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

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

[Out]

x/a^5-1/9*cos(d*x+c)^4*sin(d*x+c)/d/(a+a*cos(d*x+c))^5-13/63*cos(d*x+c)^3*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^4-34
/105*cos(d*x+c)^2*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c))^3+173/315*sin(d*x+c)/a^3/d/(a+a*cos(d*x+c))^2-661/315*sin(
d*x+c)/d/(a^5+a^5*cos(d*x+c))

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Rubi [A]  time = 0.39, antiderivative size = 168, normalized size of antiderivative = 1.00, 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 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]

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 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 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 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 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]

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*}

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Mathematica [A]  time = 0.49, 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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fricas [A]  time = 0.71, size = 188, normalized size = 1.12 \[ \frac {315 \, d x \cos \left (d x + c\right )^{5} + 1575 \, d x \cos \left (d x + c\right )^{4} + 3150 \, d x \cos \left (d x + c\right )^{3} + 3150 \, d x \cos \left (d x + c\right )^{2} + 1575 \, d x \cos \left (d x + c\right ) + 315 \, d x - {\left (863 \, \cos \left (d x + c\right )^{4} + 2740 \, \cos \left (d x + c\right )^{3} + 3549 \, \cos \left (d x + c\right )^{2} + 2125 \, \cos \left (d x + c\right ) + 488\right )} \sin \left (d x + c\right )}{315 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]

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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giac [A]  time = 0.55, size = 100, normalized size = 0.60 \[ \frac {\frac {5040 \, {\left (d x + c\right )}}{a^{5}} - \frac {35 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 270 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1008 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2730 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9765 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{5040 \, d} \]

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

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maple [A]  time = 0.05, size = 113, normalized size = 0.67 \[ -\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{144 d \,a^{5}}+\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{5}}-\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{5 d \,a^{5}}+\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \,a^{5}}-\frac {31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5/(a+a*cos(d*x+c))^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 = 0.70, size = 132, normalized size = 0.79 \[ -\frac {\frac {\frac {9765 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2730 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1008 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {270 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac {10080 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{5040 \, d} \]

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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mupad [B]  time = 0.48, size = 125, normalized size = 0.74 \[ \frac {x}{a^5}-\frac {\frac {863\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{315}-\frac {356\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{315}+\frac {169\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{420}-\frac {41\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{504}+\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{144}}{a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a^5 - (sin(c/2 + (d*x)/2)/144 - (41*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2))/504 + (169*cos(c/2 + (d*x)/2)^4
*sin(c/2 + (d*x)/2))/420 - (356*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2))/315 + (863*cos(c/2 + (d*x)/2)^8*sin(c
/2 + (d*x)/2))/315)/(a^5*d*cos(c/2 + (d*x)/2)^9)

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sympy [A]  time = 28.10, size = 116, normalized size = 0.69 \[ \begin {cases} \frac {x}{a^{5}} - \frac {\tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{144 a^{5} d} + \frac {3 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{5} d} - \frac {\tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5 a^{5} d} + \frac {13 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{5} d} - \frac {31 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16 a^{5} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{5}{\relax (c )}}{\left (a \cos {\relax (c )} + a\right )^{5}} & \text {otherwise} \end {cases} \]

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]

Piecewise((x/a**5 - tan(c/2 + d*x/2)**9/(144*a**5*d) + 3*tan(c/2 + d*x/2)**7/(56*a**5*d) - tan(c/2 + d*x/2)**5
/(5*a**5*d) + 13*tan(c/2 + d*x/2)**3/(24*a**5*d) - 31*tan(c/2 + d*x/2)/(16*a**5*d), Ne(d, 0)), (x*cos(c)**5/(a
*cos(c) + a)**5, True))

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