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

Optimal. Leaf size=183 \[ -\frac{8 \cos ^3(c+d x)}{9009 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}-\frac{8 \cos ^3(c+d x)}{3003 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac{20 \cos ^3(c+d x)}{3003 a^3 d (a \sin (c+d x)+a)^5}-\frac{20 \cos ^3(c+d x)}{1287 a^2 d (a \sin (c+d x)+a)^6}-\frac{5 \cos ^3(c+d x)}{143 a d (a \sin (c+d x)+a)^7}-\frac{\cos ^3(c+d x)}{13 d (a \sin (c+d x)+a)^8} \]

[Out]

-Cos[c + d*x]^3/(13*d*(a + a*Sin[c + d*x])^8) - (5*Cos[c + d*x]^3)/(143*a*d*(a + a*Sin[c + d*x])^7) - (20*Cos[
c + d*x]^3)/(1287*a^2*d*(a + a*Sin[c + d*x])^6) - (20*Cos[c + d*x]^3)/(3003*a^3*d*(a + a*Sin[c + d*x])^5) - (8
*Cos[c + d*x]^3)/(3003*d*(a^2 + a^2*Sin[c + d*x])^4) - (8*Cos[c + d*x]^3)/(9009*a^2*d*(a^2 + a^2*Sin[c + d*x])
^3)

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Rubi [A]  time = 0.271703, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2672, 2671} \[ -\frac{8 \cos ^3(c+d x)}{9009 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}-\frac{8 \cos ^3(c+d x)}{3003 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac{20 \cos ^3(c+d x)}{3003 a^3 d (a \sin (c+d x)+a)^5}-\frac{20 \cos ^3(c+d x)}{1287 a^2 d (a \sin (c+d x)+a)^6}-\frac{5 \cos ^3(c+d x)}{143 a d (a \sin (c+d x)+a)^7}-\frac{\cos ^3(c+d x)}{13 d (a \sin (c+d x)+a)^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cos[c + d*x]^3/(13*d*(a + a*Sin[c + d*x])^8) - (5*Cos[c + d*x]^3)/(143*a*d*(a + a*Sin[c + d*x])^7) - (20*Cos[
c + d*x]^3)/(1287*a^2*d*(a + a*Sin[c + d*x])^6) - (20*Cos[c + d*x]^3)/(3003*a^3*d*(a + a*Sin[c + d*x])^5) - (8
*Cos[c + d*x]^3)/(3003*d*(a^2 + a^2*Sin[c + d*x])^4) - (8*Cos[c + d*x]^3)/(9009*a^2*d*(a^2 + a^2*Sin[c + d*x])
^3)

Rule 2672

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

Rule 2671

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

Rubi steps

\begin{align*} \int \frac{\cos ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=-\frac{\cos ^3(c+d x)}{13 d (a+a \sin (c+d x))^8}+\frac{5 \int \frac{\cos ^2(c+d x)}{(a+a \sin (c+d x))^7} \, dx}{13 a}\\ &=-\frac{\cos ^3(c+d x)}{13 d (a+a \sin (c+d x))^8}-\frac{5 \cos ^3(c+d x)}{143 a d (a+a \sin (c+d x))^7}+\frac{20 \int \frac{\cos ^2(c+d x)}{(a+a \sin (c+d x))^6} \, dx}{143 a^2}\\ &=-\frac{\cos ^3(c+d x)}{13 d (a+a \sin (c+d x))^8}-\frac{5 \cos ^3(c+d x)}{143 a d (a+a \sin (c+d x))^7}-\frac{20 \cos ^3(c+d x)}{1287 a^2 d (a+a \sin (c+d x))^6}+\frac{20 \int \frac{\cos ^2(c+d x)}{(a+a \sin (c+d x))^5} \, dx}{429 a^3}\\ &=-\frac{\cos ^3(c+d x)}{13 d (a+a \sin (c+d x))^8}-\frac{5 \cos ^3(c+d x)}{143 a d (a+a \sin (c+d x))^7}-\frac{20 \cos ^3(c+d x)}{1287 a^2 d (a+a \sin (c+d x))^6}-\frac{20 \cos ^3(c+d x)}{3003 a^3 d (a+a \sin (c+d x))^5}+\frac{40 \int \frac{\cos ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx}{3003 a^4}\\ &=-\frac{\cos ^3(c+d x)}{13 d (a+a \sin (c+d x))^8}-\frac{5 \cos ^3(c+d x)}{143 a d (a+a \sin (c+d x))^7}-\frac{20 \cos ^3(c+d x)}{1287 a^2 d (a+a \sin (c+d x))^6}-\frac{20 \cos ^3(c+d x)}{3003 a^3 d (a+a \sin (c+d x))^5}-\frac{8 \cos ^3(c+d x)}{3003 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac{8 \int \frac{\cos ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx}{3003 a^5}\\ &=-\frac{\cos ^3(c+d x)}{13 d (a+a \sin (c+d x))^8}-\frac{5 \cos ^3(c+d x)}{143 a d (a+a \sin (c+d x))^7}-\frac{20 \cos ^3(c+d x)}{1287 a^2 d (a+a \sin (c+d x))^6}-\frac{20 \cos ^3(c+d x)}{3003 a^3 d (a+a \sin (c+d x))^5}-\frac{8 \cos ^3(c+d x)}{9009 a^5 d (a+a \sin (c+d x))^3}-\frac{8 \cos ^3(c+d x)}{3003 d \left (a^2+a^2 \sin (c+d x)\right )^4}\\ \end{align*}

Mathematica [A]  time = 0.121781, size = 78, normalized size = 0.43 \[ -\frac{\left (8 \sin ^5(c+d x)+64 \sin ^4(c+d x)+236 \sin ^3(c+d x)+544 \sin ^2(c+d x)+911 \sin (c+d x)+1240\right ) \cos ^3(c+d x)}{9009 a^8 d (\sin (c+d x)+1)^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cos[c + d*x]^3*(1240 + 911*Sin[c + d*x] + 544*Sin[c + d*x]^2 + 236*Sin[c + d*x]^3 + 64*Sin[c + d*x]^4 + 8*Si
n[c + d*x]^5))/(9009*a^8*d*(1 + Sin[c + d*x])^8)

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Maple [A]  time = 0.148, size = 205, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{d{a}^{8}} \left ( 432\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-10}-{\frac{4528}{7\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{7}}}+64\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-12}+{\frac{1336}{3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{6}}}-{\frac{94}{3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{3}}}-{\frac{5840}{9\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{9}}}- \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-1}-{\frac{128}{13\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{13}}}-240\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-5}+7\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-2}+100\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-4}-{\frac{2272}{11\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{11}}}+736\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-8} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d/a^8*(432/(tan(1/2*d*x+1/2*c)+1)^10-4528/7/(tan(1/2*d*x+1/2*c)+1)^7+64/(tan(1/2*d*x+1/2*c)+1)^12+1336/3/(ta
n(1/2*d*x+1/2*c)+1)^6-94/3/(tan(1/2*d*x+1/2*c)+1)^3-5840/9/(tan(1/2*d*x+1/2*c)+1)^9-1/(tan(1/2*d*x+1/2*c)+1)-1
28/13/(tan(1/2*d*x+1/2*c)+1)^13-240/(tan(1/2*d*x+1/2*c)+1)^5+7/(tan(1/2*d*x+1/2*c)+1)^2+100/(tan(1/2*d*x+1/2*c
)+1)^4-2272/11/(tan(1/2*d*x+1/2*c)+1)^11+736/(tan(1/2*d*x+1/2*c)+1)^8)

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Maxima [B]  time = 1.06583, size = 738, normalized size = 4.03 \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)^2/(a+a*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

-2/9009*(7111*sin(d*x + c)/(cos(d*x + c) + 1) + 51675*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 171457*sin(d*x + c
)^3/(cos(d*x + c) + 1)^3 + 451165*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 785070*sin(d*x + c)^5/(cos(d*x + c) +
1)^5 + 1076790*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 1051050*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 810810*sin(
d*x + c)^8/(cos(d*x + c) + 1)^8 + 435435*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 183183*sin(d*x + c)^10/(cos(d*x
+ c) + 1)^10 + 45045*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 + 9009*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 124
0)/((a^8 + 13*a^8*sin(d*x + c)/(cos(d*x + c) + 1) + 78*a^8*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 286*a^8*sin(d
*x + c)^3/(cos(d*x + c) + 1)^3 + 715*a^8*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1287*a^8*sin(d*x + c)^5/(cos(d*
x + c) + 1)^5 + 1716*a^8*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 1716*a^8*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 +
1287*a^8*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 715*a^8*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 286*a^8*sin(d*x +
c)^10/(cos(d*x + c) + 1)^10 + 78*a^8*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 + 13*a^8*sin(d*x + c)^12/(cos(d*x
+ c) + 1)^12 + a^8*sin(d*x + c)^13/(cos(d*x + c) + 1)^13)*d)

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Fricas [A]  time = 1.72716, size = 896, normalized size = 4.9 \begin{align*} \frac{8 \, \cos \left (d x + c\right )^{7} - 48 \, \cos \left (d x + c\right )^{6} - 196 \, \cos \left (d x + c\right )^{5} + 280 \, \cos \left (d x + c\right )^{4} + 735 \, \cos \left (d x + c\right )^{3} - 378 \, \cos \left (d x + c\right )^{2} -{\left (8 \, \cos \left (d x + c\right )^{6} + 56 \, \cos \left (d x + c\right )^{5} - 140 \, \cos \left (d x + c\right )^{4} - 420 \, \cos \left (d x + c\right )^{3} + 315 \, \cos \left (d x + c\right )^{2} + 693 \, \cos \left (d x + c\right ) + 1386\right )} \sin \left (d x + c\right ) + 693 \, \cos \left (d x + c\right ) + 1386}{9009 \,{\left (a^{8} d \cos \left (d x + c\right )^{7} + 7 \, a^{8} d \cos \left (d x + c\right )^{6} - 18 \, a^{8} d \cos \left (d x + c\right )^{5} - 56 \, a^{8} d \cos \left (d x + c\right )^{4} + 48 \, a^{8} d \cos \left (d x + c\right )^{3} + 112 \, a^{8} d \cos \left (d x + c\right )^{2} - 32 \, a^{8} d \cos \left (d x + c\right ) - 64 \, a^{8} d +{\left (a^{8} d \cos \left (d x + c\right )^{6} - 6 \, a^{8} d \cos \left (d x + c\right )^{5} - 24 \, a^{8} d \cos \left (d x + c\right )^{4} + 32 \, a^{8} d \cos \left (d x + c\right )^{3} + 80 \, a^{8} d \cos \left (d x + c\right )^{2} - 32 \, a^{8} d \cos \left (d x + c\right ) - 64 \, a^{8} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9009*(8*cos(d*x + c)^7 - 48*cos(d*x + c)^6 - 196*cos(d*x + c)^5 + 280*cos(d*x + c)^4 + 735*cos(d*x + c)^3 -
378*cos(d*x + c)^2 - (8*cos(d*x + c)^6 + 56*cos(d*x + c)^5 - 140*cos(d*x + c)^4 - 420*cos(d*x + c)^3 + 315*cos
(d*x + c)^2 + 693*cos(d*x + c) + 1386)*sin(d*x + c) + 693*cos(d*x + c) + 1386)/(a^8*d*cos(d*x + c)^7 + 7*a^8*d
*cos(d*x + c)^6 - 18*a^8*d*cos(d*x + c)^5 - 56*a^8*d*cos(d*x + c)^4 + 48*a^8*d*cos(d*x + c)^3 + 112*a^8*d*cos(
d*x + c)^2 - 32*a^8*d*cos(d*x + c) - 64*a^8*d + (a^8*d*cos(d*x + c)^6 - 6*a^8*d*cos(d*x + c)^5 - 24*a^8*d*cos(
d*x + c)^4 + 32*a^8*d*cos(d*x + c)^3 + 80*a^8*d*cos(d*x + c)^2 - 32*a^8*d*cos(d*x + c) - 64*a^8*d)*sin(d*x + c
))

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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)**2/(a+a*sin(d*x+c))**8,x)

[Out]

Timed out

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Giac [A]  time = 1.18117, size = 239, normalized size = 1.31 \begin{align*} -\frac{2 \,{\left (9009 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} + 45045 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 183183 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 435435 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 810810 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 1051050 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 1076790 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 785070 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 451165 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 171457 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 51675 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 7111 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1240\right )}}{9009 \, a^{8} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9009*(9009*tan(1/2*d*x + 1/2*c)^12 + 45045*tan(1/2*d*x + 1/2*c)^11 + 183183*tan(1/2*d*x + 1/2*c)^10 + 43543
5*tan(1/2*d*x + 1/2*c)^9 + 810810*tan(1/2*d*x + 1/2*c)^8 + 1051050*tan(1/2*d*x + 1/2*c)^7 + 1076790*tan(1/2*d*
x + 1/2*c)^6 + 785070*tan(1/2*d*x + 1/2*c)^5 + 451165*tan(1/2*d*x + 1/2*c)^4 + 171457*tan(1/2*d*x + 1/2*c)^3 +
51675*tan(1/2*d*x + 1/2*c)^2 + 7111*tan(1/2*d*x + 1/2*c) + 1240)/(a^8*d*(tan(1/2*d*x + 1/2*c) + 1)^13)