3.723 \(\int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=185 \[ \frac {2 \cos ^9(c+d x)}{9 a^2 d}-\frac {4 \cos ^7(c+d x)}{7 a^2 d}+\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {\sin ^5(c+d x) \cos ^5(c+d x)}{10 a^2 d}-\frac {3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 a^2 d}-\frac {3 \sin (c+d x) \cos ^5(c+d x)}{32 a^2 d}+\frac {3 \sin (c+d x) \cos ^3(c+d x)}{128 a^2 d}+\frac {9 \sin (c+d x) \cos (c+d x)}{256 a^2 d}+\frac {9 x}{256 a^2} \]

[Out]

9/256*x/a^2+2/5*cos(d*x+c)^5/a^2/d-4/7*cos(d*x+c)^7/a^2/d+2/9*cos(d*x+c)^9/a^2/d+9/256*cos(d*x+c)*sin(d*x+c)/a
^2/d+3/128*cos(d*x+c)^3*sin(d*x+c)/a^2/d-3/32*cos(d*x+c)^5*sin(d*x+c)/a^2/d-3/16*cos(d*x+c)^5*sin(d*x+c)^3/a^2
/d-1/10*cos(d*x+c)^5*sin(d*x+c)^5/a^2/d

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Rubi [A]  time = 0.46, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2875, 2873, 2568, 2635, 8, 2565, 270} \[ \frac {2 \cos ^9(c+d x)}{9 a^2 d}-\frac {4 \cos ^7(c+d x)}{7 a^2 d}+\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {\sin ^5(c+d x) \cos ^5(c+d x)}{10 a^2 d}-\frac {3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 a^2 d}-\frac {3 \sin (c+d x) \cos ^5(c+d x)}{32 a^2 d}+\frac {3 \sin (c+d x) \cos ^3(c+d x)}{128 a^2 d}+\frac {9 \sin (c+d x) \cos (c+d x)}{256 a^2 d}+\frac {9 x}{256 a^2} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^8*Sin[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]

[Out]

(9*x)/(256*a^2) + (2*Cos[c + d*x]^5)/(5*a^2*d) - (4*Cos[c + d*x]^7)/(7*a^2*d) + (2*Cos[c + d*x]^9)/(9*a^2*d) +
(9*Cos[c + d*x]*Sin[c + d*x])/(256*a^2*d) + (3*Cos[c + d*x]^3*Sin[c + d*x])/(128*a^2*d) - (3*Cos[c + d*x]^5*S
in[c + d*x])/(32*a^2*d) - (3*Cos[c + d*x]^5*Sin[c + d*x]^3)/(16*a^2*d) - (Cos[c + d*x]^5*Sin[c + d*x]^5)/(10*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 ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \cos ^4(c+d x) \sin ^4(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac {\int \left (a^2 \cos ^4(c+d x) \sin ^4(c+d x)-2 a^2 \cos ^4(c+d x) \sin ^5(c+d x)+a^2 \cos ^4(c+d x) \sin ^6(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \cos ^4(c+d x) \sin ^4(c+d x) \, dx}{a^2}+\frac {\int \cos ^4(c+d x) \sin ^6(c+d x) \, dx}{a^2}-\frac {2 \int \cos ^4(c+d x) \sin ^5(c+d x) \, dx}{a^2}\\ &=-\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)}{10 a^2 d}+\frac {3 \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx}{8 a^2}+\frac {\int \cos ^4(c+d x) \sin ^4(c+d x) \, dx}{2 a^2}+\frac {2 \operatorname {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac {\cos ^5(c+d x) \sin (c+d x)}{16 a^2 d}-\frac {3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 a^2 d}-\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{10 a^2 d}+\frac {\int \cos ^4(c+d x) \, dx}{16 a^2}+\frac {3 \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx}{16 a^2}+\frac {2 \operatorname {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {4 \cos ^7(c+d x)}{7 a^2 d}+\frac {2 \cos ^9(c+d x)}{9 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{64 a^2 d}-\frac {3 \cos ^5(c+d x) \sin (c+d x)}{32 a^2 d}-\frac {3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 a^2 d}-\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{10 a^2 d}+\frac {\int \cos ^4(c+d x) \, dx}{32 a^2}+\frac {3 \int \cos ^2(c+d x) \, dx}{64 a^2}\\ &=\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {4 \cos ^7(c+d x)}{7 a^2 d}+\frac {2 \cos ^9(c+d x)}{9 a^2 d}+\frac {3 \cos (c+d x) \sin (c+d x)}{128 a^2 d}+\frac {3 \cos ^3(c+d x) \sin (c+d x)}{128 a^2 d}-\frac {3 \cos ^5(c+d x) \sin (c+d x)}{32 a^2 d}-\frac {3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 a^2 d}-\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{10 a^2 d}+\frac {3 \int 1 \, dx}{128 a^2}+\frac {3 \int \cos ^2(c+d x) \, dx}{128 a^2}\\ &=\frac {3 x}{128 a^2}+\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {4 \cos ^7(c+d x)}{7 a^2 d}+\frac {2 \cos ^9(c+d x)}{9 a^2 d}+\frac {9 \cos (c+d x) \sin (c+d x)}{256 a^2 d}+\frac {3 \cos ^3(c+d x) \sin (c+d x)}{128 a^2 d}-\frac {3 \cos ^5(c+d x) \sin (c+d x)}{32 a^2 d}-\frac {3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 a^2 d}-\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{10 a^2 d}+\frac {3 \int 1 \, dx}{256 a^2}\\ &=\frac {9 x}{256 a^2}+\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {4 \cos ^7(c+d x)}{7 a^2 d}+\frac {2 \cos ^9(c+d x)}{9 a^2 d}+\frac {9 \cos (c+d x) \sin (c+d x)}{256 a^2 d}+\frac {3 \cos ^3(c+d x) \sin (c+d x)}{128 a^2 d}-\frac {3 \cos ^5(c+d x) \sin (c+d x)}{32 a^2 d}-\frac {3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 a^2 d}-\frac {\cos ^5(c+d x) \sin ^5(c+d x)}{10 a^2 d}\\ \end {align*}

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Mathematica [B]  time = 8.38, size = 585, normalized size = 3.16 \[ \frac {45360 d x \sin \left (\frac {c}{2}\right )-30240 \sin \left (\frac {c}{2}+d x\right )+30240 \sin \left (\frac {3 c}{2}+d x\right )-1260 \sin \left (\frac {3 c}{2}+2 d x\right )-1260 \sin \left (\frac {5 c}{2}+2 d x\right )-6720 \sin \left (\frac {5 c}{2}+3 d x\right )+6720 \sin \left (\frac {7 c}{2}+3 d x\right )-7560 \sin \left (\frac {7 c}{2}+4 d x\right )-7560 \sin \left (\frac {9 c}{2}+4 d x\right )+4032 \sin \left (\frac {9 c}{2}+5 d x\right )-4032 \sin \left (\frac {11 c}{2}+5 d x\right )+630 \sin \left (\frac {11 c}{2}+6 d x\right )+630 \sin \left (\frac {13 c}{2}+6 d x\right )+720 \sin \left (\frac {13 c}{2}+7 d x\right )-720 \sin \left (\frac {15 c}{2}+7 d x\right )+945 \sin \left (\frac {15 c}{2}+8 d x\right )+945 \sin \left (\frac {17 c}{2}+8 d x\right )-560 \sin \left (\frac {17 c}{2}+9 d x\right )+560 \sin \left (\frac {19 c}{2}+9 d x\right )-126 \sin \left (\frac {19 c}{2}+10 d x\right )-126 \sin \left (\frac {21 c}{2}+10 d x\right )-2520 \cos \left (\frac {c}{2}\right ) (187 c-18 d x)+30240 \cos \left (\frac {c}{2}+d x\right )+30240 \cos \left (\frac {3 c}{2}+d x\right )-1260 \cos \left (\frac {3 c}{2}+2 d x\right )+1260 \cos \left (\frac {5 c}{2}+2 d x\right )+6720 \cos \left (\frac {5 c}{2}+3 d x\right )+6720 \cos \left (\frac {7 c}{2}+3 d x\right )-7560 \cos \left (\frac {7 c}{2}+4 d x\right )+7560 \cos \left (\frac {9 c}{2}+4 d x\right )-4032 \cos \left (\frac {9 c}{2}+5 d x\right )-4032 \cos \left (\frac {11 c}{2}+5 d x\right )+630 \cos \left (\frac {11 c}{2}+6 d x\right )-630 \cos \left (\frac {13 c}{2}+6 d x\right )-720 \cos \left (\frac {13 c}{2}+7 d x\right )-720 \cos \left (\frac {15 c}{2}+7 d x\right )+945 \cos \left (\frac {15 c}{2}+8 d x\right )-945 \cos \left (\frac {17 c}{2}+8 d x\right )+560 \cos \left (\frac {17 c}{2}+9 d x\right )+560 \cos \left (\frac {19 c}{2}+9 d x\right )-126 \cos \left (\frac {19 c}{2}+10 d x\right )+126 \cos \left (\frac {21 c}{2}+10 d x\right )-471240 c \sin \left (\frac {c}{2}\right )+327180 \sin \left (\frac {c}{2}\right )}{1290240 a^2 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]

[Out]

(-2520*(187*c - 18*d*x)*Cos[c/2] + 30240*Cos[c/2 + d*x] + 30240*Cos[(3*c)/2 + d*x] - 1260*Cos[(3*c)/2 + 2*d*x]
+ 1260*Cos[(5*c)/2 + 2*d*x] + 6720*Cos[(5*c)/2 + 3*d*x] + 6720*Cos[(7*c)/2 + 3*d*x] - 7560*Cos[(7*c)/2 + 4*d*
x] + 7560*Cos[(9*c)/2 + 4*d*x] - 4032*Cos[(9*c)/2 + 5*d*x] - 4032*Cos[(11*c)/2 + 5*d*x] + 630*Cos[(11*c)/2 + 6
*d*x] - 630*Cos[(13*c)/2 + 6*d*x] - 720*Cos[(13*c)/2 + 7*d*x] - 720*Cos[(15*c)/2 + 7*d*x] + 945*Cos[(15*c)/2 +
8*d*x] - 945*Cos[(17*c)/2 + 8*d*x] + 560*Cos[(17*c)/2 + 9*d*x] + 560*Cos[(19*c)/2 + 9*d*x] - 126*Cos[(19*c)/2
+ 10*d*x] + 126*Cos[(21*c)/2 + 10*d*x] + 327180*Sin[c/2] - 471240*c*Sin[c/2] + 45360*d*x*Sin[c/2] - 30240*Sin
[c/2 + d*x] + 30240*Sin[(3*c)/2 + d*x] - 1260*Sin[(3*c)/2 + 2*d*x] - 1260*Sin[(5*c)/2 + 2*d*x] - 6720*Sin[(5*c
)/2 + 3*d*x] + 6720*Sin[(7*c)/2 + 3*d*x] - 7560*Sin[(7*c)/2 + 4*d*x] - 7560*Sin[(9*c)/2 + 4*d*x] + 4032*Sin[(9
*c)/2 + 5*d*x] - 4032*Sin[(11*c)/2 + 5*d*x] + 630*Sin[(11*c)/2 + 6*d*x] + 630*Sin[(13*c)/2 + 6*d*x] + 720*Sin[
(13*c)/2 + 7*d*x] - 720*Sin[(15*c)/2 + 7*d*x] + 945*Sin[(15*c)/2 + 8*d*x] + 945*Sin[(17*c)/2 + 8*d*x] - 560*Si
n[(17*c)/2 + 9*d*x] + 560*Sin[(19*c)/2 + 9*d*x] - 126*Sin[(19*c)/2 + 10*d*x] - 126*Sin[(21*c)/2 + 10*d*x])/(12
90240*a^2*d*(Cos[c/2] + Sin[c/2]))

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fricas [A]  time = 0.46, size = 100, normalized size = 0.54 \[ \frac {17920 \, \cos \left (d x + c\right )^{9} - 46080 \, \cos \left (d x + c\right )^{7} + 32256 \, \cos \left (d x + c\right )^{5} + 2835 \, d x - 63 \, {\left (128 \, \cos \left (d x + c\right )^{9} - 496 \, \cos \left (d x + c\right )^{7} + 488 \, \cos \left (d x + c\right )^{5} - 30 \, \cos \left (d x + c\right )^{3} - 45 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/80640*(17920*cos(d*x + c)^9 - 46080*cos(d*x + c)^7 + 32256*cos(d*x + c)^5 + 2835*d*x - 63*(128*cos(d*x + c)^
9 - 496*cos(d*x + c)^7 + 488*cos(d*x + c)^5 - 30*cos(d*x + c)^3 - 45*cos(d*x + c))*sin(d*x + c))/(a^2*d)

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giac [A]  time = 0.26, size = 257, normalized size = 1.39 \[ \frac {\frac {2835 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (2835 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{19} + 27405 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} - 139356 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 860160 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 618660 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 430080 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 1609650 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 516096 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 1609650 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1290240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 618660 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 368640 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 139356 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 184320 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 27405 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2835 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4096\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{10} a^{2}}}{80640 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/80640*(2835*(d*x + c)/a^2 + 2*(2835*tan(1/2*d*x + 1/2*c)^19 + 27405*tan(1/2*d*x + 1/2*c)^17 - 139356*tan(1/2
*d*x + 1/2*c)^15 + 860160*tan(1/2*d*x + 1/2*c)^14 - 618660*tan(1/2*d*x + 1/2*c)^13 - 430080*tan(1/2*d*x + 1/2*
c)^12 + 1609650*tan(1/2*d*x + 1/2*c)^11 + 516096*tan(1/2*d*x + 1/2*c)^10 - 1609650*tan(1/2*d*x + 1/2*c)^9 + 12
90240*tan(1/2*d*x + 1/2*c)^8 + 618660*tan(1/2*d*x + 1/2*c)^7 - 368640*tan(1/2*d*x + 1/2*c)^6 + 139356*tan(1/2*
d*x + 1/2*c)^5 + 184320*tan(1/2*d*x + 1/2*c)^4 - 27405*tan(1/2*d*x + 1/2*c)^3 + 40960*tan(1/2*d*x + 1/2*c)^2 -
2835*tan(1/2*d*x + 1/2*c) + 4096)/((tan(1/2*d*x + 1/2*c)^2 + 1)^10*a^2))/d

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maple [B]  time = 0.46, size = 619, normalized size = 3.35 \[ \frac {32}{315 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {64 \left (\tan ^{2}\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 )^{10}}-\frac {87 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {32 \left (\tan ^{4}\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 )^{10}}+\frac {553 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {64 \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 )^{10}}+\frac {491 \left (\tan ^{7}\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 )^{10}}+\frac {32 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {2555 \left (\tan ^{9}\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 )^{10}}+\frac {64 \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 )^{10}}+\frac {2555 \left (\tan ^{11}\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 )^{10}}-\frac {32 \left (\tan ^{12}\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 )^{10}}-\frac {491 \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 )^{10}}+\frac {64 \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 )^{10}}-\frac {553 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {87 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {9 \left (\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {9 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d \,a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^8*sin(d*x+c)^4/(a+a*sin(d*x+c))^2,x)

[Out]

32/315/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^10-9/128/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)+64/63/d/a^
2/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^2-87/128/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)
^3+32/7/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^4+553/160/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1
/2*d*x+1/2*c)^5-64/7/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^6+491/32/d/a^2/(1+tan(1/2*d*x+1/2*c)
^2)^10*tan(1/2*d*x+1/2*c)^7+32/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^8-2555/64/d/a^2/(1+tan(1/2
*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^9+64/5/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^10+2555/64/d/
a^2/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^11-32/3/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c
)^12-491/32/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^13+64/3/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^10*tan
(1/2*d*x+1/2*c)^14-553/160/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^15+87/128/d/a^2/(1+tan(1/2*d*x
+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^17+9/128/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^10*tan(1/2*d*x+1/2*c)^19+9/128/d/a^2*
arctan(tan(1/2*d*x+1/2*c))

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maxima [B]  time = 0.44, size = 605, normalized size = 3.27 \[ -\frac {\frac {\frac {2835 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {40960 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {27405 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {184320 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {139356 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {368640 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {618660 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {1290240 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {1609650 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {516096 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {1609650 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {430080 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {618660 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {860160 \, \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {139356 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} - \frac {27405 \, \sin \left (d x + c\right )^{17}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{17}} - \frac {2835 \, \sin \left (d x + c\right )^{19}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{19}} - 4096}{a^{2} + \frac {10 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {45 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {120 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {210 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {252 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {210 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {120 \, a^{2} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {45 \, a^{2} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} + \frac {10 \, a^{2} \sin \left (d x + c\right )^{18}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{18}} + \frac {a^{2} \sin \left (d x + c\right )^{20}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{20}}} - \frac {2835 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{40320 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^4/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/40320*((2835*sin(d*x + c)/(cos(d*x + c) + 1) - 40960*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 27405*sin(d*x +
c)^3/(cos(d*x + c) + 1)^3 - 184320*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 139356*sin(d*x + c)^5/(cos(d*x + c) +
1)^5 + 368640*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 618660*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 1290240*sin(
d*x + c)^8/(cos(d*x + c) + 1)^8 + 1609650*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 516096*sin(d*x + c)^10/(cos(d*
x + c) + 1)^10 - 1609650*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 + 430080*sin(d*x + c)^12/(cos(d*x + c) + 1)^12
+ 618660*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 860160*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + 139356*sin(d*x
+ c)^15/(cos(d*x + c) + 1)^15 - 27405*sin(d*x + c)^17/(cos(d*x + c) + 1)^17 - 2835*sin(d*x + c)^19/(cos(d*x +
c) + 1)^19 - 4096)/(a^2 + 10*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 45*a^2*sin(d*x + c)^4/(cos(d*x + c) +
1)^4 + 120*a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 210*a^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 252*a^2*sin
(d*x + c)^10/(cos(d*x + c) + 1)^10 + 210*a^2*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 120*a^2*sin(d*x + c)^14/(
cos(d*x + c) + 1)^14 + 45*a^2*sin(d*x + c)^16/(cos(d*x + c) + 1)^16 + 10*a^2*sin(d*x + c)^18/(cos(d*x + c) + 1
)^18 + a^2*sin(d*x + c)^20/(cos(d*x + c) + 1)^20) - 2835*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d

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mupad [B]  time = 11.63, size = 250, normalized size = 1.35 \[ \frac {9\,x}{256\,a^2}+\frac {\frac {9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}+\frac {87\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}-\frac {553\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{160}+\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{3}-\frac {491\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}-\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}+\frac {2555\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{5}-\frac {2555\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {491\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}-\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{7}+\frac {553\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{7}-\frac {87\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128}+\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{63}-\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}+\frac {32}{315}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^8*sin(c + d*x)^4)/(a + a*sin(c + d*x))^2,x)

[Out]

(9*x)/(256*a^2) + ((64*tan(c/2 + (d*x)/2)^2)/63 - (9*tan(c/2 + (d*x)/2))/128 - (87*tan(c/2 + (d*x)/2)^3)/128 +
(32*tan(c/2 + (d*x)/2)^4)/7 + (553*tan(c/2 + (d*x)/2)^5)/160 - (64*tan(c/2 + (d*x)/2)^6)/7 + (491*tan(c/2 + (
d*x)/2)^7)/32 + 32*tan(c/2 + (d*x)/2)^8 - (2555*tan(c/2 + (d*x)/2)^9)/64 + (64*tan(c/2 + (d*x)/2)^10)/5 + (255
5*tan(c/2 + (d*x)/2)^11)/64 - (32*tan(c/2 + (d*x)/2)^12)/3 - (491*tan(c/2 + (d*x)/2)^13)/32 + (64*tan(c/2 + (d
*x)/2)^14)/3 - (553*tan(c/2 + (d*x)/2)^15)/160 + (87*tan(c/2 + (d*x)/2)^17)/128 + (9*tan(c/2 + (d*x)/2)^19)/12
8 + 32/315)/(a^2*d*(tan(c/2 + (d*x)/2)^2 + 1)^10)

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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)**4/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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