3.85 \(\int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^5} \, dx\)

Optimal. Leaf size=155 \[ \frac{83 \sin (c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}-\frac{142 \sin (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}+\frac{67 \sin (c+d x)}{315 a^2 d (a \cos (c+d x)+a)^3}-\frac{\sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{11 \sin (c+d x) \cos ^2(c+d x)}{63 a d (a \cos (c+d x)+a)^4} \]

[Out]

-(Cos[c + d*x]^3*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (11*Cos[c + d*x]^2*Sin[c + d*x])/(63*a*d*(a + a*
Cos[c + d*x])^4) + (67*Sin[c + d*x])/(315*a^2*d*(a + a*Cos[c + d*x])^3) - (142*Sin[c + d*x])/(315*a^3*d*(a + a
*Cos[c + d*x])^2) + (83*Sin[c + d*x])/(315*d*(a^5 + a^5*Cos[c + d*x]))

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Rubi [A]  time = 0.300067, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2765, 2977, 2968, 3019, 2750, 2648} \[ \frac{83 \sin (c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}-\frac{142 \sin (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}+\frac{67 \sin (c+d x)}{315 a^2 d (a \cos (c+d x)+a)^3}-\frac{\sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{11 \sin (c+d x) \cos ^2(c+d x)}{63 a d (a \cos (c+d x)+a)^4} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4/(a + a*Cos[c + d*x])^5,x]

[Out]

-(Cos[c + d*x]^3*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (11*Cos[c + d*x]^2*Sin[c + d*x])/(63*a*d*(a + a*
Cos[c + d*x])^4) + (67*Sin[c + d*x])/(315*a^2*d*(a + a*Cos[c + d*x])^3) - (142*Sin[c + d*x])/(315*a^3*d*(a + a
*Cos[c + d*x])^2) + (83*Sin[c + d*x])/(315*d*(a^5 + a^5*Cos[c + d*x]))

Rule 2765

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

Rule 2977

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]
)^n)/(a*f*(2*m + 1)), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n -
1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ
[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3019

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_
.)*(x_)]^2), x_Symbol] :> Simp[((A*b - a*B + b*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(a*f*(2*m + 1)), x] + D
ist[1/(a^2*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[a*A*(m + 1) + m*(b*B - a*C) + b*C*(2*m + 1)*Sin[e
+ f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]

Rule 2750

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

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^5} \, dx &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{\int \frac{\cos ^2(c+d x) (3 a-8 a \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx}{9 a^2}\\ &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{11 \cos ^2(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac{\int \frac{\cos (c+d x) \left (22 a^2-45 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{11 \cos ^2(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac{\int \frac{22 a^2 \cos (c+d x)-45 a^2 \cos ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{11 \cos ^2(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac{67 \sin (c+d x)}{315 a^2 d (a+a \cos (c+d x))^3}+\frac{\int \frac{-201 a^3+225 a^3 \cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{11 \cos ^2(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac{67 \sin (c+d x)}{315 a^2 d (a+a \cos (c+d x))^3}-\frac{142 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}+\frac{83 \int \frac{1}{a+a \cos (c+d x)} \, dx}{315 a^4}\\ &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{11 \cos ^2(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac{67 \sin (c+d x)}{315 a^2 d (a+a \cos (c+d x))^3}-\frac{142 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}+\frac{83 \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.263809, size = 138, normalized size = 0.89 \[ \frac{\sec \left (\frac{c}{2}\right ) \left (-5040 \sin \left (c+\frac{d x}{2}\right )+3612 \sin \left (c+\frac{3 d x}{2}\right )-3360 \sin \left (2 c+\frac{3 d x}{2}\right )+1728 \sin \left (2 c+\frac{5 d x}{2}\right )-1260 \sin \left (3 c+\frac{5 d x}{2}\right )+432 \sin \left (3 c+\frac{7 d x}{2}\right )-315 \sin \left (4 c+\frac{7 d x}{2}\right )+83 \sin \left (4 c+\frac{9 d x}{2}\right )+5418 \sin \left (\frac{d x}{2}\right )\right ) \sec ^9\left (\frac{1}{2} (c+d x)\right )}{80640 a^5 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^4/(a + a*Cos[c + d*x])^5,x]

[Out]

(Sec[c/2]*Sec[(c + d*x)/2]^9*(5418*Sin[(d*x)/2] - 5040*Sin[c + (d*x)/2] + 3612*Sin[c + (3*d*x)/2] - 3360*Sin[2
*c + (3*d*x)/2] + 1728*Sin[2*c + (5*d*x)/2] - 1260*Sin[3*c + (5*d*x)/2] + 432*Sin[3*c + (7*d*x)/2] - 315*Sin[4
*c + (7*d*x)/2] + 83*Sin[4*c + (9*d*x)/2]))/(80640*a^5*d)

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Maple [A]  time = 0.039, size = 71, normalized size = 0.5 \begin{align*}{\frac{1}{16\,d{a}^{5}} \left ({\frac{1}{9} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}-{\frac{4}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{6}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{4}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+\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)^4/(a+cos(d*x+c)*a)^5,x)

[Out]

1/16/d/a^5*(1/9*tan(1/2*d*x+1/2*c)^9-4/7*tan(1/2*d*x+1/2*c)^7+6/5*tan(1/2*d*x+1/2*c)^5-4/3*tan(1/2*d*x+1/2*c)^
3+tan(1/2*d*x+1/2*c))

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Maxima [A]  time = 1.13768, size = 144, normalized size = 0.93 \begin{align*} \frac{\frac{315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{420 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{378 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{180 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{5040 \, a^{5} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4/(a+a*cos(d*x+c))^5,x, algorithm="maxima")

[Out]

1/5040*(315*sin(d*x + c)/(cos(d*x + c) + 1) - 420*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 378*sin(d*x + c)^5/(co
s(d*x + c) + 1)^5 - 180*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 35*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/(a^5*d)

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Fricas [A]  time = 1.57359, size = 316, normalized size = 2.04 \begin{align*} \frac{{\left (83 \, \cos \left (d x + c\right )^{4} + 100 \, \cos \left (d x + c\right )^{3} + 84 \, \cos \left (d x + c\right )^{2} + 40 \, \cos \left (d x + c\right ) + 8\right )} \sin \left (d x + c\right )}{315 \,{\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4/(a+a*cos(d*x+c))^5,x, algorithm="fricas")

[Out]

1/315*(83*cos(d*x + c)^4 + 100*cos(d*x + c)^3 + 84*cos(d*x + c)^2 + 40*cos(d*x + c) + 8)*sin(d*x + c)/(a^5*d*c
os(d*x + c)^5 + 5*a^5*d*cos(d*x + c)^4 + 10*a^5*d*cos(d*x + c)^3 + 10*a^5*d*cos(d*x + c)^2 + 5*a^5*d*cos(d*x +
c) + a^5*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)**4/(a+a*cos(d*x+c))**5,x)

[Out]

Timed out

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Giac [A]  time = 1.4085, size = 97, normalized size = 0.63 \begin{align*} \frac{35 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 180 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 378 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 420 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 315 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{5040 \, a^{5} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4/(a+a*cos(d*x+c))^5,x, algorithm="giac")

[Out]

1/5040*(35*tan(1/2*d*x + 1/2*c)^9 - 180*tan(1/2*d*x + 1/2*c)^7 + 378*tan(1/2*d*x + 1/2*c)^5 - 420*tan(1/2*d*x
+ 1/2*c)^3 + 315*tan(1/2*d*x + 1/2*c))/(a^5*d)