3.611 \(\int \frac {\cos ^3(c+d x) (1-\cos ^2(c+d x))}{(a+b \cos (c+d x))^3} \, dx\)

Optimal. Leaf size=268 \[ \frac {\left (4 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {x \left (12 a^2-b^2\right )}{2 b^5}+\frac {a \left (12 a^2-11 b^2\right ) \sin (c+d x)}{2 b^4 d \left (a^2-b^2\right )}-\frac {\left (6 a^2-5 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left (a^2-b^2\right )}+\frac {a \left (12 a^4-19 a^2 b^2+6 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}+\frac {\sin (c+d x) \cos ^3(c+d x)}{2 b d (a+b \cos (c+d x))^2} \]

[Out]

-1/2*(12*a^2-b^2)*x/b^5+a*(12*a^4-19*a^2*b^2+6*b^4)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(
3/2)/b^5/(a+b)^(3/2)/d+1/2*a*(12*a^2-11*b^2)*sin(d*x+c)/b^4/(a^2-b^2)/d-1/2*(6*a^2-5*b^2)*cos(d*x+c)*sin(d*x+c
)/b^3/(a^2-b^2)/d+1/2*cos(d*x+c)^3*sin(d*x+c)/b/d/(a+b*cos(d*x+c))^2+1/2*(4*a^2-3*b^2)*cos(d*x+c)^2*sin(d*x+c)
/b^2/(a^2-b^2)/d/(a+b*cos(d*x+c))

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Rubi [A]  time = 0.75, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3048, 3049, 3023, 2735, 2659, 205} \[ \frac {a \left (12 a^2-11 b^2\right ) \sin (c+d x)}{2 b^4 d \left (a^2-b^2\right )}+\frac {a \left (-19 a^2 b^2+12 a^4+6 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}+\frac {\left (4 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\left (6 a^2-5 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left (a^2-b^2\right )}-\frac {x \left (12 a^2-b^2\right )}{2 b^5}+\frac {\sin (c+d x) \cos ^3(c+d x)}{2 b d (a+b \cos (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^3*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]

[Out]

-((12*a^2 - b^2)*x)/(2*b^5) + (a*(12*a^4 - 19*a^2*b^2 + 6*b^4)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a +
b]])/((a - b)^(3/2)*b^5*(a + b)^(3/2)*d) + (a*(12*a^2 - 11*b^2)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)*d) - ((6*a^2
- 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(2*b*d*(a + b*Cos[c
+ d*x])^2) + ((4*a^2 - 3*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[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 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]

Rule 3048

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

Rule 3049

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

Rubi steps

\begin {align*} \int \frac {\cos ^3(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx &=\frac {\cos ^3(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}-\frac {\int \frac {\cos ^2(c+d x) \left (-3 \left (a^2-b^2\right )+4 \left (a^2-b^2\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=\frac {\cos ^3(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}+\frac {\left (4 a^2-3 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {\cos (c+d x) \left (2 \left (4 a^4-7 a^2 b^2+3 b^4\right )-a b \left (a^2-b^2\right ) \cos (c+d x)-2 \left (6 a^2-5 b^2\right ) \left (a^2-b^2\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}+\frac {\left (4 a^2-3 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {-2 a \left (6 a^4-11 a^2 b^2+5 b^4\right )+2 b \left (2 a^4-3 a^2 b^2+b^4\right ) \cos (c+d x)+2 a \left (12 a^2-11 b^2\right ) \left (a^2-b^2\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{4 b^3 \left (a^2-b^2\right )^2}\\ &=\frac {a \left (12 a^2-11 b^2\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}-\frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}+\frac {\left (4 a^2-3 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {-2 a b \left (6 a^4-11 a^2 b^2+5 b^4\right )-2 \left (a^2-b^2\right )^2 \left (12 a^2-b^2\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{4 b^4 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (12 a^2-b^2\right ) x}{2 b^5}+\frac {a \left (12 a^2-11 b^2\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}-\frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}+\frac {\left (4 a^2-3 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\left (a \left (12 a^4-19 a^2 b^2+6 b^4\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{2 b^5 \left (a^2-b^2\right )}\\ &=-\frac {\left (12 a^2-b^2\right ) x}{2 b^5}+\frac {a \left (12 a^2-11 b^2\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}-\frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}+\frac {\left (4 a^2-3 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\left (a \left (12 a^4-19 a^2 b^2+6 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^5 \left (a^2-b^2\right ) d}\\ &=-\frac {\left (12 a^2-b^2\right ) x}{2 b^5}+\frac {a \left (12 a^4-19 a^2 b^2+6 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} b^5 (a+b)^{3/2} d}+\frac {a \left (12 a^2-11 b^2\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}-\frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}+\frac {\left (4 a^2-3 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 5.06, size = 374, normalized size = 1.40 \[ \frac {\frac {-96 a^6 c-96 a^6 d x+96 a^5 b \sin (c+d x)+72 a^4 b^2 \sin (2 (c+d x))+56 a^4 b^2 c+56 a^4 b^2 d x-80 a^3 b^3 \sin (c+d x)+8 a^3 b^3 \sin (3 (c+d x))-70 a^2 b^4 \sin (2 (c+d x))-a^2 b^4 \sin (4 (c+d x))+44 a^2 b^4 c+44 a^2 b^4 d x-16 a b \left (12 a^4-13 a^2 b^2+b^4\right ) (c+d x) \cos (c+d x)-4 b^2 \left (12 a^4-13 a^2 b^2+b^4\right ) (c+d x) \cos (2 (c+d x))-8 a b^5 \sin (c+d x)-8 a b^5 \sin (3 (c+d x))+2 b^6 \sin (2 (c+d x))+b^6 \sin (4 (c+d x))-4 b^6 c-4 b^6 d x}{(a+b \cos (c+d x))^2}-\frac {16 a \left (12 a^4-19 a^2 b^2+6 b^4\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\sqrt {b^2-a^2}}}{16 b^5 d (a-b) (a+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^3*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]

[Out]

((-16*a*(12*a^4 - 19*a^2*b^2 + 6*b^4)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] +
(-96*a^6*c + 56*a^4*b^2*c + 44*a^2*b^4*c - 4*b^6*c - 96*a^6*d*x + 56*a^4*b^2*d*x + 44*a^2*b^4*d*x - 4*b^6*d*x
- 16*a*b*(12*a^4 - 13*a^2*b^2 + b^4)*(c + d*x)*Cos[c + d*x] - 4*b^2*(12*a^4 - 13*a^2*b^2 + b^4)*(c + d*x)*Cos
[2*(c + d*x)] + 96*a^5*b*Sin[c + d*x] - 80*a^3*b^3*Sin[c + d*x] - 8*a*b^5*Sin[c + d*x] + 72*a^4*b^2*Sin[2*(c +
d*x)] - 70*a^2*b^4*Sin[2*(c + d*x)] + 2*b^6*Sin[2*(c + d*x)] + 8*a^3*b^3*Sin[3*(c + d*x)] - 8*a*b^5*Sin[3*(c
+ d*x)] - a^2*b^4*Sin[4*(c + d*x)] + b^6*Sin[4*(c + d*x)])/(a + b*Cos[c + d*x])^2)/(16*(a - b)*b^5*(a + b)*d)

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fricas [A]  time = 0.58, size = 983, normalized size = 3.67 \[ \left [-\frac {2 \, {\left (12 \, a^{6} b^{2} - 25 \, a^{4} b^{4} + 14 \, a^{2} b^{6} - b^{8}\right )} d x \cos \left (d x + c\right )^{2} + 4 \, {\left (12 \, a^{7} b - 25 \, a^{5} b^{3} + 14 \, a^{3} b^{5} - a b^{7}\right )} d x \cos \left (d x + c\right ) + 2 \, {\left (12 \, a^{8} - 25 \, a^{6} b^{2} + 14 \, a^{4} b^{4} - a^{2} b^{6}\right )} d x - {\left (12 \, a^{7} - 19 \, a^{5} b^{2} + 6 \, a^{3} b^{4} + {\left (12 \, a^{5} b^{2} - 19 \, a^{3} b^{4} + 6 \, a b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (12 \, a^{6} b - 19 \, a^{4} b^{3} + 6 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - 2 \, {\left (12 \, a^{7} b - 23 \, a^{5} b^{3} + 11 \, a^{3} b^{5} - {\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (18 \, a^{6} b^{2} - 35 \, a^{4} b^{4} + 17 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{4} b^{7} - 2 \, a^{2} b^{9} + b^{11}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{6} - 2 \, a^{3} b^{8} + a b^{10}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b^{5} - 2 \, a^{4} b^{7} + a^{2} b^{9}\right )} d\right )}}, -\frac {{\left (12 \, a^{6} b^{2} - 25 \, a^{4} b^{4} + 14 \, a^{2} b^{6} - b^{8}\right )} d x \cos \left (d x + c\right )^{2} + 2 \, {\left (12 \, a^{7} b - 25 \, a^{5} b^{3} + 14 \, a^{3} b^{5} - a b^{7}\right )} d x \cos \left (d x + c\right ) + {\left (12 \, a^{8} - 25 \, a^{6} b^{2} + 14 \, a^{4} b^{4} - a^{2} b^{6}\right )} d x - {\left (12 \, a^{7} - 19 \, a^{5} b^{2} + 6 \, a^{3} b^{4} + {\left (12 \, a^{5} b^{2} - 19 \, a^{3} b^{4} + 6 \, a b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (12 \, a^{6} b - 19 \, a^{4} b^{3} + 6 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) - {\left (12 \, a^{7} b - 23 \, a^{5} b^{3} + 11 \, a^{3} b^{5} - {\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (18 \, a^{6} b^{2} - 35 \, a^{4} b^{4} + 17 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b^{7} - 2 \, a^{2} b^{9} + b^{11}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{6} - 2 \, a^{3} b^{8} + a b^{10}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b^{5} - 2 \, a^{4} b^{7} + a^{2} b^{9}\right )} d\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(1-cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x, algorithm="fricas")

[Out]

[-1/4*(2*(12*a^6*b^2 - 25*a^4*b^4 + 14*a^2*b^6 - b^8)*d*x*cos(d*x + c)^2 + 4*(12*a^7*b - 25*a^5*b^3 + 14*a^3*b
^5 - a*b^7)*d*x*cos(d*x + c) + 2*(12*a^8 - 25*a^6*b^2 + 14*a^4*b^4 - a^2*b^6)*d*x - (12*a^7 - 19*a^5*b^2 + 6*a
^3*b^4 + (12*a^5*b^2 - 19*a^3*b^4 + 6*a*b^6)*cos(d*x + c)^2 + 2*(12*a^6*b - 19*a^4*b^3 + 6*a^2*b^5)*cos(d*x +
c))*sqrt(-a^2 + b^2)*log((2*a*b*cos(d*x + c) + (2*a^2 - b^2)*cos(d*x + c)^2 - 2*sqrt(-a^2 + b^2)*(a*cos(d*x +
c) + b)*sin(d*x + c) - a^2 + 2*b^2)/(b^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + a^2)) - 2*(12*a^7*b - 23*a^5*b^
3 + 11*a^3*b^5 - (a^4*b^4 - 2*a^2*b^6 + b^8)*cos(d*x + c)^3 + 4*(a^5*b^3 - 2*a^3*b^5 + a*b^7)*cos(d*x + c)^2 +
(18*a^6*b^2 - 35*a^4*b^4 + 17*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^4*b^7 - 2*a^2*b^9 + b^11)*d*cos(d*x +
c)^2 + 2*(a^5*b^6 - 2*a^3*b^8 + a*b^10)*d*cos(d*x + c) + (a^6*b^5 - 2*a^4*b^7 + a^2*b^9)*d), -1/2*((12*a^6*b^2
- 25*a^4*b^4 + 14*a^2*b^6 - b^8)*d*x*cos(d*x + c)^2 + 2*(12*a^7*b - 25*a^5*b^3 + 14*a^3*b^5 - a*b^7)*d*x*cos(
d*x + c) + (12*a^8 - 25*a^6*b^2 + 14*a^4*b^4 - a^2*b^6)*d*x - (12*a^7 - 19*a^5*b^2 + 6*a^3*b^4 + (12*a^5*b^2 -
19*a^3*b^4 + 6*a*b^6)*cos(d*x + c)^2 + 2*(12*a^6*b - 19*a^4*b^3 + 6*a^2*b^5)*cos(d*x + c))*sqrt(a^2 - b^2)*ar
ctan(-(a*cos(d*x + c) + b)/(sqrt(a^2 - b^2)*sin(d*x + c))) - (12*a^7*b - 23*a^5*b^3 + 11*a^3*b^5 - (a^4*b^4 -
2*a^2*b^6 + b^8)*cos(d*x + c)^3 + 4*(a^5*b^3 - 2*a^3*b^5 + a*b^7)*cos(d*x + c)^2 + (18*a^6*b^2 - 35*a^4*b^4 +
17*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^4*b^7 - 2*a^2*b^9 + b^11)*d*cos(d*x + c)^2 + 2*(a^5*b^6 - 2*a^3*b^
8 + a*b^10)*d*cos(d*x + c) + (a^6*b^5 - 2*a^4*b^7 + a^2*b^9)*d)]

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giac [B]  time = 3.06, size = 1194, normalized size = 4.46 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(1-cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x, algorithm="giac")

[Out]

1/2*((24*a^7*b^4 - 12*a^6*b^5 - 56*a^5*b^6 + 25*a^4*b^7 + 39*a^3*b^8 - 14*a^2*b^9 - 7*a*b^10 + b^11 + 12*a^4*a
bs(-a^2*b^5 + b^7) - 6*a^3*b*abs(-a^2*b^5 + b^7) - 13*a^2*b^2*abs(-a^2*b^5 + b^7) + 5*a*b^3*abs(-a^2*b^5 + b^7
) + b^4*abs(-a^2*b^5 + b^7))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(2*tan(1/2*d*x + 1/2*c)/sqrt((4*a^3*b^4
- 4*a*b^6 + sqrt(-16*(a^3*b^4 + a^2*b^5 - a*b^6 - b^7)*(a^3*b^4 - a^2*b^5 - a*b^6 + b^7) + 16*(a^3*b^4 - a*b^
6)^2))/(a^3*b^4 - a^2*b^5 - a*b^6 + b^7))))/(a^3*b^4*abs(-a^2*b^5 + b^7) - a*b^6*abs(-a^2*b^5 + b^7) + (a^2*b^
5 - b^7)^2) - ((12*a^4 - 6*a^3*b - 13*a^2*b^2 + 5*a*b^3 + b^4)*sqrt(a^2 - b^2)*abs(-a^2*b^5 + b^7)*abs(-a + b)
- (24*a^7*b^4 - 12*a^6*b^5 - 56*a^5*b^6 + 25*a^4*b^7 + 39*a^3*b^8 - 14*a^2*b^9 - 7*a*b^10 + b^11)*sqrt(a^2 -
b^2)*abs(-a + b))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(2*tan(1/2*d*x + 1/2*c)/sqrt((4*a^3*b^4 - 4*a*b^6
- sqrt(-16*(a^3*b^4 + a^2*b^5 - a*b^6 - b^7)*(a^3*b^4 - a^2*b^5 - a*b^6 + b^7) + 16*(a^3*b^4 - a*b^6)^2))/(a^3
*b^4 - a^2*b^5 - a*b^6 + b^7))))/((a^2*b^5 - b^7)^2*(a^2 - 2*a*b + b^2) - (a^5*b^4 - 2*a^4*b^5 + 2*a^2*b^7 - a
*b^8)*abs(-a^2*b^5 + b^7)) + 2*(12*a^5*tan(1/2*d*x + 1/2*c)^7 - 18*a^4*b*tan(1/2*d*x + 1/2*c)^7 - 7*a^3*b^2*ta
n(1/2*d*x + 1/2*c)^7 + 18*a^2*b^3*tan(1/2*d*x + 1/2*c)^7 - 4*a*b^4*tan(1/2*d*x + 1/2*c)^7 - b^5*tan(1/2*d*x +
1/2*c)^7 + 36*a^5*tan(1/2*d*x + 1/2*c)^5 - 18*a^4*b*tan(1/2*d*x + 1/2*c)^5 - 37*a^3*b^2*tan(1/2*d*x + 1/2*c)^5
+ 14*a^2*b^3*tan(1/2*d*x + 1/2*c)^5 + 4*a*b^4*tan(1/2*d*x + 1/2*c)^5 + 3*b^5*tan(1/2*d*x + 1/2*c)^5 + 36*a^5*
tan(1/2*d*x + 1/2*c)^3 + 18*a^4*b*tan(1/2*d*x + 1/2*c)^3 - 37*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - 14*a^2*b^3*tan(
1/2*d*x + 1/2*c)^3 + 4*a*b^4*tan(1/2*d*x + 1/2*c)^3 - 3*b^5*tan(1/2*d*x + 1/2*c)^3 + 12*a^5*tan(1/2*d*x + 1/2*
c) + 18*a^4*b*tan(1/2*d*x + 1/2*c) - 7*a^3*b^2*tan(1/2*d*x + 1/2*c) - 18*a^2*b^3*tan(1/2*d*x + 1/2*c) - 4*a*b^
4*tan(1/2*d*x + 1/2*c) + b^5*tan(1/2*d*x + 1/2*c))/((a^2*b^4 - b^6)*(a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x
+ 1/2*c)^4 + 2*a*tan(1/2*d*x + 1/2*c)^2 + a + b)^2))/d

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maple [B]  time = 0.10, size = 704, normalized size = 2.63 \[ \frac {6 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,b^{4} \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )^{2} \left (a +b \right )}-\frac {a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,b^{3} \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )^{2} \left (a +b \right )}-\frac {6 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,b^{2} \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )^{2} \left (a +b \right )}+\frac {6 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,b^{4} \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )^{2} \left (a -b \right )}+\frac {a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,b^{3} \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )^{2} \left (a -b \right )}-\frac {6 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,b^{2} \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )^{2} \left (a -b \right )}+\frac {12 a^{5} \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d \,b^{5} \left (a^{2}-b^{2}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {19 a^{3} \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d \,b^{3} \left (a^{2}-b^{2}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {6 a \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d b \left (a^{2}-b^{2}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {6 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,b^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,b^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {12 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{d \,b^{5}}+\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*(1-cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x)

[Out]

6/d*a^4/b^4/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)*tan(1/2*d*x+1/2*c)^3-1/d*a^3/b^3/(a*ta
n(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)*tan(1/2*d*x+1/2*c)^3-6/d*a^2/b^2/(a*tan(1/2*d*x+1/2*c)^
2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)*tan(1/2*d*x+1/2*c)^3+6/d*a^4/b^4/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2
*c)^2*b+a+b)^2/(a-b)*tan(1/2*d*x+1/2*c)+1/d*a^3/b^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a-b
)*tan(1/2*d*x+1/2*c)-6/d*a^2/b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a-b)*tan(1/2*d*x+1/2*c
)+12/d*a^5/b^5/(a^2-b^2)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))-19/d*a^3/b^3
/(a^2-b^2)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))+6/d*a/b/(a^2-b^2)/((a-b)*(
a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))+6/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*
x+1/2*c)^3*a+1/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3+6/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/
2*d*x+1/2*c)*a-1/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)-12/d/b^5*arctan(tan(1/2*d*x+1/2*c))*a^2+1
/d/b^3*arctan(tan(1/2*d*x+1/2*c))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(1-cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?

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mupad [B]  time = 8.84, size = 4038, normalized size = 15.07 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(cos(c + d*x)^3*(cos(c + d*x)^2 - 1))/(a + b*cos(c + d*x))^3,x)

[Out]

((tan(c/2 + (d*x)/2)*(6*a^3*b - 5*a*b^3 + 12*a^4 + b^4 - 13*a^2*b^2))/(a*b^4 - b^5) + (tan(c/2 + (d*x)/2)^3*(4
*a*b^4 + 18*a^4*b + 36*a^5 - 3*b^5 - 14*a^2*b^3 - 37*a^3*b^2))/((a*b^4 - b^5)*(a + b)) + (tan(c/2 + (d*x)/2)^5
*(4*a*b^4 - 18*a^4*b + 36*a^5 + 3*b^5 + 14*a^2*b^3 - 37*a^3*b^2))/((a*b^4 - b^5)*(a + b)) + (tan(c/2 + (d*x)/2
)^7*(5*a*b^3 - 6*a^3*b + 12*a^4 + b^4 - 13*a^2*b^2))/(b^4*(a + b)))/(d*(2*a*b + tan(c/2 + (d*x)/2)^4*(6*a^2 -
2*b^2) + tan(c/2 + (d*x)/2)^2*(4*a*b + 4*a^2) - tan(c/2 + (d*x)/2)^6*(4*a*b - 4*a^2) + tan(c/2 + (d*x)/2)^8*(a
^2 - 2*a*b + b^2) + a^2 + b^2)) + (atan((((a^2*12i - b^2*1i)*((((4*(24*a*b^16 - 4*b^17 + 36*a^2*b^15 - 100*a^3
*b^14 - 56*a^4*b^13 + 124*a^5*b^12 + 24*a^6*b^11 - 48*a^7*b^10))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (4*ta
n(c/2 + (d*x)/2)*(a^2*12i - b^2*1i)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^
10))/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))*(a^2*12i - b^2*1i))/(2*b^5) + (8*tan(c/2 + (d*x)/2)*(288*a^10
- 288*a^9*b - 2*a*b^9 + b^10 + 11*a^2*b^8 + 52*a^3*b^7 - 61*a^4*b^6 - 386*a^5*b^5 + 386*a^6*b^4 + 624*a^7*b^3
- 624*a^8*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8))*1i)/(2*b^5) - ((a^2*12i - b^2*1i)*((((4*(24*a*b^16 - 4*b^
17 + 36*a^2*b^15 - 100*a^3*b^14 - 56*a^4*b^13 + 124*a^5*b^12 + 24*a^6*b^11 - 48*a^7*b^10))/(a*b^14 + b^15 - a^
2*b^13 - a^3*b^12) + (4*tan(c/2 + (d*x)/2)*(a^2*12i - b^2*1i)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^
12 + 8*a^5*b^11 - 8*a^6*b^10))/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))*(a^2*12i - b^2*1i))/(2*b^5) - (8*tan
(c/2 + (d*x)/2)*(288*a^10 - 288*a^9*b - 2*a*b^9 + b^10 + 11*a^2*b^8 + 52*a^3*b^7 - 61*a^4*b^6 - 386*a^5*b^5 +
386*a^6*b^4 + 624*a^7*b^3 - 624*a^8*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8))*1i)/(2*b^5))/((8*(6*a*b^10 + 86
4*a^10*b - 1728*a^11 + 30*a^2*b^9 - 169*a^3*b^8 - 491*a^4*b^7 + 1495*a^5*b^6 + 1746*a^6*b^5 - 4356*a^7*b^4 - 2
160*a^8*b^3 + 4752*a^9*b^2))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + ((a^2*12i - b^2*1i)*((((4*(24*a*b^16 - 4*
b^17 + 36*a^2*b^15 - 100*a^3*b^14 - 56*a^4*b^13 + 124*a^5*b^12 + 24*a^6*b^11 - 48*a^7*b^10))/(a*b^14 + b^15 -
a^2*b^13 - a^3*b^12) - (4*tan(c/2 + (d*x)/2)*(a^2*12i - b^2*1i)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*
b^12 + 8*a^5*b^11 - 8*a^6*b^10))/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))*(a^2*12i - b^2*1i))/(2*b^5) + (8*t
an(c/2 + (d*x)/2)*(288*a^10 - 288*a^9*b - 2*a*b^9 + b^10 + 11*a^2*b^8 + 52*a^3*b^7 - 61*a^4*b^6 - 386*a^5*b^5
+ 386*a^6*b^4 + 624*a^7*b^3 - 624*a^8*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))/(2*b^5) + ((a^2*12i - b^2*1i
)*((((4*(24*a*b^16 - 4*b^17 + 36*a^2*b^15 - 100*a^3*b^14 - 56*a^4*b^13 + 124*a^5*b^12 + 24*a^6*b^11 - 48*a^7*b
^10))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + (4*tan(c/2 + (d*x)/2)*(a^2*12i - b^2*1i)*(8*a*b^15 - 8*a^2*b^14
- 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))*(a^2*12i -
b^2*1i))/(2*b^5) - (8*tan(c/2 + (d*x)/2)*(288*a^10 - 288*a^9*b - 2*a*b^9 + b^10 + 11*a^2*b^8 + 52*a^3*b^7 - 61
*a^4*b^6 - 386*a^5*b^5 + 386*a^6*b^4 + 624*a^7*b^3 - 624*a^8*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))/(2*b^
5)))*(a^2*12i - b^2*1i)*1i)/(b^5*d) + (a*atan(((a*((8*tan(c/2 + (d*x)/2)*(288*a^10 - 288*a^9*b - 2*a*b^9 + b^1
0 + 11*a^2*b^8 + 52*a^3*b^7 - 61*a^4*b^6 - 386*a^5*b^5 + 386*a^6*b^4 + 624*a^7*b^3 - 624*a^8*b^2))/(a*b^10 + b
^11 - a^2*b^9 - a^3*b^8) + (a*((4*(24*a*b^16 - 4*b^17 + 36*a^2*b^15 - 100*a^3*b^14 - 56*a^4*b^13 + 124*a^5*b^1
2 + 24*a^6*b^11 - 48*a^7*b^10))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (4*a*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a
- b)^3)^(1/2)*(12*a^4 + 6*b^4 - 19*a^2*b^2)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 -
8*a^6*b^10))/((a*b^10 + b^11 - a^2*b^9 - a^3*b^8)*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*(-(a + b)^3*(a -
b)^3)^(1/2)*(12*a^4 + 6*b^4 - 19*a^2*b^2))/(2*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*(-(a + b)^3*(a - b)^
3)^(1/2)*(12*a^4 + 6*b^4 - 19*a^2*b^2)*1i)/(2*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)) + (a*((8*tan(c/2 + (d*
x)/2)*(288*a^10 - 288*a^9*b - 2*a*b^9 + b^10 + 11*a^2*b^8 + 52*a^3*b^7 - 61*a^4*b^6 - 386*a^5*b^5 + 386*a^6*b^
4 + 624*a^7*b^3 - 624*a^8*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) - (a*((4*(24*a*b^16 - 4*b^17 + 36*a^2*b^15
- 100*a^3*b^14 - 56*a^4*b^13 + 124*a^5*b^12 + 24*a^6*b^11 - 48*a^7*b^10))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^1
2) + (4*a*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a - b)^3)^(1/2)*(12*a^4 + 6*b^4 - 19*a^2*b^2)*(8*a*b^15 - 8*a^2*b^14
- 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/((a*b^10 + b^11 - a^2*b^9 - a^3*b^8)*(b^11 - 3*a^2*b^
9 + 3*a^4*b^7 - a^6*b^5)))*(-(a + b)^3*(a - b)^3)^(1/2)*(12*a^4 + 6*b^4 - 19*a^2*b^2))/(2*(b^11 - 3*a^2*b^9 +
3*a^4*b^7 - a^6*b^5)))*(-(a + b)^3*(a - b)^3)^(1/2)*(12*a^4 + 6*b^4 - 19*a^2*b^2)*1i)/(2*(b^11 - 3*a^2*b^9 + 3
*a^4*b^7 - a^6*b^5)))/((8*(6*a*b^10 + 864*a^10*b - 1728*a^11 + 30*a^2*b^9 - 169*a^3*b^8 - 491*a^4*b^7 + 1495*a
^5*b^6 + 1746*a^6*b^5 - 4356*a^7*b^4 - 2160*a^8*b^3 + 4752*a^9*b^2))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + (
a*((8*tan(c/2 + (d*x)/2)*(288*a^10 - 288*a^9*b - 2*a*b^9 + b^10 + 11*a^2*b^8 + 52*a^3*b^7 - 61*a^4*b^6 - 386*a
^5*b^5 + 386*a^6*b^4 + 624*a^7*b^3 - 624*a^8*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) + (a*((4*(24*a*b^16 - 4
*b^17 + 36*a^2*b^15 - 100*a^3*b^14 - 56*a^4*b^13 + 124*a^5*b^12 + 24*a^6*b^11 - 48*a^7*b^10))/(a*b^14 + b^15 -
a^2*b^13 - a^3*b^12) - (4*a*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a - b)^3)^(1/2)*(12*a^4 + 6*b^4 - 19*a^2*b^2)*(8*
a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/((a*b^10 + b^11 - a^2*b^9 - a^3*b^
8)*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*(-(a + b)^3*(a - b)^3)^(1/2)*(12*a^4 + 6*b^4 - 19*a^2*b^2))/(2*(
b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*(-(a + b)^3*(a - b)^3)^(1/2)*(12*a^4 + 6*b^4 - 19*a^2*b^2))/(2*(b^11
- 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)) - (a*((8*tan(c/2 + (d*x)/2)*(288*a^10 - 288*a^9*b - 2*a*b^9 + b^10 + 11*a
^2*b^8 + 52*a^3*b^7 - 61*a^4*b^6 - 386*a^5*b^5 + 386*a^6*b^4 + 624*a^7*b^3 - 624*a^8*b^2))/(a*b^10 + b^11 - a^
2*b^9 - a^3*b^8) - (a*((4*(24*a*b^16 - 4*b^17 + 36*a^2*b^15 - 100*a^3*b^14 - 56*a^4*b^13 + 124*a^5*b^12 + 24*a
^6*b^11 - 48*a^7*b^10))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + (4*a*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a - b)^3)
^(1/2)*(12*a^4 + 6*b^4 - 19*a^2*b^2)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b
^10))/((a*b^10 + b^11 - a^2*b^9 - a^3*b^8)*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*(-(a + b)^3*(a - b)^3)^(
1/2)*(12*a^4 + 6*b^4 - 19*a^2*b^2))/(2*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*(-(a + b)^3*(a - b)^3)^(1/2)
*(12*a^4 + 6*b^4 - 19*a^2*b^2))/(2*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5))))*(-(a + b)^3*(a - b)^3)^(1/2)*(1
2*a^4 + 6*b^4 - 19*a^2*b^2)*1i)/(d*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**3*(1-cos(d*x+c)**2)/(a+b*cos(d*x+c))**3,x)

[Out]

Timed out

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