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

Optimal. Leaf size=124 \[ -\frac{2 \cos ^7(c+d x)}{35 a^2 d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{15 a^2 d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{12 a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac{x}{8 a^2}-\frac{\cos ^9(c+d x)}{5 d (a \sin (c+d x)+a)^2} \]

[Out]

-x/(8*a^2) - (2*Cos[c + d*x]^7)/(35*a^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) - (Cos[c + d*x]^3*Sin[c + d
*x])/(12*a^2*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(15*a^2*d) - Cos[c + d*x]^9/(5*d*(a + a*Sin[c + d*x])^2)

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Rubi [A]  time = 0.137529, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2859, 2682, 2635, 8} \[ -\frac{2 \cos ^7(c+d x)}{35 a^2 d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{15 a^2 d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{12 a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac{x}{8 a^2}-\frac{\cos ^9(c+d x)}{5 d (a \sin (c+d x)+a)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x/(8*a^2) - (2*Cos[c + d*x]^7)/(35*a^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) - (Cos[c + d*x]^3*Sin[c + d
*x])/(12*a^2*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(15*a^2*d) - Cos[c + d*x]^9/(5*d*(a + a*Sin[c + d*x])^2)

Rule 2859

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[((b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*(2*m +
p + 1)), x] + Dist[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^
(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[
m + p], 0]) && NeQ[2*m + p + 1, 0]

Rule 2682

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

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]

Rubi steps

\begin{align*} \int \frac{\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx &=-\frac{\cos ^9(c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac{2 \int \frac{\cos ^8(c+d x)}{a+a \sin (c+d x)} \, dx}{5 a}\\ &=-\frac{2 \cos ^7(c+d x)}{35 a^2 d}-\frac{\cos ^9(c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac{2 \int \cos ^6(c+d x) \, dx}{5 a^2}\\ &=-\frac{2 \cos ^7(c+d x)}{35 a^2 d}-\frac{\cos ^5(c+d x) \sin (c+d x)}{15 a^2 d}-\frac{\cos ^9(c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac{\int \cos ^4(c+d x) \, dx}{3 a^2}\\ &=-\frac{2 \cos ^7(c+d x)}{35 a^2 d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{12 a^2 d}-\frac{\cos ^5(c+d x) \sin (c+d x)}{15 a^2 d}-\frac{\cos ^9(c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac{\int \cos ^2(c+d x) \, dx}{4 a^2}\\ &=-\frac{2 \cos ^7(c+d x)}{35 a^2 d}-\frac{\cos (c+d x) \sin (c+d x)}{8 a^2 d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{12 a^2 d}-\frac{\cos ^5(c+d x) \sin (c+d x)}{15 a^2 d}-\frac{\cos ^9(c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac{\int 1 \, dx}{8 a^2}\\ &=-\frac{x}{8 a^2}-\frac{2 \cos ^7(c+d x)}{35 a^2 d}-\frac{\cos (c+d x) \sin (c+d x)}{8 a^2 d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{12 a^2 d}-\frac{\cos ^5(c+d x) \sin (c+d x)}{15 a^2 d}-\frac{\cos ^9(c+d x)}{5 d (a+a \sin (c+d x))^2}\\ \end{align*}

Mathematica [B]  time = 4.86483, size = 418, normalized size = 3.37 \[ -\frac{1680 d x \sin \left (\frac{c}{2}\right )-1155 \sin \left (\frac{c}{2}+d x\right )+1155 \sin \left (\frac{3 c}{2}+d x\right )+210 \sin \left (\frac{3 c}{2}+2 d x\right )+210 \sin \left (\frac{5 c}{2}+2 d x\right )-525 \sin \left (\frac{5 c}{2}+3 d x\right )+525 \sin \left (\frac{7 c}{2}+3 d x\right )-210 \sin \left (\frac{7 c}{2}+4 d x\right )-210 \sin \left (\frac{9 c}{2}+4 d x\right )-63 \sin \left (\frac{9 c}{2}+5 d x\right )+63 \sin \left (\frac{11 c}{2}+5 d x\right )-70 \sin \left (\frac{11 c}{2}+6 d x\right )-70 \sin \left (\frac{13 c}{2}+6 d x\right )+15 \sin \left (\frac{13 c}{2}+7 d x\right )-15 \sin \left (\frac{15 c}{2}+7 d x\right )+70 \cos \left (\frac{c}{2}\right ) (24 d x+7)+1155 \cos \left (\frac{c}{2}+d x\right )+1155 \cos \left (\frac{3 c}{2}+d x\right )+210 \cos \left (\frac{3 c}{2}+2 d x\right )-210 \cos \left (\frac{5 c}{2}+2 d x\right )+525 \cos \left (\frac{5 c}{2}+3 d x\right )+525 \cos \left (\frac{7 c}{2}+3 d x\right )-210 \cos \left (\frac{7 c}{2}+4 d x\right )+210 \cos \left (\frac{9 c}{2}+4 d x\right )+63 \cos \left (\frac{9 c}{2}+5 d x\right )+63 \cos \left (\frac{11 c}{2}+5 d x\right )-70 \cos \left (\frac{11 c}{2}+6 d x\right )+70 \cos \left (\frac{13 c}{2}+6 d x\right )-15 \cos \left (\frac{13 c}{2}+7 d x\right )-15 \cos \left (\frac{15 c}{2}+7 d x\right )-490 \sin \left (\frac{c}{2}\right )}{13440 a^2 d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(70*(7 + 24*d*x)*Cos[c/2] + 1155*Cos[c/2 + d*x] + 1155*Cos[(3*c)/2 + d*x] + 210*Cos[(3*c)/2 + 2*d*x] - 210*Co
s[(5*c)/2 + 2*d*x] + 525*Cos[(5*c)/2 + 3*d*x] + 525*Cos[(7*c)/2 + 3*d*x] - 210*Cos[(7*c)/2 + 4*d*x] + 210*Cos[
(9*c)/2 + 4*d*x] + 63*Cos[(9*c)/2 + 5*d*x] + 63*Cos[(11*c)/2 + 5*d*x] - 70*Cos[(11*c)/2 + 6*d*x] + 70*Cos[(13*
c)/2 + 6*d*x] - 15*Cos[(13*c)/2 + 7*d*x] - 15*Cos[(15*c)/2 + 7*d*x] - 490*Sin[c/2] + 1680*d*x*Sin[c/2] - 1155*
Sin[c/2 + d*x] + 1155*Sin[(3*c)/2 + d*x] + 210*Sin[(3*c)/2 + 2*d*x] + 210*Sin[(5*c)/2 + 2*d*x] - 525*Sin[(5*c)
/2 + 3*d*x] + 525*Sin[(7*c)/2 + 3*d*x] - 210*Sin[(7*c)/2 + 4*d*x] - 210*Sin[(9*c)/2 + 4*d*x] - 63*Sin[(9*c)/2
+ 5*d*x] + 63*Sin[(11*c)/2 + 5*d*x] - 70*Sin[(11*c)/2 + 6*d*x] - 70*Sin[(13*c)/2 + 6*d*x] + 15*Sin[(13*c)/2 +
7*d*x] - 15*Sin[(15*c)/2 + 7*d*x])/(13440*a^2*d*(Cos[c/2] + Sin[c/2]))

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Maple [B]  time = 0.089, size = 449, normalized size = 3.6 \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)/(a+a*sin(d*x+c))^2,x)

[Out]

-1/4/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^13-2/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2
*c)^12+11/3/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^11-8/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*
d*x+1/2*c)^10-31/12/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^9-2/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*t
an(1/2*d*x+1/2*c)^8-16/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^6+31/12/d/a^2/(1+tan(1/2*d*x+1/2*c)
^2)^7*tan(1/2*d*x+1/2*c)^5-14/5/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^4-11/3/d/a^2/(1+tan(1/2*d*
x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^3-8/5/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)^2+1/4/d/a^2/(1+tan(
1/2*d*x+1/2*c)^2)^7*tan(1/2*d*x+1/2*c)-18/35/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^7-1/4/d/a^2*arctan(tan(1/2*d*x+1/2
*c))

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Maxima [B]  time = 1.67944, size = 589, normalized size = 4.75 \begin{align*} \frac{\frac{\frac{105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{672 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{1540 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{1176 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{1085 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{6720 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{840 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{1085 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{3360 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac{1540 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac{840 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac{105 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - 216}{a^{2} + \frac{7 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{21 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{35 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{35 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{21 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac{7 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac{a^{2} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}}} - \frac{105 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*((105*sin(d*x + c)/(cos(d*x + c) + 1) - 672*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 1540*sin(d*x + c)^3/(c
os(d*x + c) + 1)^3 - 1176*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1085*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 672
0*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 840*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 1085*sin(d*x + c)^9/(cos(d*x
+ c) + 1)^9 - 3360*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 1540*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 840*s
in(d*x + c)^12/(cos(d*x + c) + 1)^12 - 105*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 216)/(a^2 + 7*a^2*sin(d*x +
c)^2/(cos(d*x + c) + 1)^2 + 21*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 35*a^2*sin(d*x + c)^6/(cos(d*x + c)
+ 1)^6 + 35*a^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 21*a^2*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 7*a^2*sin
(d*x + c)^12/(cos(d*x + c) + 1)^12 + a^2*sin(d*x + c)^14/(cos(d*x + c) + 1)^14) - 105*arctan(sin(d*x + c)/(cos
(d*x + c) + 1))/a^2)/d

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Fricas [A]  time = 1.15135, size = 189, normalized size = 1.52 \begin{align*} \frac{120 \, \cos \left (d x + c\right )^{7} - 336 \, \cos \left (d x + c\right )^{5} - 105 \, d x + 35 \,{\left (8 \, \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 )}{840 \, 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)/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/840*(120*cos(d*x + c)^7 - 336*cos(d*x + c)^5 - 105*d*x + 35*(8*cos(d*x + c)^5 - 2*cos(d*x + c)^3 - 3*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)/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.28466, size = 259, normalized size = 2.09 \begin{align*} -\frac{\frac{105 \,{\left (d x + c\right )}}{a^{2}} + \frac{2 \,{\left (105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 840 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} - 1540 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 3360 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 1085 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 840 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 6720 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 1085 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1176 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 1540 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 672 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 216\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{7} a^{2}}}{840 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/840*(105*(d*x + c)/a^2 + 2*(105*tan(1/2*d*x + 1/2*c)^13 + 840*tan(1/2*d*x + 1/2*c)^12 - 1540*tan(1/2*d*x +
1/2*c)^11 + 3360*tan(1/2*d*x + 1/2*c)^10 + 1085*tan(1/2*d*x + 1/2*c)^9 + 840*tan(1/2*d*x + 1/2*c)^8 + 6720*tan
(1/2*d*x + 1/2*c)^6 - 1085*tan(1/2*d*x + 1/2*c)^5 + 1176*tan(1/2*d*x + 1/2*c)^4 + 1540*tan(1/2*d*x + 1/2*c)^3
+ 672*tan(1/2*d*x + 1/2*c)^2 - 105*tan(1/2*d*x + 1/2*c) + 216)/((tan(1/2*d*x + 1/2*c)^2 + 1)^7*a^2))/d