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

Optimal. Leaf size=168 \[ -\frac{34 \sin (c+d x) \cos ^2(c+d x)}{105 a^2 d (a \cos (c+d x)+a)^3}-\frac{661 \sin (c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}+\frac{173 \sin (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}+\frac{x}{a^5}-\frac{\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{13 \sin (c+d x) \cos ^3(c+d x)}{63 a d (a \cos (c+d x)+a)^4} \]

[Out]

x/a^5 - (Cos[c + d*x]^4*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (13*Cos[c + d*x]^3*Sin[c + d*x])/(63*a*d*
(a + a*Cos[c + d*x])^4) - (34*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^3) + (173*Sin[c + d
*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) - (661*Sin[c + d*x])/(315*d*(a^5 + a^5*Cos[c + d*x]))

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Rubi [A]  time = 0.392132, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2765, 2977, 2968, 3019, 2735, 2648} \[ -\frac{34 \sin (c+d x) \cos ^2(c+d x)}{105 a^2 d (a \cos (c+d x)+a)^3}-\frac{661 \sin (c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}+\frac{173 \sin (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}+\frac{x}{a^5}-\frac{\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{13 \sin (c+d x) \cos ^3(c+d x)}{63 a d (a \cos (c+d x)+a)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/a^5 - (Cos[c + d*x]^4*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (13*Cos[c + d*x]^3*Sin[c + d*x])/(63*a*d*
(a + a*Cos[c + d*x])^4) - (34*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^3) + (173*Sin[c + d
*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) - (661*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 2735

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

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 ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx &=-\frac{\cos ^4(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{\int \frac{\cos ^3(c+d x) (4 a-9 a \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx}{9 a^2}\\ &=-\frac{\cos ^4(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{13 \cos ^3(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac{\int \frac{\cos ^2(c+d x) \left (39 a^2-63 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac{\cos ^4(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{13 \cos ^3(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac{34 \cos ^2(c+d x) \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}-\frac{\int \frac{\cos (c+d x) \left (204 a^3-315 a^3 \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac{\cos ^4(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{13 \cos ^3(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac{34 \cos ^2(c+d x) \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}-\frac{\int \frac{204 a^3 \cos (c+d x)-315 a^3 \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac{\cos ^4(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{13 \cos ^3(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac{34 \cos ^2(c+d x) \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}+\frac{173 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}+\frac{\int \frac{-1038 a^4+945 a^4 \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{945 a^8}\\ &=\frac{x}{a^5}-\frac{\cos ^4(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{13 \cos ^3(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac{34 \cos ^2(c+d x) \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}+\frac{173 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac{661 \int \frac{1}{a+a \cos (c+d x)} \, dx}{315 a^4}\\ &=\frac{x}{a^5}-\frac{\cos ^4(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{13 \cos ^3(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac{34 \cos ^2(c+d x) \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}+\frac{173 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac{661 \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.487604, size = 280, normalized size = 1.67 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec ^9\left (\frac{1}{2} (c+d x)\right ) \left (100800 \sin \left (c+\frac{d x}{2}\right )-88284 \sin \left (c+\frac{3 d x}{2}\right )+56700 \sin \left (2 c+\frac{3 d x}{2}\right )-43236 \sin \left (2 c+\frac{5 d x}{2}\right )+18900 \sin \left (3 c+\frac{5 d x}{2}\right )-12384 \sin \left (3 c+\frac{7 d x}{2}\right )+3150 \sin \left (4 c+\frac{7 d x}{2}\right )-1726 \sin \left (4 c+\frac{9 d x}{2}\right )+39690 d x \cos \left (c+\frac{d x}{2}\right )+26460 d x \cos \left (c+\frac{3 d x}{2}\right )+26460 d x \cos \left (2 c+\frac{3 d x}{2}\right )+11340 d x \cos \left (2 c+\frac{5 d x}{2}\right )+11340 d x \cos \left (3 c+\frac{5 d x}{2}\right )+2835 d x \cos \left (3 c+\frac{7 d x}{2}\right )+2835 d x \cos \left (4 c+\frac{7 d x}{2}\right )+315 d x \cos \left (4 c+\frac{9 d x}{2}\right )+315 d x \cos \left (5 c+\frac{9 d x}{2}\right )-116676 \sin \left (\frac{d x}{2}\right )+39690 d x \cos \left (\frac{d x}{2}\right )\right )}{161280 a^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[c/2]*Sec[(c + d*x)/2]^9*(39690*d*x*Cos[(d*x)/2] + 39690*d*x*Cos[c + (d*x)/2] + 26460*d*x*Cos[c + (3*d*x)/
2] + 26460*d*x*Cos[2*c + (3*d*x)/2] + 11340*d*x*Cos[2*c + (5*d*x)/2] + 11340*d*x*Cos[3*c + (5*d*x)/2] + 2835*d
*x*Cos[3*c + (7*d*x)/2] + 2835*d*x*Cos[4*c + (7*d*x)/2] + 315*d*x*Cos[4*c + (9*d*x)/2] + 315*d*x*Cos[5*c + (9*
d*x)/2] - 116676*Sin[(d*x)/2] + 100800*Sin[c + (d*x)/2] - 88284*Sin[c + (3*d*x)/2] + 56700*Sin[2*c + (3*d*x)/2
] - 43236*Sin[2*c + (5*d*x)/2] + 18900*Sin[3*c + (5*d*x)/2] - 12384*Sin[3*c + (7*d*x)/2] + 3150*Sin[4*c + (7*d
*x)/2] - 1726*Sin[4*c + (9*d*x)/2]))/(161280*a^5*d)

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Maple [A]  time = 0.042, size = 113, normalized size = 0.7 \begin{align*} -{\frac{1}{144\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}+{\frac{3}{56\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{1}{5\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{13}{24\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{31}{16\,d{a}^{5}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5/(a+cos(d*x+c)*a)^5,x)

[Out]

-1/144/d/a^5*tan(1/2*d*x+1/2*c)^9+3/56/d/a^5*tan(1/2*d*x+1/2*c)^7-1/5/d/a^5*tan(1/2*d*x+1/2*c)^5+13/24/d/a^5*t
an(1/2*d*x+1/2*c)^3-31/16/d/a^5*tan(1/2*d*x+1/2*c)+2/d/a^5*arctan(tan(1/2*d*x+1/2*c))

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Maxima [A]  time = 1.70098, size = 178, normalized size = 1.06 \begin{align*} -\frac{\frac{\frac{9765 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{2730 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{1008 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{270 \, \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}}}{a^{5}} - \frac{10080 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{5040 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5040*((9765*sin(d*x + c)/(cos(d*x + c) + 1) - 2730*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 1008*sin(d*x + c)^
5/(cos(d*x + c) + 1)^5 - 270*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
- 10080*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^5)/d

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Fricas [A]  time = 1.65424, size = 514, normalized size = 3.06 \begin{align*} \frac{315 \, d x \cos \left (d x + c\right )^{5} + 1575 \, d x \cos \left (d x + c\right )^{4} + 3150 \, d x \cos \left (d x + c\right )^{3} + 3150 \, d x \cos \left (d x + c\right )^{2} + 1575 \, d x \cos \left (d x + c\right ) + 315 \, d x -{\left (863 \, \cos \left (d x + c\right )^{4} + 2740 \, \cos \left (d x + c\right )^{3} + 3549 \, \cos \left (d x + c\right )^{2} + 2125 \, \cos \left (d x + c\right ) + 488\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)^5/(a+a*cos(d*x+c))^5,x, algorithm="fricas")

[Out]

1/315*(315*d*x*cos(d*x + c)^5 + 1575*d*x*cos(d*x + c)^4 + 3150*d*x*cos(d*x + c)^3 + 3150*d*x*cos(d*x + c)^2 +
1575*d*x*cos(d*x + c) + 315*d*x - (863*cos(d*x + c)^4 + 2740*cos(d*x + c)^3 + 3549*cos(d*x + c)^2 + 2125*cos(d
*x + c) + 488)*sin(d*x + c))/(a^5*d*cos(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)**5/(a+a*cos(d*x+c))**5,x)

[Out]

Timed out

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Giac [A]  time = 1.33333, size = 135, normalized size = 0.8 \begin{align*} \frac{\frac{5040 \,{\left (d x + c\right )}}{a^{5}} - \frac{35 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 270 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 1008 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 2730 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9765 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{45}}}{5040 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5040*(5040*(d*x + c)/a^5 - (35*a^40*tan(1/2*d*x + 1/2*c)^9 - 270*a^40*tan(1/2*d*x + 1/2*c)^7 + 1008*a^40*tan
(1/2*d*x + 1/2*c)^5 - 2730*a^40*tan(1/2*d*x + 1/2*c)^3 + 9765*a^40*tan(1/2*d*x + 1/2*c))/a^45)/d