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

Optimal. Leaf size=103 \[ \frac{7 \cos ^5(c+d x)}{15 a^3 d}+\frac{7 \sin (c+d x) \cos ^3(c+d x)}{12 a^3 d}+\frac{7 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac{7 x}{8 a^3}+\frac{2 \cos ^7(c+d x)}{3 a d (a \sin (c+d x)+a)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.109426, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2680, 2682, 2635, 8} \[ \frac{7 \cos ^5(c+d x)}{15 a^3 d}+\frac{7 \sin (c+d x) \cos ^3(c+d x)}{12 a^3 d}+\frac{7 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac{7 x}{8 a^3}+\frac{2 \cos ^7(c+d x)}{3 a d (a \sin (c+d x)+a)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 1.09445, size = 141, normalized size = 1.37 \[ -\frac{\left (\sqrt{\sin (c+d x)+1} \left (24 \sin ^5(c+d x)-114 \sin ^4(c+d x)+202 \sin ^3(c+d x)-127 \sin ^2(c+d x)-121 \sin (c+d x)+136\right )-210 \sin ^{-1}\left (\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right ) \sqrt{1-\sin (c+d x)}\right ) \cos ^9(c+d x)}{120 a^3 d (\sin (c+d x)-1)^5 (\sin (c+d x)+1)^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cos[c + d*x]^9*(-210*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*
(136 - 121*Sin[c + d*x] - 127*Sin[c + d*x]^2 + 202*Sin[c + d*x]^3 - 114*Sin[c + d*x]^4 + 24*Sin[c + d*x]^5)))/
(120*a^3*d*(-1 + Sin[c + d*x])^5*(1 + Sin[c + d*x])^(9/2))

________________________________________________________________________________________

Maple [B]  time = 0.083, size = 313, normalized size = 3. \begin{align*} -{\frac{1}{4\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}+6\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}}{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{5}}}-{\frac{13}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}+16\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}}{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{5}}}+{\frac{20}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}+{\frac{13}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}+{\frac{16}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}+{\frac{1}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}+{\frac{34}{15\,d{a}^{3}} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}+{\frac{7}{4\,d{a}^{3}}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \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))^3,x)

[Out]

-1/4/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^9+6/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*
c)^8-13/2/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^7+16/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*
x+1/2*c)^6+20/3/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^4+13/2/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^5*ta
n(1/2*d*x+1/2*c)^3+16/3/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^2+1/4/d/a^3/(1+tan(1/2*d*x+1/2*c)^
2)^5*tan(1/2*d*x+1/2*c)+34/15/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^5+7/4/d/a^3*arctan(tan(1/2*d*x+1/2*c))

________________________________________________________________________________________

Maxima [B]  time = 1.47899, size = 419, normalized size = 4.07 \begin{align*} \frac{\frac{\frac{15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{320 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{390 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{400 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{960 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{390 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{360 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{15 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + 136}{a^{3} + \frac{5 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{10 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{10 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{5 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} + \frac{105 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{60 \, 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))^3,x, algorithm="maxima")

[Out]

1/60*((15*sin(d*x + c)/(cos(d*x + c) + 1) + 320*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 390*sin(d*x + c)^3/(cos(
d*x + c) + 1)^3 + 400*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 960*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 390*sin(
d*x + c)^7/(cos(d*x + c) + 1)^7 + 360*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 15*sin(d*x + c)^9/(cos(d*x + c) +
1)^9 + 136)/(a^3 + 5*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 10*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 10
*a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 5*a^3*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + a^3*sin(d*x + c)^10/(co
s(d*x + c) + 1)^10) + 105*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d

________________________________________________________________________________________

Fricas [A]  time = 1.98222, size = 163, normalized size = 1.58 \begin{align*} -\frac{24 \, \cos \left (d x + c\right )^{5} - 160 \, \cos \left (d x + c\right )^{3} - 105 \, d x + 15 \,{\left (6 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, a^{3} 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))^3,x, algorithm="fricas")

[Out]

-1/120*(24*cos(d*x + c)^5 - 160*cos(d*x + c)^3 - 105*d*x + 15*(6*cos(d*x + c)^3 - 7*cos(d*x + c))*sin(d*x + c)
)/(a^3*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)**8/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.16394, size = 189, normalized size = 1.83 \begin{align*} \frac{\frac{105 \,{\left (d x + c\right )}}{a^{3}} - \frac{2 \,{\left (15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 360 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 390 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 960 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 400 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 390 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 320 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 136\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{5} a^{3}}}{120 \, 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))^3,x, algorithm="giac")

[Out]

1/120*(105*(d*x + c)/a^3 - 2*(15*tan(1/2*d*x + 1/2*c)^9 - 360*tan(1/2*d*x + 1/2*c)^8 + 390*tan(1/2*d*x + 1/2*c
)^7 - 960*tan(1/2*d*x + 1/2*c)^6 - 400*tan(1/2*d*x + 1/2*c)^4 - 390*tan(1/2*d*x + 1/2*c)^3 - 320*tan(1/2*d*x +
1/2*c)^2 - 15*tan(1/2*d*x + 1/2*c) - 136)/((tan(1/2*d*x + 1/2*c)^2 + 1)^5*a^3))/d