x*Sin[c + (d*x)/2] + 12*d*x*Sin[c + (3*d*x)/2] + 9*Sin[2*c + (3*d*x)/2] + 3*Sin[2*c
+ (5*d*x)/2]))/(12288*a^2*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (5*(180*d*x*Cos[
(d*x)/2] - 21*Cos[c + (d*x)/2] + 147*Cos[c + (3*d*x)/2] - 60*d*x*Cos[2*c + (3*d*x)/2] - 15*Cos[3*c + (5*d*x)/2
] + 3*Cos[3*c + (7*d*x)/2] + Cos[5*c + (9*d*x)/2] - 201*Sin[(d*x)/2] + 180*d*x*Sin[c + (d*x)/2] + 60*d*x*Sin[c
+ (3*d*x)/2] + 73*Sin[2*c + (3*d*x)/2] + 15*Sin[2*c + (5*d*x)/2] + 3*Sin[4*c + (7*d*x)/2] - Sin[4*c + (9*d*x)
/2]))/(12288*a^2*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (7*(2520*d*x*Cos[(d*x)/2]
+ 165*Cos[c + (d*x)/2] + 1905*Cos[c + (3*d*x)/2] - 840*d*x*Cos[2*c + (3*d*x)/2] - 210*Cos[3*c + (5*d*x)/2] + 4
2*Cos[3*c + (7*d*x)/2] + 14*Cos[5*c + (9*d*x)/2] - 6*Cos[5*c + (11*d*x)/2] - 3*Cos[7*c + (13*d*x)/2] - 2355*Si
n[(d*x)/2] + 2520*d*x*Sin[c + (d*x)/2] + 840*d*x*Sin[c + (3*d*x)/2] + 1175*Sin[2*c + (3*d*x)/2] + 210*Sin[2*c
+ (5*d*x)/2] + 42*Sin[4*c + (7*d*x)/2] - 14*Sin[4*c + (9*d*x)/2] - 6*Sin[6*c + (11*d*x)/2] + 3*Sin[6*c + (13*d
*x)/2]))/(30720*a^2*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (7560*d*x*Cos[(d*x)/2]
+ 1239*Cos[c + (d*x)/2] + 5467*Cos[c + (3*d*x)/2] - 2520*d*x*Cos[2*c + (3*d*x)/2] - 630*Cos[3*c + (5*d*x)/2] +
126*Cos[3*c + (7*d*x)/2] + 42*Cos[5*c + (9*d*x)/2] - 18*Cos[5*c + (11*d*x)/2] - 9*Cos[7*c + (13*d*x)/2] + 5*C
os[7*c + (15*d*x)/2] + 3*Cos[9*c + (17*d*x)/2] - 6321*Sin[(d*x)/2] + 7560*d*x*Sin[c + (d*x)/2] + 2520*d*x*Sin[
c + (3*d*x)/2] + 3773*Sin[2*c + (3*d*x)/2] + 630*Sin[2*c + (5*d*x)/2] + 126*Sin[4*c + (7*d*x)/2] - 42*Sin[4*c
+ (9*d*x)/2] - 18*Sin[6*c + (11*d*x)/2] + 9*Sin[6*c + (13*d*x)/2] + 5*Sin[8*c + (15*d*x)/2] - 3*Sin[8*c + (17*
d*x)/2])/(43008*a^2*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)

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fricas [A]  time = 0.51, size = 110, normalized size = 0.54 \[ \frac {40320 \, \cos \left (d x + c\right )^{11} - 197120 \, \cos \left (d x + c\right )^{9} + 316800 \, \cos \left (d x + c\right )^{7} - 177408 \, \cos \left (d x + c\right )^{5} - 10395 \, d x + 693 \, {\left (128 \, \cos \left (d x + c\right )^{9} - 336 \, \cos \left (d x + c\right )^{7} + 248 \, \cos \left (d x + c\right )^{5} - 10 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{443520 \, a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.31, size = 270, normalized size = 1.33 \[ -\frac {\frac {10395 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (10395 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{21} + 110880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{19} + 535689 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 2365440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} - 6564096 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 8279040 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} + 8364510 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 12536832 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 20579328 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 8364510 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 2534400 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 6564096 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 506880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 535689 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 957440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 110880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 191488 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10395 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17408\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{11} a^{2}}}{443520 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [B]  time = 0.51, size = 653, normalized size = 3.22 \[ -\frac {272}{3465 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {272 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {272 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {773 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{320 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {16 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {148 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {80 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {1207 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {464 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {848 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {1207 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {112 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {148 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {32 \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {773 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{320 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}-\frac {3 \left (\tan ^{21}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d \,a^{2} \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}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-272/3465/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^11+3/64/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)-272/315/
d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^2+1/2/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c
)^3-272/63/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^4+773/320/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^11*ta
n(1/2*d*x+1/2*c)^5-16/7/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^6-148/5/d/a^2/(1+tan(1/2*d*x+1/2*
c)^2)^11*tan(1/2*d*x+1/2*c)^7+80/7/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^8+1207/32/d/a^2/(1+tan
(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^9-464/5/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^10+848/1
5/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^12-1207/32/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*
x+1/2*c)^13-112/3/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^14+148/5/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)
^11*tan(1/2*d*x+1/2*c)^15-32/3/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^16-773/320/d/a^2/(1+tan(1/
2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^17-1/2/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^19-3/64/d/a^
2/(1+tan(1/2*d*x+1/2*c)^2)^11*tan(1/2*d*x+1/2*c)^21-3/64/d/a^2*arctan(tan(1/2*d*x+1/2*c))

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maxima [B]  time = 0.49, size = 648, normalized size = 3.19 \[ \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} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 11.51, size = 264, normalized size = 1.30 \[ -\frac {3\,x}{128\,a^2}-\frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{64}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{2}+\frac {773\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{320}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{3}-\frac {148\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{5}+\frac {112\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{3}+\frac {1207\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}-\frac {848\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{15}+\frac {464\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{5}-\frac {1207\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{32}-\frac {80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{7}+\frac {148\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}+\frac {16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{7}-\frac {773\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320}+\frac {272\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{63}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {272\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{315}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {272}{3465}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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