3.83 \(\int \frac {\cos ^6(c+d x)}{(a+a \cos (c+d x))^5} \, dx\)

Optimal. Leaf size=191 \[ \frac {181 \sin (c+d x)}{63 a^5 d}+\frac {5 \sin (c+d x)}{d \left (a^5 \cos (c+d x)+a^5\right )}-\frac {5 x}{a^5}-\frac {67 \sin (c+d x) \cos ^2(c+d x)}{63 a^3 d (a \cos (c+d x)+a)^2}-\frac {29 \sin (c+d x) \cos ^3(c+d x)}{63 a^2 d (a \cos (c+d x)+a)^3}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac {5 \sin (c+d x) \cos ^4(c+d x)}{21 a d (a \cos (c+d x)+a)^4} \]

[Out]

-5*x/a^5+181/63*sin(d*x+c)/a^5/d-1/9*cos(d*x+c)^5*sin(d*x+c)/d/(a+a*cos(d*x+c))^5-5/21*cos(d*x+c)^4*sin(d*x+c)
/a/d/(a+a*cos(d*x+c))^4-29/63*cos(d*x+c)^3*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c))^3-67/63*cos(d*x+c)^2*sin(d*x+c)/a
^3/d/(a+a*cos(d*x+c))^2+5*sin(d*x+c)/d/(a^5+a^5*cos(d*x+c))

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Rubi [A]  time = 0.49, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2765, 2977, 2968, 3023, 12, 2735, 2648} \[ \frac {181 \sin (c+d x)}{63 a^5 d}-\frac {29 \sin (c+d x) \cos ^3(c+d x)}{63 a^2 d (a \cos (c+d x)+a)^3}-\frac {67 \sin (c+d x) \cos ^2(c+d x)}{63 a^3 d (a \cos (c+d x)+a)^2}+\frac {5 \sin (c+d x)}{d \left (a^5 \cos (c+d x)+a^5\right )}-\frac {5 x}{a^5}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac {5 \sin (c+d x) \cos ^4(c+d x)}{21 a d (a \cos (c+d x)+a)^4} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6/(a + a*Cos[c + d*x])^5,x]

[Out]

(-5*x)/a^5 + (181*Sin[c + d*x])/(63*a^5*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (5*C
os[c + d*x]^4*Sin[c + d*x])/(21*a*d*(a + a*Cos[c + d*x])^4) - (29*Cos[c + d*x]^3*Sin[c + d*x])/(63*a^2*d*(a +
a*Cos[c + d*x])^3) - (67*Cos[c + d*x]^2*Sin[c + d*x])/(63*a^3*d*(a + a*Cos[c + d*x])^2) + (5*Sin[c + d*x])/(d*
(a^5 + a^5*Cos[c + d*x]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 3023

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

Rubi steps

\begin {align*} \int \frac {\cos ^6(c+d x)}{(a+a \cos (c+d x))^5} \, dx &=-\frac {\cos ^5(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {\int \frac {\cos ^4(c+d x) (5 a-10 a \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx}{9 a^2}\\ &=-\frac {\cos ^5(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {5 \cos ^4(c+d x) \sin (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac {\int \frac {\cos ^3(c+d x) \left (60 a^2-85 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac {\cos ^5(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {5 \cos ^4(c+d x) \sin (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac {29 \cos ^3(c+d x) \sin (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac {\int \frac {\cos ^2(c+d x) \left (435 a^3-570 a^3 \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac {\cos ^5(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {5 \cos ^4(c+d x) \sin (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac {29 \cos ^3(c+d x) \sin (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac {67 \cos ^2(c+d x) \sin (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}-\frac {\int \frac {\cos (c+d x) \left (2010 a^4-2715 a^4 \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{945 a^8}\\ &=-\frac {\cos ^5(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {5 \cos ^4(c+d x) \sin (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac {29 \cos ^3(c+d x) \sin (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac {67 \cos ^2(c+d x) \sin (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}-\frac {\int \frac {2010 a^4 \cos (c+d x)-2715 a^4 \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{945 a^8}\\ &=\frac {181 \sin (c+d x)}{63 a^5 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {5 \cos ^4(c+d x) \sin (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac {29 \cos ^3(c+d x) \sin (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac {67 \cos ^2(c+d x) \sin (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}-\frac {\int \frac {4725 a^5 \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{945 a^9}\\ &=\frac {181 \sin (c+d x)}{63 a^5 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {5 \cos ^4(c+d x) \sin (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac {29 \cos ^3(c+d x) \sin (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac {67 \cos ^2(c+d x) \sin (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}-\frac {5 \int \frac {\cos (c+d x)}{a+a \cos (c+d x)} \, dx}{a^4}\\ &=-\frac {5 x}{a^5}+\frac {181 \sin (c+d x)}{63 a^5 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {5 \cos ^4(c+d x) \sin (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac {29 \cos ^3(c+d x) \sin (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac {67 \cos ^2(c+d x) \sin (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}+\frac {5 \int \frac {1}{a+a \cos (c+d x)} \, dx}{a^4}\\ &=-\frac {5 x}{a^5}+\frac {181 \sin (c+d x)}{63 a^5 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {5 \cos ^4(c+d x) \sin (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac {29 \cos ^3(c+d x) \sin (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac {67 \cos ^2(c+d x) \sin (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}+\frac {5 \sin (c+d x)}{d \left (a^5+a^5 \cos (c+d x)\right )}\\ \end {align*}

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Mathematica [A]  time = 0.73, size = 319, normalized size = 1.67 \[ -\frac {\sec \left (\frac {c}{2}\right ) \sec ^9\left (\frac {1}{2} (c+d x)\right ) \left (143010 \sin \left (c+\frac {d x}{2}\right )-138726 \sin \left (c+\frac {3 d x}{2}\right )+73290 \sin \left (2 c+\frac {3 d x}{2}\right )-70389 \sin \left (2 c+\frac {5 d x}{2}\right )+20475 \sin \left (3 c+\frac {5 d x}{2}\right )-21141 \sin \left (3 c+\frac {7 d x}{2}\right )+1575 \sin \left (4 c+\frac {7 d x}{2}\right )-3091 \sin \left (4 c+\frac {9 d x}{2}\right )-567 \sin \left (5 c+\frac {9 d x}{2}\right )-63 \sin \left (5 c+\frac {11 d x}{2}\right )-63 \sin \left (6 c+\frac {11 d x}{2}\right )+79380 d x \cos \left (c+\frac {d x}{2}\right )+52920 d x \cos \left (c+\frac {3 d x}{2}\right )+52920 d x \cos \left (2 c+\frac {3 d x}{2}\right )+22680 d x \cos \left (2 c+\frac {5 d x}{2}\right )+22680 d x \cos \left (3 c+\frac {5 d x}{2}\right )+5670 d x \cos \left (3 c+\frac {7 d x}{2}\right )+5670 d x \cos \left (4 c+\frac {7 d x}{2}\right )+630 d x \cos \left (4 c+\frac {9 d x}{2}\right )+630 d x \cos \left (5 c+\frac {9 d x}{2}\right )-175014 \sin \left (\frac {d x}{2}\right )+79380 d x \cos \left (\frac {d x}{2}\right )\right )}{64512 a^5 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6/(a + a*Cos[c + d*x])^5,x]

[Out]

-1/64512*(Sec[c/2]*Sec[(c + d*x)/2]^9*(79380*d*x*Cos[(d*x)/2] + 79380*d*x*Cos[c + (d*x)/2] + 52920*d*x*Cos[c +
(3*d*x)/2] + 52920*d*x*Cos[2*c + (3*d*x)/2] + 22680*d*x*Cos[2*c + (5*d*x)/2] + 22680*d*x*Cos[3*c + (5*d*x)/2]
+ 5670*d*x*Cos[3*c + (7*d*x)/2] + 5670*d*x*Cos[4*c + (7*d*x)/2] + 630*d*x*Cos[4*c + (9*d*x)/2] + 630*d*x*Cos[
5*c + (9*d*x)/2] - 175014*Sin[(d*x)/2] + 143010*Sin[c + (d*x)/2] - 138726*Sin[c + (3*d*x)/2] + 73290*Sin[2*c +
(3*d*x)/2] - 70389*Sin[2*c + (5*d*x)/2] + 20475*Sin[3*c + (5*d*x)/2] - 21141*Sin[3*c + (7*d*x)/2] + 1575*Sin[
4*c + (7*d*x)/2] - 3091*Sin[4*c + (9*d*x)/2] - 567*Sin[5*c + (9*d*x)/2] - 63*Sin[5*c + (11*d*x)/2] - 63*Sin[6*
c + (11*d*x)/2]))/(a^5*d)

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fricas [A]  time = 1.19, size = 198, normalized size = 1.04 \[ -\frac {315 \, d x \cos \left (d x + c\right )^{5} + 1575 \, d x \cos \left (d x + c\right )^{4} + 3150 \, d x \cos \left (d x + c\right )^{3} + 3150 \, d x \cos \left (d x + c\right )^{2} + 1575 \, d x \cos \left (d x + c\right ) + 315 \, d x - {\left (63 \, \cos \left (d x + c\right )^{5} + 946 \, \cos \left (d x + c\right )^{4} + 2840 \, \cos \left (d x + c\right )^{3} + 3633 \, \cos \left (d x + c\right )^{2} + 2165 \, \cos \left (d x + c\right ) + 496\right )} \sin \left (d x + c\right )}{63 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6/(a+a*cos(d*x+c))^5,x, algorithm="fricas")

[Out]

-1/63*(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 - (63*cos(d*x + c)^5 + 946*cos(d*x + c)^4 + 2840*cos(d*x + c)^3 + 3633*cos(d*x
+ c)^2 + 2165*cos(d*x + c) + 496)*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 = 0.74, size = 129, normalized size = 0.68 \[ -\frac {\frac {5040 \, {\left (d x + c\right )}}{a^{5}} - \frac {2016 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{5}} - \frac {7 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 72 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 378 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1512 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8127 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{1008 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6/(a+a*cos(d*x+c))^5,x, algorithm="giac")

[Out]

-1/1008*(5040*(d*x + c)/a^5 - 2016*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 + 1)*a^5) - (7*a^40*tan(1/2*d
*x + 1/2*c)^9 - 72*a^40*tan(1/2*d*x + 1/2*c)^7 + 378*a^40*tan(1/2*d*x + 1/2*c)^5 - 1512*a^40*tan(1/2*d*x + 1/2
*c)^3 + 8127*a^40*tan(1/2*d*x + 1/2*c))/a^45)/d

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maple [A]  time = 0.06, size = 145, normalized size = 0.76 \[ \frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{144 d \,a^{5}}-\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{14 d \,a^{5}}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{5}}-\frac {3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{5}}+\frac {129 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {10 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6/(a+a*cos(d*x+c))^5,x)

[Out]

1/144/d/a^5*tan(1/2*d*x+1/2*c)^9-1/14/d/a^5*tan(1/2*d*x+1/2*c)^7+3/8/d/a^5*tan(1/2*d*x+1/2*c)^5-3/2/d/a^5*tan(
1/2*d*x+1/2*c)^3+129/16/d/a^5*tan(1/2*d*x+1/2*c)+2/d/a^5*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)-10/d/a^5*
arctan(tan(1/2*d*x+1/2*c))

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maxima [A]  time = 0.77, size = 178, normalized size = 0.93 \[ \frac {\frac {2016 \, \sin \left (d x + c\right )}{{\left (a^{5} + \frac {a^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {8127 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1512 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {378 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {72 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {7 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac {10080 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{1008 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6/(a+a*cos(d*x+c))^5,x, algorithm="maxima")

[Out]

1/1008*(2016*sin(d*x + c)/((a^5 + a^5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)) + (8127*sin(d*x
+ c)/(cos(d*x + c) + 1) - 1512*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 378*sin(d*x + c)^5/(cos(d*x + c) + 1)^5
- 72*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 7*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 = 0.51, size = 159, normalized size = 0.83 \[ \frac {7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-100\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+636\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-2512\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+10096\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+2016\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-5040\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (c+d\,x\right )}{1008\,a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6/(a + a*cos(c + d*x))^5,x)

[Out]

(7*sin(c/2 + (d*x)/2) - 100*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2) + 636*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)
/2) - 2512*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2) + 10096*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2) + 2016*cos(
c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2) - 5040*cos(c/2 + (d*x)/2)^9*(c + d*x))/(1008*a^5*d*cos(c/2 + (d*x)/2)^9)

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sympy [A]  time = 42.52, size = 320, normalized size = 1.68 \[ \begin {cases} - \frac {5040 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1008 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1008 a^{5} d} - \frac {5040 d x}{1008 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1008 a^{5} d} + \frac {7 \tan ^{11}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1008 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1008 a^{5} d} - \frac {65 \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1008 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1008 a^{5} d} + \frac {306 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1008 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1008 a^{5} d} - \frac {1134 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1008 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1008 a^{5} d} + \frac {6615 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1008 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1008 a^{5} d} + \frac {10143 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1008 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1008 a^{5} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{6}{\relax (c )}}{\left (a \cos {\relax (c )} + a\right )^{5}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6/(a+a*cos(d*x+c))**5,x)

[Out]

Piecewise((-5040*d*x*tan(c/2 + d*x/2)**2/(1008*a**5*d*tan(c/2 + d*x/2)**2 + 1008*a**5*d) - 5040*d*x/(1008*a**5
*d*tan(c/2 + d*x/2)**2 + 1008*a**5*d) + 7*tan(c/2 + d*x/2)**11/(1008*a**5*d*tan(c/2 + d*x/2)**2 + 1008*a**5*d)
- 65*tan(c/2 + d*x/2)**9/(1008*a**5*d*tan(c/2 + d*x/2)**2 + 1008*a**5*d) + 306*tan(c/2 + d*x/2)**7/(1008*a**5
*d*tan(c/2 + d*x/2)**2 + 1008*a**5*d) - 1134*tan(c/2 + d*x/2)**5/(1008*a**5*d*tan(c/2 + d*x/2)**2 + 1008*a**5*
d) + 6615*tan(c/2 + d*x/2)**3/(1008*a**5*d*tan(c/2 + d*x/2)**2 + 1008*a**5*d) + 10143*tan(c/2 + d*x/2)/(1008*a
**5*d*tan(c/2 + d*x/2)**2 + 1008*a**5*d), Ne(d, 0)), (x*cos(c)**6/(a*cos(c) + a)**5, True))

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