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

Optimal. Leaf size=127 \[ \frac{2 \cos ^5(c+d x)}{5 a^3 d (a \sin (c+d x)+a)^5}-\frac{2 \cos ^3(c+d x)}{3 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}+\frac{2 \cos (c+d x)}{d \left (a^8 \sin (c+d x)+a^8\right )}+\frac{x}{a^8}-\frac{2 \cos ^7(c+d x)}{7 a d (a \sin (c+d x)+a)^7} \]

[Out]

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

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Rubi [A]  time = 0.182357, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2680, 8} \[ \frac{2 \cos ^5(c+d x)}{5 a^3 d (a \sin (c+d x)+a)^5}-\frac{2 \cos ^3(c+d x)}{3 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}+\frac{2 \cos (c+d x)}{d \left (a^8 \sin (c+d x)+a^8\right )}+\frac{x}{a^8}-\frac{2 \cos ^7(c+d x)}{7 a d (a \sin (c+d x)+a)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2680

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

Rule 8

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

Rubi steps

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

Mathematica [B]  time = 6.07156, size = 275, normalized size = 2.17 \[ -\frac{2 \sqrt{2} \left (\frac{1}{2} (1-\sin (c+d x))-1\right )^4 \left (\frac{\sin ^{-1}\left (\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right ) \sqrt{1-\sin (c+d x)}}{\sqrt{2} \sqrt{\frac{1}{2} (\sin (c+d x)-1)+1}}+\frac{(1-\sin (c+d x))^4}{112 \left (\frac{1}{2} (1-\sin (c+d x))-1\right )^4}+\frac{(1-\sin (c+d x))^3}{40 \left (\frac{1}{2} (1-\sin (c+d x))-1\right )^3}+\frac{(1-\sin (c+d x))^2}{12 \left (\frac{1}{2} (1-\sin (c+d x))-1\right )^2}+\frac{1-\sin (c+d x)}{2 \left (\frac{1}{2} (1-\sin (c+d x))-1\right )}\right ) \cos ^9(c+d x)}{a^8 d \left (\frac{1}{2} (\sin (c+d x)-1)+1\right )^{7/2} (1-\sin (c+d x))^5 (\sin (c+d x)+1)^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2]*Cos[c + d*x]^9*(-1 + (1 - Sin[c + d*x])/2)^4*((ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin
[c + d*x]])/(Sqrt[2]*Sqrt[1 + (-1 + Sin[c + d*x])/2]) + (1 - Sin[c + d*x])/(2*(-1 + (1 - Sin[c + d*x])/2)) + (
1 - Sin[c + d*x])^2/(12*(-1 + (1 - Sin[c + d*x])/2)^2) + (1 - Sin[c + d*x])^3/(40*(-1 + (1 - Sin[c + d*x])/2)^
3) + (1 - Sin[c + d*x])^4/(112*(-1 + (1 - Sin[c + d*x])/2)^4)))/(a^8*d*(1 + (-1 + Sin[c + d*x])/2)^(7/2)*(1 -
Sin[c + d*x])^5*(1 + Sin[c + d*x])^(9/2))

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Maple [A]  time = 0.125, size = 146, normalized size = 1.2 \begin{align*} 2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{8}}}-{\frac{256}{7\,d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-7}}+128\,{\frac{1}{d{a}^{8} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{6}}}-{\frac{896}{5\,d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-5}}+128\,{\frac{1}{d{a}^{8} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{4}}}-{\frac{160}{3\,d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}+16\,{\frac{1}{d{a}^{8} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d/a^8*arctan(tan(1/2*d*x+1/2*c))-256/7/d/a^8/(tan(1/2*d*x+1/2*c)+1)^7+128/d/a^8/(tan(1/2*d*x+1/2*c)+1)^6-896
/5/d/a^8/(tan(1/2*d*x+1/2*c)+1)^5+128/d/a^8/(tan(1/2*d*x+1/2*c)+1)^4-160/3/d/a^8/(tan(1/2*d*x+1/2*c)+1)^3+16/d
/a^8/(tan(1/2*d*x+1/2*c)+1)^2

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Maxima [B]  time = 1.51588, size = 398, normalized size = 3.13 \begin{align*} \frac{2 \,{\left (\frac{8 \,{\left (\frac{133 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{294 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{490 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{175 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{105 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 19\right )}}{a^{8} + \frac{7 \, a^{8} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{21 \, a^{8} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{35 \, a^{8} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{35 \, a^{8} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{21 \, a^{8} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{7 \, a^{8} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{a^{8} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}} + \frac{105 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{8}}\right )}}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(8*(133*sin(d*x + c)/(cos(d*x + c) + 1) + 294*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 490*sin(d*x + c)^3/(
cos(d*x + c) + 1)^3 + 175*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 105*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 19)/
(a^8 + 7*a^8*sin(d*x + c)/(cos(d*x + c) + 1) + 21*a^8*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 35*a^8*sin(d*x + c
)^3/(cos(d*x + c) + 1)^3 + 35*a^8*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 21*a^8*sin(d*x + c)^5/(cos(d*x + c) +
1)^5 + 7*a^8*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a^8*sin(d*x + c)^7/(cos(d*x + c) + 1)^7) + 105*arctan(sin(d
*x + c)/(cos(d*x + c) + 1))/a^8)/d

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Fricas [B]  time = 1.71823, size = 655, normalized size = 5.16 \begin{align*} \frac{{\left (105 \, d x - 352\right )} \cos \left (d x + c\right )^{4} -{\left (315 \, d x + 568\right )} \cos \left (d x + c\right )^{3} - 24 \,{\left (35 \, d x - 31\right )} \cos \left (d x + c\right )^{2} + 840 \, d x + 60 \,{\left (7 \, d x + 12\right )} \cos \left (d x + c\right ) -{\left ({\left (105 \, d x + 352\right )} \cos \left (d x + c\right )^{3} + 12 \,{\left (35 \, d x - 18\right )} \cos \left (d x + c\right )^{2} - 840 \, d x - 60 \,{\left (7 \, d x + 16\right )} \cos \left (d x + c\right ) - 240\right )} \sin \left (d x + c\right ) - 240}{105 \,{\left (a^{8} d \cos \left (d x + c\right )^{4} - 3 \, a^{8} d \cos \left (d x + c\right )^{3} - 8 \, a^{8} d \cos \left (d x + c\right )^{2} + 4 \, a^{8} d \cos \left (d x + c\right ) + 8 \, a^{8} d -{\left (a^{8} d \cos \left (d x + c\right )^{3} + 4 \, a^{8} d \cos \left (d x + c\right )^{2} - 4 \, a^{8} d \cos \left (d x + c\right ) - 8 \, 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)^8/(a+a*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/105*((105*d*x - 352)*cos(d*x + c)^4 - (315*d*x + 568)*cos(d*x + c)^3 - 24*(35*d*x - 31)*cos(d*x + c)^2 + 840
*d*x + 60*(7*d*x + 12)*cos(d*x + c) - ((105*d*x + 352)*cos(d*x + c)^3 + 12*(35*d*x - 18)*cos(d*x + c)^2 - 840*
d*x - 60*(7*d*x + 16)*cos(d*x + c) - 240)*sin(d*x + c) - 240)/(a^8*d*cos(d*x + c)^4 - 3*a^8*d*cos(d*x + c)^3 -
8*a^8*d*cos(d*x + c)^2 + 4*a^8*d*cos(d*x + c) + 8*a^8*d - (a^8*d*cos(d*x + c)^3 + 4*a^8*d*cos(d*x + c)^2 - 4*
a^8*d*cos(d*x + c) - 8*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)**8/(a+a*sin(d*x+c))**8,x)

[Out]

Timed out

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Giac [A]  time = 1.20658, size = 134, normalized size = 1.06 \begin{align*} \frac{\frac{105 \,{\left (d x + c\right )}}{a^{8}} + \frac{16 \,{\left (105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 175 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 490 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 294 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 133 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 19\right )}}{a^{8}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{7}}}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(105*(d*x + c)/a^8 + 16*(105*tan(1/2*d*x + 1/2*c)^5 + 175*tan(1/2*d*x + 1/2*c)^4 + 490*tan(1/2*d*x + 1/2
*c)^3 + 294*tan(1/2*d*x + 1/2*c)^2 + 133*tan(1/2*d*x + 1/2*c) + 19)/(a^8*(tan(1/2*d*x + 1/2*c) + 1)^7))/d