3.741 \(\int \frac{\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=131 \[ -\frac{7 \cos ^5(c+d x)}{30 a^3 d}-\frac{\cos ^7(c+d x)}{6 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac{7 \sin (c+d x) \cos ^3(c+d x)}{24 a^3 d}-\frac{7 \sin (c+d x) \cos (c+d x)}{16 a^3 d}-\frac{7 x}{16 a^3}-\frac{\cos ^9(c+d x)}{3 d (a \sin (c+d x)+a)^3} \]

[Out]

(-7*x)/(16*a^3) - (7*Cos[c + d*x]^5)/(30*a^3*d) - (7*Cos[c + d*x]*Sin[c + d*x])/(16*a^3*d) - (7*Cos[c + d*x]^3
*Sin[c + d*x])/(24*a^3*d) - Cos[c + d*x]^9/(3*d*(a + a*Sin[c + d*x])^3) - Cos[c + d*x]^7/(6*d*(a^3 + a^3*Sin[c
+ d*x]))

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Rubi [A]  time = 0.16982, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2859, 2679, 2682, 2635, 8} \[ -\frac{7 \cos ^5(c+d x)}{30 a^3 d}-\frac{\cos ^7(c+d x)}{6 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac{7 \sin (c+d x) \cos ^3(c+d x)}{24 a^3 d}-\frac{7 \sin (c+d x) \cos (c+d x)}{16 a^3 d}-\frac{7 x}{16 a^3}-\frac{\cos ^9(c+d x)}{3 d (a \sin (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^8*Sin[c + d*x])/(a + a*Sin[c + d*x])^3,x]

[Out]

(-7*x)/(16*a^3) - (7*Cos[c + d*x]^5)/(30*a^3*d) - (7*Cos[c + d*x]*Sin[c + d*x])/(16*a^3*d) - (7*Cos[c + d*x]^3
*Sin[c + d*x])/(24*a^3*d) - Cos[c + d*x]^9/(3*d*(a + a*Sin[c + d*x])^3) - Cos[c + d*x]^7/(6*d*(a^3 + a^3*Sin[c
+ d*x]))

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 2679

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

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))^3} \, dx &=-\frac{\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac{\int \frac{\cos ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{a}\\ &=-\frac{\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac{\cos ^7(c+d x)}{6 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac{7 \int \frac{\cos ^6(c+d x)}{a+a \sin (c+d x)} \, dx}{6 a^2}\\ &=-\frac{7 \cos ^5(c+d x)}{30 a^3 d}-\frac{\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac{\cos ^7(c+d x)}{6 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac{7 \int \cos ^4(c+d x) \, dx}{6 a^3}\\ &=-\frac{7 \cos ^5(c+d x)}{30 a^3 d}-\frac{7 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}-\frac{\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac{\cos ^7(c+d x)}{6 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac{7 \int \cos ^2(c+d x) \, dx}{8 a^3}\\ &=-\frac{7 \cos ^5(c+d x)}{30 a^3 d}-\frac{7 \cos (c+d x) \sin (c+d x)}{16 a^3 d}-\frac{7 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}-\frac{\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac{\cos ^7(c+d x)}{6 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac{7 \int 1 \, dx}{16 a^3}\\ &=-\frac{7 x}{16 a^3}-\frac{7 \cos ^5(c+d x)}{30 a^3 d}-\frac{7 \cos (c+d x) \sin (c+d x)}{16 a^3 d}-\frac{7 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}-\frac{\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac{\cos ^7(c+d x)}{6 d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end{align*}

Mathematica [B]  time = 2.0293, size = 366, normalized size = 2.79 \[ \frac{-840 d x \sin \left (\frac{c}{2}\right )+600 \sin \left (\frac{c}{2}+d x\right )-600 \sin \left (\frac{3 c}{2}+d x\right )+15 \sin \left (\frac{3 c}{2}+2 d x\right )+15 \sin \left (\frac{5 c}{2}+2 d x\right )+140 \sin \left (\frac{5 c}{2}+3 d x\right )-140 \sin \left (\frac{7 c}{2}+3 d x\right )+105 \sin \left (\frac{7 c}{2}+4 d x\right )+105 \sin \left (\frac{9 c}{2}+4 d x\right )-36 \sin \left (\frac{9 c}{2}+5 d x\right )+36 \sin \left (\frac{11 c}{2}+5 d x\right )-5 \sin \left (\frac{11 c}{2}+6 d x\right )-5 \sin \left (\frac{13 c}{2}+6 d x\right )-21 \cos \left (\frac{c}{2}\right ) (40 d x+1)-600 \cos \left (\frac{c}{2}+d x\right )-600 \cos \left (\frac{3 c}{2}+d x\right )+15 \cos \left (\frac{3 c}{2}+2 d x\right )-15 \cos \left (\frac{5 c}{2}+2 d x\right )-140 \cos \left (\frac{5 c}{2}+3 d x\right )-140 \cos \left (\frac{7 c}{2}+3 d x\right )+105 \cos \left (\frac{7 c}{2}+4 d x\right )-105 \cos \left (\frac{9 c}{2}+4 d x\right )+36 \cos \left (\frac{9 c}{2}+5 d x\right )+36 \cos \left (\frac{11 c}{2}+5 d x\right )-5 \cos \left (\frac{11 c}{2}+6 d x\right )+5 \cos \left (\frac{13 c}{2}+6 d x\right )+21 \sin \left (\frac{c}{2}\right )}{1920 a^3 d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^8*Sin[c + d*x])/(a + a*Sin[c + d*x])^3,x]

[Out]

(-21*(1 + 40*d*x)*Cos[c/2] - 600*Cos[c/2 + d*x] - 600*Cos[(3*c)/2 + d*x] + 15*Cos[(3*c)/2 + 2*d*x] - 15*Cos[(5
*c)/2 + 2*d*x] - 140*Cos[(5*c)/2 + 3*d*x] - 140*Cos[(7*c)/2 + 3*d*x] + 105*Cos[(7*c)/2 + 4*d*x] - 105*Cos[(9*c
)/2 + 4*d*x] + 36*Cos[(9*c)/2 + 5*d*x] + 36*Cos[(11*c)/2 + 5*d*x] - 5*Cos[(11*c)/2 + 6*d*x] + 5*Cos[(13*c)/2 +
6*d*x] + 21*Sin[c/2] - 840*d*x*Sin[c/2] + 600*Sin[c/2 + d*x] - 600*Sin[(3*c)/2 + d*x] + 15*Sin[(3*c)/2 + 2*d*
x] + 15*Sin[(5*c)/2 + 2*d*x] + 140*Sin[(5*c)/2 + 3*d*x] - 140*Sin[(7*c)/2 + 3*d*x] + 105*Sin[(7*c)/2 + 4*d*x]
+ 105*Sin[(9*c)/2 + 4*d*x] - 36*Sin[(9*c)/2 + 5*d*x] + 36*Sin[(11*c)/2 + 5*d*x] - 5*Sin[(11*c)/2 + 6*d*x] - 5*
Sin[(13*c)/2 + 6*d*x])/(1920*a^3*d*(Cos[c/2] + Sin[c/2]))

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Maple [B]  time = 0.089, size = 415, normalized size = 3.2 \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))^3,x)

[Out]

-7/8/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^11-2/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2
*c)^10+73/24/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^9-18/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2
*d*x+1/2*c)^8+37/4/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^7-44/3/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6
*tan(1/2*d*x+1/2*c)^6-37/4/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^5-4/d/a^3/(1+tan(1/2*d*x+1/2*c)
^2)^6*tan(1/2*d*x+1/2*c)^4-73/24/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^3-34/5/d/a^3/(1+tan(1/2*d
*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^2+7/8/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)-22/15/d/a^3/(1+tan
(1/2*d*x+1/2*c)^2)^6-7/8/d/a^3*arctan(tan(1/2*d*x+1/2*c))

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Maxima [B]  time = 1.55939, size = 531, normalized size = 4.05 \begin{align*} \frac{\frac{\frac{105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{816 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{365 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{480 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{1110 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{1760 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{1110 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{2160 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{365 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{240 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac{105 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 176}{a^{3} + \frac{6 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{15 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{20 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{15 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{6 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac{a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac{105 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{120 \, 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))^3,x, algorithm="maxima")

[Out]

1/120*((105*sin(d*x + c)/(cos(d*x + c) + 1) - 816*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 365*sin(d*x + c)^3/(co
s(d*x + c) + 1)^3 - 480*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 1110*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1760*
sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 1110*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 2160*sin(d*x + c)^8/(cos(d*x
+ c) + 1)^8 + 365*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 240*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 105*sin(d*
x + c)^11/(cos(d*x + c) + 1)^11 - 176)/(a^3 + 6*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 15*a^3*sin(d*x + c)^
4/(cos(d*x + c) + 1)^4 + 20*a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 15*a^3*sin(d*x + c)^8/(cos(d*x + c) + 1)
^8 + 6*a^3*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + a^3*sin(d*x + c)^12/(cos(d*x + c) + 1)^12) - 105*arctan(sin
(d*x + c)/(cos(d*x + c) + 1))/a^3)/d

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Fricas [A]  time = 1.14432, size = 190, normalized size = 1.45 \begin{align*} \frac{144 \, \cos \left (d x + c\right )^{5} - 320 \, \cos \left (d x + c\right )^{3} - 105 \, d x - 5 \,{\left (8 \, \cos \left (d x + c\right )^{5} - 50 \, \cos \left (d x + c\right )^{3} + 21 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, a^{3} 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))^3,x, algorithm="fricas")

[Out]

1/240*(144*cos(d*x + c)^5 - 320*cos(d*x + c)^3 - 105*d*x - 5*(8*cos(d*x + c)^5 - 50*cos(d*x + c)^3 + 21*cos(d*
x + c))*sin(d*x + c))/(a^3*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))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.27153, size = 242, normalized size = 1.85 \begin{align*} -\frac{\frac{105 \,{\left (d x + c\right )}}{a^{3}} + \frac{2 \,{\left (105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 240 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 365 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 2160 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 1110 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 1760 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 1110 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 480 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 365 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 816 \, \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 ) + 176\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{6} a^{3}}}{240 \, 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))^3,x, algorithm="giac")

[Out]

-1/240*(105*(d*x + c)/a^3 + 2*(105*tan(1/2*d*x + 1/2*c)^11 + 240*tan(1/2*d*x + 1/2*c)^10 - 365*tan(1/2*d*x + 1
/2*c)^9 + 2160*tan(1/2*d*x + 1/2*c)^8 - 1110*tan(1/2*d*x + 1/2*c)^7 + 1760*tan(1/2*d*x + 1/2*c)^6 + 1110*tan(1
/2*d*x + 1/2*c)^5 + 480*tan(1/2*d*x + 1/2*c)^4 + 365*tan(1/2*d*x + 1/2*c)^3 + 816*tan(1/2*d*x + 1/2*c)^2 - 105
*tan(1/2*d*x + 1/2*c) + 176)/((tan(1/2*d*x + 1/2*c)^2 + 1)^6*a^3))/d