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*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)

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Rubi [A]  time = 0.457852, antiderivative size = 185, normalized size of antiderivative = 1., 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 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]

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 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 8

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

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 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]

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*}

Mathematica [B]  time = 8.43935, 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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Maple [B]  time = 0.125, size = 619, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}

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 = 1.58117, size = 817, normalized size = 4.42 \begin{align*} \text{result too large to display} \end{align*}

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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Fricas [A]  time = 1.18664, size = 292, normalized size = 1.58 \begin{align*} \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} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [A]  time = 1.28953, size = 347, normalized size = 1.88 \begin{align*} \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} \end{align*}

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