3.624 \(\int \frac{\cos ^6(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=159 \[ \frac{\cos ^9(c+d x)}{9 a d}-\frac{2 \cos ^7(c+d x)}{7 a d}+\frac{\cos ^5(c+d x)}{5 a d}-\frac{\sin ^3(c+d x) \cos ^5(c+d x)}{8 a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{16 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{64 a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{128 a d}+\frac{3 x}{128 a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.215867, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2839, 2568, 2635, 8, 2565, 270} \[ \frac{\cos ^9(c+d x)}{9 a d}-\frac{2 \cos ^7(c+d x)}{7 a d}+\frac{\cos ^5(c+d x)}{5 a d}-\frac{\sin ^3(c+d x) \cos ^5(c+d x)}{8 a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{16 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{64 a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{128 a d}+\frac{3 x}{128 a} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^6*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]

[Out]

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

Rule 2839

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]

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

Mathematica [B]  time = 8.5865, size = 429, normalized size = 2.7 \[ \frac{15120 d x \sin \left (\frac{c}{2}\right )-7560 \sin \left (\frac{c}{2}+d x\right )+7560 \sin \left (\frac{3 c}{2}+d x\right )-1680 \sin \left (\frac{5 c}{2}+3 d x\right )+1680 \sin \left (\frac{7 c}{2}+3 d x\right )-2520 \sin \left (\frac{7 c}{2}+4 d x\right )-2520 \sin \left (\frac{9 c}{2}+4 d x\right )+1008 \sin \left (\frac{9 c}{2}+5 d x\right )-1008 \sin \left (\frac{11 c}{2}+5 d x\right )+180 \sin \left (\frac{13 c}{2}+7 d x\right )-180 \sin \left (\frac{15 c}{2}+7 d x\right )+315 \sin \left (\frac{15 c}{2}+8 d x\right )+315 \sin \left (\frac{17 c}{2}+8 d x\right )-140 \sin \left (\frac{17 c}{2}+9 d x\right )+140 \sin \left (\frac{19 c}{2}+9 d x\right )+2520 \cos \left (\frac{c}{2}\right ) (5 c+6 d x)+7560 \cos \left (\frac{c}{2}+d x\right )+7560 \cos \left (\frac{3 c}{2}+d x\right )+1680 \cos \left (\frac{5 c}{2}+3 d x\right )+1680 \cos \left (\frac{7 c}{2}+3 d x\right )-2520 \cos \left (\frac{7 c}{2}+4 d x\right )+2520 \cos \left (\frac{9 c}{2}+4 d x\right )-1008 \cos \left (\frac{9 c}{2}+5 d x\right )-1008 \cos \left (\frac{11 c}{2}+5 d x\right )-180 \cos \left (\frac{13 c}{2}+7 d x\right )-180 \cos \left (\frac{15 c}{2}+7 d x\right )+315 \cos \left (\frac{15 c}{2}+8 d x\right )-315 \cos \left (\frac{17 c}{2}+8 d x\right )+140 \cos \left (\frac{17 c}{2}+9 d x\right )+140 \cos \left (\frac{19 c}{2}+9 d x\right )+12600 c \sin \left (\frac{c}{2}\right )+12600 \sin \left (\frac{c}{2}\right )}{645120 a d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]

[Out]

(2520*(5*c + 6*d*x)*Cos[c/2] + 7560*Cos[c/2 + d*x] + 7560*Cos[(3*c)/2 + d*x] + 1680*Cos[(5*c)/2 + 3*d*x] + 168
0*Cos[(7*c)/2 + 3*d*x] - 2520*Cos[(7*c)/2 + 4*d*x] + 2520*Cos[(9*c)/2 + 4*d*x] - 1008*Cos[(9*c)/2 + 5*d*x] - 1
008*Cos[(11*c)/2 + 5*d*x] - 180*Cos[(13*c)/2 + 7*d*x] - 180*Cos[(15*c)/2 + 7*d*x] + 315*Cos[(15*c)/2 + 8*d*x]
- 315*Cos[(17*c)/2 + 8*d*x] + 140*Cos[(17*c)/2 + 9*d*x] + 140*Cos[(19*c)/2 + 9*d*x] + 12600*Sin[c/2] + 12600*c
*Sin[c/2] + 15120*d*x*Sin[c/2] - 7560*Sin[c/2 + d*x] + 7560*Sin[(3*c)/2 + d*x] - 1680*Sin[(5*c)/2 + 3*d*x] + 1
680*Sin[(7*c)/2 + 3*d*x] - 2520*Sin[(7*c)/2 + 4*d*x] - 2520*Sin[(9*c)/2 + 4*d*x] + 1008*Sin[(9*c)/2 + 5*d*x] -
1008*Sin[(11*c)/2 + 5*d*x] + 180*Sin[(13*c)/2 + 7*d*x] - 180*Sin[(15*c)/2 + 7*d*x] + 315*Sin[(15*c)/2 + 8*d*x
] + 315*Sin[(17*c)/2 + 8*d*x] - 140*Sin[(17*c)/2 + 9*d*x] + 140*Sin[(19*c)/2 + 9*d*x])/(645120*a*d*(Cos[c/2] +
Sin[c/2]))

________________________________________________________________________________________

Maple [B]  time = 0.106, size = 517, normalized size = 3.3 \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)^6*sin(d*x+c)^4/(a+a*sin(d*x+c)),x)

[Out]

16/315/d/a/(1+tan(1/2*d*x+1/2*c)^2)^9-3/64/d/a/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)+16/35/d/a/(1+tan(
1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^2-13/32/d/a/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^3+64/35/d/a/(
1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^4+155/32/d/a/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^5-32/5
/d/a/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^6-169/32/d/a/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^
7+112/5/d/a/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^8-16/d/a/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*
c)^10+169/32/d/a/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^11+32/3/d/a/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*
d*x+1/2*c)^12-155/32/d/a/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^13+13/32/d/a/(1+tan(1/2*d*x+1/2*c)^2)^9
*tan(1/2*d*x+1/2*c)^15+3/64/d/a/(1+tan(1/2*d*x+1/2*c)^2)^9*tan(1/2*d*x+1/2*c)^17+3/64/a/d*arctan(tan(1/2*d*x+1
/2*c))

________________________________________________________________________________________

Maxima [B]  time = 1.55172, size = 678, normalized size = 4.26 \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)^6*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/20160*((945*sin(d*x + c)/(cos(d*x + c) + 1) - 9216*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 8190*sin(d*x + c)^
3/(cos(d*x + c) + 1)^3 - 36864*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 97650*sin(d*x + c)^5/(cos(d*x + c) + 1)^5
+ 129024*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 106470*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 451584*sin(d*x +
c)^8/(cos(d*x + c) + 1)^8 + 322560*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 106470*sin(d*x + c)^11/(cos(d*x + c
) + 1)^11 - 215040*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 97650*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 8190*
sin(d*x + c)^15/(cos(d*x + c) + 1)^15 - 945*sin(d*x + c)^17/(cos(d*x + c) + 1)^17 - 1024)/(a + 9*a*sin(d*x + c
)^2/(cos(d*x + c) + 1)^2 + 36*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 84*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6
+ 126*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 126*a*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 84*a*sin(d*x + c)
^12/(cos(d*x + c) + 1)^12 + 36*a*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + 9*a*sin(d*x + c)^16/(cos(d*x + c) + 1
)^16 + a*sin(d*x + c)^18/(cos(d*x + c) + 1)^18) - 945*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d

________________________________________________________________________________________

Fricas [A]  time = 1.19707, size = 252, normalized size = 1.58 \begin{align*} \frac{4480 \, \cos \left (d x + c\right )^{9} - 11520 \, \cos \left (d x + c\right )^{7} + 8064 \, \cos \left (d x + c\right )^{5} + 945 \, d x + 315 \,{\left (16 \, \cos \left (d x + c\right )^{7} - 24 \, \cos \left (d x + c\right )^{5} + 2 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{40320 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/40320*(4480*cos(d*x + c)^9 - 11520*cos(d*x + c)^7 + 8064*cos(d*x + c)^5 + 945*d*x + 315*(16*cos(d*x + c)^7 -
24*cos(d*x + c)^5 + 2*cos(d*x + c)^3 + 3*cos(d*x + c))*sin(d*x + c))/(a*d)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.31384, size = 294, normalized size = 1.85 \begin{align*} \frac{\frac{945 \,{\left (d x + c\right )}}{a} + \frac{2 \,{\left (945 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{17} + 8190 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} - 97650 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 215040 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} + 106470 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 322560 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 451584 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 106470 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 129024 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 97650 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 36864 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 8190 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9216 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 945 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1024\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{9} a}}{40320 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/40320*(945*(d*x + c)/a + 2*(945*tan(1/2*d*x + 1/2*c)^17 + 8190*tan(1/2*d*x + 1/2*c)^15 - 97650*tan(1/2*d*x +
1/2*c)^13 + 215040*tan(1/2*d*x + 1/2*c)^12 + 106470*tan(1/2*d*x + 1/2*c)^11 - 322560*tan(1/2*d*x + 1/2*c)^10
+ 451584*tan(1/2*d*x + 1/2*c)^8 - 106470*tan(1/2*d*x + 1/2*c)^7 - 129024*tan(1/2*d*x + 1/2*c)^6 + 97650*tan(1/
2*d*x + 1/2*c)^5 + 36864*tan(1/2*d*x + 1/2*c)^4 - 8190*tan(1/2*d*x + 1/2*c)^3 + 9216*tan(1/2*d*x + 1/2*c)^2 -
945*tan(1/2*d*x + 1/2*c) + 1024)/((tan(1/2*d*x + 1/2*c)^2 + 1)^9*a))/d