3.601 \(\int \frac {\cos ^4(c+d x) (1-\cos ^2(c+d x))}{(a+b \cos (c+d x))^2} \, dx\)

Optimal. Leaf size=237 \[ \frac {a \left (15 a^2-2 b^2\right ) \sin (c+d x)}{3 b^5 d}-\frac {\left (20 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{8 b^4 d}-\frac {x \left (40 a^4-12 a^2 b^2-b^4\right )}{8 b^6}+\frac {2 a^3 \left (5 a^2-4 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^6 d \sqrt {a-b} \sqrt {a+b}}+\frac {5 a \sin (c+d x) \cos ^2(c+d x)}{3 b^3 d}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}-\frac {5 \sin (c+d x) \cos ^3(c+d x)}{4 b^2 d} \]

[Out]

-1/8*(40*a^4-12*a^2*b^2-b^4)*x/b^6+1/3*a*(15*a^2-2*b^2)*sin(d*x+c)/b^5/d-1/8*(20*a^2-b^2)*cos(d*x+c)*sin(d*x+c
)/b^4/d+5/3*a*cos(d*x+c)^2*sin(d*x+c)/b^3/d-5/4*cos(d*x+c)^3*sin(d*x+c)/b^2/d+cos(d*x+c)^4*sin(d*x+c)/b/d/(a+b
*cos(d*x+c))+2*a^3*(5*a^2-4*b^2)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/b^6/d/(a-b)^(1/2)/(a+b)^(1
/2)

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Rubi [A]  time = 0.90, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3048, 3050, 3049, 3023, 2735, 2659, 205} \[ \frac {a \left (15 a^2-2 b^2\right ) \sin (c+d x)}{3 b^5 d}+\frac {2 a^3 \left (5 a^2-4 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^6 d \sqrt {a-b} \sqrt {a+b}}-\frac {\left (20 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{8 b^4 d}-\frac {x \left (-12 a^2 b^2+40 a^4-b^4\right )}{8 b^6}+\frac {5 a \sin (c+d x) \cos ^2(c+d x)}{3 b^3 d}+\frac {\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}-\frac {5 \sin (c+d x) \cos ^3(c+d x)}{4 b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^4*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]

[Out]

-((40*a^4 - 12*a^2*b^2 - b^4)*x)/(8*b^6) + (2*a^3*(5*a^2 - 4*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a
+ b]])/(Sqrt[a - b]*b^6*Sqrt[a + b]*d) + (a*(15*a^2 - 2*b^2)*Sin[c + d*x])/(3*b^5*d) - ((20*a^2 - b^2)*Cos[c
+ d*x]*Sin[c + d*x])/(8*b^4*d) + (5*a*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^3*d) - (5*Cos[c + d*x]^3*Sin[c + d*x])
/(4*b^2*d) + (Cos[c + d*x]^4*Sin[c + d*x])/(b*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])))

Rule 3050

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (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)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*
x] + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, 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 ^4(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx &=\frac {\cos ^4(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac {\int \frac {\cos ^3(c+d x) \left (-4 \left (a^2-b^2\right )+5 \left (a^2-b^2\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{b \left (a^2-b^2\right )}\\ &=-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{4 b^2 d}+\frac {\cos ^4(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac {\int \frac {\cos ^2(c+d x) \left (15 a \left (a^2-b^2\right )-b \left (a^2-b^2\right ) \cos (c+d x)-20 a \left (a^2-b^2\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{4 b^2 \left (a^2-b^2\right )}\\ &=\frac {5 a \cos ^2(c+d x) \sin (c+d x)}{3 b^3 d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{4 b^2 d}+\frac {\cos ^4(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (-40 a^2 \left (a^2-b^2\right )+5 a b \left (a^2-b^2\right ) \cos (c+d x)+3 \left (a^2-b^2\right ) \left (20 a^2-b^2\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{12 b^3 \left (a^2-b^2\right )}\\ &=-\frac {\left (20 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^4 d}+\frac {5 a \cos ^2(c+d x) \sin (c+d x)}{3 b^3 d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{4 b^2 d}+\frac {\cos ^4(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac {\int \frac {3 a \left (20 a^4-21 a^2 b^2+b^4\right )-b \left (a^2-b^2\right ) \left (20 a^2+3 b^2\right ) \cos (c+d x)-8 a \left (15 a^2-2 b^2\right ) \left (a^2-b^2\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{24 b^4 \left (a^2-b^2\right )}\\ &=\frac {a \left (15 a^2-2 b^2\right ) \sin (c+d x)}{3 b^5 d}-\frac {\left (20 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^4 d}+\frac {5 a \cos ^2(c+d x) \sin (c+d x)}{3 b^3 d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{4 b^2 d}+\frac {\cos ^4(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac {\int \frac {3 a b \left (20 a^4-21 a^2 b^2+b^4\right )+3 \left (40 a^6-52 a^4 b^2+11 a^2 b^4+b^6\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{24 b^5 \left (a^2-b^2\right )}\\ &=-\frac {\left (40 a^4-12 a^2 b^2-b^4\right ) x}{8 b^6}+\frac {a \left (15 a^2-2 b^2\right ) \sin (c+d x)}{3 b^5 d}-\frac {\left (20 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^4 d}+\frac {5 a \cos ^2(c+d x) \sin (c+d x)}{3 b^3 d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{4 b^2 d}+\frac {\cos ^4(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}+\frac {\left (a^3 \left (5 a^2-4 b^2\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{b^6}\\ &=-\frac {\left (40 a^4-12 a^2 b^2-b^4\right ) x}{8 b^6}+\frac {a \left (15 a^2-2 b^2\right ) \sin (c+d x)}{3 b^5 d}-\frac {\left (20 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^4 d}+\frac {5 a \cos ^2(c+d x) \sin (c+d x)}{3 b^3 d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{4 b^2 d}+\frac {\cos ^4(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}+\frac {\left (2 a^3 \left (5 a^2-4 b^2\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^6 d}\\ &=-\frac {\left (40 a^4-12 a^2 b^2-b^4\right ) x}{8 b^6}+\frac {2 a^3 \left (5 a^2-4 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^6 \sqrt {a+b} d}+\frac {a \left (15 a^2-2 b^2\right ) \sin (c+d x)}{3 b^5 d}-\frac {\left (20 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^4 d}+\frac {5 a \cos ^2(c+d x) \sin (c+d x)}{3 b^3 d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{4 b^2 d}+\frac {\cos ^4(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 3.88, size = 271, normalized size = 1.14 \[ \frac {\frac {-960 a^5 c-960 a^5 d x+240 a^3 b^2 \sin (2 (c+d x))+288 a^3 b^2 c+288 a^3 b^2 d x-40 a^2 b^3 \sin (3 (c+d x))+24 a^2 b \left (40 a^2-7 b^2\right ) \sin (c+d x)+24 b \left (-40 a^4+12 a^2 b^2+b^4\right ) (c+d x) \cos (c+d x)-32 a b^4 \sin (2 (c+d x))+10 a b^4 \sin (4 (c+d x))+24 a b^4 c+24 a b^4 d x-3 b^5 \sin (3 (c+d x))-3 b^5 \sin (5 (c+d x))}{a+b \cos (c+d x)}-\frac {384 a^3 \left (5 a^2-4 b^2\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\sqrt {b^2-a^2}}}{192 b^6 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^4*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]

[Out]

((-384*a^3*(5*a^2 - 4*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (-960*a^5*
c + 288*a^3*b^2*c + 24*a*b^4*c - 960*a^5*d*x + 288*a^3*b^2*d*x + 24*a*b^4*d*x + 24*b*(-40*a^4 + 12*a^2*b^2 + b
^4)*(c + d*x)*Cos[c + d*x] + 24*a^2*b*(40*a^2 - 7*b^2)*Sin[c + d*x] + 240*a^3*b^2*Sin[2*(c + d*x)] - 32*a*b^4*
Sin[2*(c + d*x)] - 40*a^2*b^3*Sin[3*(c + d*x)] - 3*b^5*Sin[3*(c + d*x)] + 10*a*b^4*Sin[4*(c + d*x)] - 3*b^5*Si
n[5*(c + d*x)])/(a + b*Cos[c + d*x]))/(192*b^6*d)

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fricas [A]  time = 0.55, size = 731, normalized size = 3.08 \[ \left [-\frac {3 \, {\left (40 \, a^{6} b - 52 \, a^{4} b^{3} + 11 \, a^{2} b^{5} + b^{7}\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (40 \, a^{7} - 52 \, a^{5} b^{2} + 11 \, a^{3} b^{4} + a b^{6}\right )} d x - 12 \, {\left (5 \, a^{6} - 4 \, a^{4} b^{2} + {\left (5 \, a^{5} b - 4 \, a^{3} b^{3}\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 ) - {\left (120 \, a^{6} b - 136 \, a^{4} b^{3} + 16 \, a^{2} b^{5} - 6 \, {\left (a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{4} + 10 \, {\left (a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{3} - {\left (20 \, a^{4} b^{3} - 23 \, a^{2} b^{5} + 3 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (60 \, a^{5} b^{2} - 73 \, a^{3} b^{4} + 13 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left ({\left (a^{2} b^{7} - b^{9}\right )} d \cos \left (d x + c\right ) + {\left (a^{3} b^{6} - a b^{8}\right )} d\right )}}, -\frac {3 \, {\left (40 \, a^{6} b - 52 \, a^{4} b^{3} + 11 \, a^{2} b^{5} + b^{7}\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (40 \, a^{7} - 52 \, a^{5} b^{2} + 11 \, a^{3} b^{4} + a b^{6}\right )} d x - 24 \, {\left (5 \, a^{6} - 4 \, a^{4} b^{2} + {\left (5 \, a^{5} b - 4 \, a^{3} b^{3}\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 (120 \, a^{6} b - 136 \, a^{4} b^{3} + 16 \, a^{2} b^{5} - 6 \, {\left (a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{4} + 10 \, {\left (a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{3} - {\left (20 \, a^{4} b^{3} - 23 \, a^{2} b^{5} + 3 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (60 \, a^{5} b^{2} - 73 \, a^{3} b^{4} + 13 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left ({\left (a^{2} b^{7} - b^{9}\right )} d \cos \left (d x + c\right ) + {\left (a^{3} b^{6} - a b^{8}\right )} d\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(1-cos(d*x+c)^2)/(a+b*cos(d*x+c))^2,x, algorithm="fricas")

[Out]

[-1/24*(3*(40*a^6*b - 52*a^4*b^3 + 11*a^2*b^5 + b^7)*d*x*cos(d*x + c) + 3*(40*a^7 - 52*a^5*b^2 + 11*a^3*b^4 +
a*b^6)*d*x - 12*(5*a^6 - 4*a^4*b^2 + (5*a^5*b - 4*a^3*b^3)*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)) - (120*a^6*b - 136*a^4*b^3 + 16*a^2*b^5 - 6*(a^2*b^5 - b^7)*cos(d*
x + c)^4 + 10*(a^3*b^4 - a*b^6)*cos(d*x + c)^3 - (20*a^4*b^3 - 23*a^2*b^5 + 3*b^7)*cos(d*x + c)^2 + (60*a^5*b^
2 - 73*a^3*b^4 + 13*a*b^6)*cos(d*x + c))*sin(d*x + c))/((a^2*b^7 - b^9)*d*cos(d*x + c) + (a^3*b^6 - a*b^8)*d),
-1/24*(3*(40*a^6*b - 52*a^4*b^3 + 11*a^2*b^5 + b^7)*d*x*cos(d*x + c) + 3*(40*a^7 - 52*a^5*b^2 + 11*a^3*b^4 +
a*b^6)*d*x - 24*(5*a^6 - 4*a^4*b^2 + (5*a^5*b - 4*a^3*b^3)*cos(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*cos(d*x +
c) + b)/(sqrt(a^2 - b^2)*sin(d*x + c))) - (120*a^6*b - 136*a^4*b^3 + 16*a^2*b^5 - 6*(a^2*b^5 - b^7)*cos(d*x +
c)^4 + 10*(a^3*b^4 - a*b^6)*cos(d*x + c)^3 - (20*a^4*b^3 - 23*a^2*b^5 + 3*b^7)*cos(d*x + c)^2 + (60*a^5*b^2 -
73*a^3*b^4 + 13*a*b^6)*cos(d*x + c))*sin(d*x + c))/((a^2*b^7 - b^9)*d*cos(d*x + c) + (a^3*b^6 - a*b^8)*d)]

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giac [A]  time = 0.95, size = 421, normalized size = 1.78 \[ \frac {\frac {48 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )} b^{5}} - \frac {3 \, {\left (40 \, a^{4} - 12 \, a^{2} b^{2} - b^{4}\right )} {\left (d x + c\right )}}{b^{6}} - \frac {48 \, {\left (5 \, a^{5} - 4 \, a^{3} b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{6}} + \frac {2 \, {\left (96 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 288 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 64 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 21 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 288 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 64 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 96 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} b^{5}}}{24 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(1-cos(d*x+c)^2)/(a+b*cos(d*x+c))^2,x, algorithm="giac")

[Out]

1/24*(48*a^4*tan(1/2*d*x + 1/2*c)/((a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 + a + b)*b^5) - 3*(40*
a^4 - 12*a^2*b^2 - b^4)*(d*x + c)/b^6 - 48*(5*a^5 - 4*a^3*b^2)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*
b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*b^6) + 2*(96
*a^3*tan(1/2*d*x + 1/2*c)^7 + 36*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 3*b^3*tan(1/2*d*x + 1/2*c)^7 + 288*a^3*tan(1/2
*d*x + 1/2*c)^5 + 36*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 64*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 21*b^3*tan(1/2*d*x + 1/2
*c)^5 + 288*a^3*tan(1/2*d*x + 1/2*c)^3 - 36*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 64*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 2
1*b^3*tan(1/2*d*x + 1/2*c)^3 + 96*a^3*tan(1/2*d*x + 1/2*c) - 36*a^2*b*tan(1/2*d*x + 1/2*c) - 3*b^3*tan(1/2*d*x
+ 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^4*b^5))/d

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maple [B]  time = 0.11, size = 708, normalized size = 2.99 \[ \frac {2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,b^{5} \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 )}+\frac {10 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^{6} \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {8 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^{4} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {8 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{d \,b^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,b^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {24 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{d \,b^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {16 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{3 d \,b^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,b^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {24 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{d \,b^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,b^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {16 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{3 d \,b^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3}}{d \,b^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,b^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {10 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{d \,b^{6}}+\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{d \,b^{4}}+\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*(1-cos(d*x+c)^2)/(a+b*cos(d*x+c))^2,x)

[Out]

2/d*a^4/b^5*tan(1/2*d*x+1/2*c)/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)+10/d*a^5/b^6/((a-b)*(a+b))^
(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))-8/d*a^3/b^4/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+
1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))+8/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7*a^3+3/d/b^4/(1+tan(1
/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7*a^2+1/4/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7+24/d/b^5
/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*a^3+3/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*a
^2-16/3/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*a-7/4/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d
*x+1/2*c)^5+24/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3*a^3-3/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^4*ta
n(1/2*d*x+1/2*c)^3*a^2+7/4/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3-16/3/d/b^3/(1+tan(1/2*d*x+1/2
*c)^2)^4*tan(1/2*d*x+1/2*c)^3*a+8/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*a^3-3/d/b^4/(1+tan(1/2*d
*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*a^2-1/4/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)-10/d/b^6*arctan(
tan(1/2*d*x+1/2*c))*a^4+3/d/b^4*arctan(tan(1/2*d*x+1/2*c))*a^2+1/4/d/b^2*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)^4*(1-cos(d*x+c)^2)/(a+b*cos(d*x+c))^2,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 = 4.78, size = 2971, normalized size = 12.54 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(cos(c + d*x)^4*(cos(c + d*x)^2 - 1))/(a + b*cos(c + d*x))^2,x)

[Out]

(atan(((((tan(c/2 + (d*x)/2)*(3*a*b^10 - 6400*a^10*b + 3200*a^11 - b^11 - 27*a^2*b^9 + 73*a^3*b^8 - 136*a^4*b^
7 + 216*a^5*b^6 - 256*a^6*b^5 - 1792*a^7*b^4 + 3840*a^8*b^3 + 1280*a^9*b^2))/(2*b^10) - (((4*b^18 - 4*a*b^17 +
44*a^2*b^16 - 172*a^3*b^15 + 48*a^4*b^14 + 240*a^5*b^13 - 160*a^6*b^12)/b^15 - (tan(c/2 + (d*x)/2)*(b^4*1i -
a^4*40i + a^2*b^2*12i)*(128*a*b^14 - 256*a^2*b^13 + 128*a^3*b^12))/(16*b^16))*(b^4*1i - a^4*40i + a^2*b^2*12i)
)/(8*b^6))*(b^4*1i - a^4*40i + a^2*b^2*12i)*1i)/(8*b^6) + (((tan(c/2 + (d*x)/2)*(3*a*b^10 - 6400*a^10*b + 3200
*a^11 - b^11 - 27*a^2*b^9 + 73*a^3*b^8 - 136*a^4*b^7 + 216*a^5*b^6 - 256*a^6*b^5 - 1792*a^7*b^4 + 3840*a^8*b^3
+ 1280*a^9*b^2))/(2*b^10) + (((4*b^18 - 4*a*b^17 + 44*a^2*b^16 - 172*a^3*b^15 + 48*a^4*b^14 + 240*a^5*b^13 -
160*a^6*b^12)/b^15 + (tan(c/2 + (d*x)/2)*(b^4*1i - a^4*40i + a^2*b^2*12i)*(128*a*b^14 - 256*a^2*b^13 + 128*a^3
*b^12))/(16*b^16))*(b^4*1i - a^4*40i + a^2*b^2*12i))/(8*b^6))*(b^4*1i - a^4*40i + a^2*b^2*12i)*1i)/(8*b^6))/((
12000*a^13*b - 8000*a^14 - 4*a^3*b^11 + 8*a^4*b^10 - 95*a^5*b^9 + 54*a^6*b^8 - 99*a^7*b^7 - 944*a^8*b^6 + 5240
*a^9*b^5 + 440*a^10*b^4 - 15800*a^11*b^3 + 7200*a^12*b^2)/b^15 - (((tan(c/2 + (d*x)/2)*(3*a*b^10 - 6400*a^10*b
+ 3200*a^11 - b^11 - 27*a^2*b^9 + 73*a^3*b^8 - 136*a^4*b^7 + 216*a^5*b^6 - 256*a^6*b^5 - 1792*a^7*b^4 + 3840*
a^8*b^3 + 1280*a^9*b^2))/(2*b^10) - (((4*b^18 - 4*a*b^17 + 44*a^2*b^16 - 172*a^3*b^15 + 48*a^4*b^14 + 240*a^5*
b^13 - 160*a^6*b^12)/b^15 - (tan(c/2 + (d*x)/2)*(b^4*1i - a^4*40i + a^2*b^2*12i)*(128*a*b^14 - 256*a^2*b^13 +
128*a^3*b^12))/(16*b^16))*(b^4*1i - a^4*40i + a^2*b^2*12i))/(8*b^6))*(b^4*1i - a^4*40i + a^2*b^2*12i))/(8*b^6)
+ (((tan(c/2 + (d*x)/2)*(3*a*b^10 - 6400*a^10*b + 3200*a^11 - b^11 - 27*a^2*b^9 + 73*a^3*b^8 - 136*a^4*b^7 +
216*a^5*b^6 - 256*a^6*b^5 - 1792*a^7*b^4 + 3840*a^8*b^3 + 1280*a^9*b^2))/(2*b^10) + (((4*b^18 - 4*a*b^17 + 44*
a^2*b^16 - 172*a^3*b^15 + 48*a^4*b^14 + 240*a^5*b^13 - 160*a^6*b^12)/b^15 + (tan(c/2 + (d*x)/2)*(b^4*1i - a^4*
40i + a^2*b^2*12i)*(128*a*b^14 - 256*a^2*b^13 + 128*a^3*b^12))/(16*b^16))*(b^4*1i - a^4*40i + a^2*b^2*12i))/(8
*b^6))*(b^4*1i - a^4*40i + a^2*b^2*12i))/(8*b^6)))*(b^4*1i - a^4*40i + a^2*b^2*12i)*1i)/(4*b^6*d) - ((tan(c/2
+ (d*x)/2)*(a*b^3 - 20*a^3*b - 40*a^4 + b^4 + 12*a^2*b^2))/(4*b^5) + (tan(c/2 + (d*x)/2)^9*(20*a^3*b - a*b^3 -
40*a^4 + b^4 + 12*a^2*b^2))/(4*b^5) + (tan(c/2 + (d*x)/2)^5*(21*b^4 - 360*a^4 + 28*a^2*b^2))/(6*b^5) - (tan(c
/2 + (d*x)/2)^3*(60*a^3*b - 23*a*b^3 + 240*a^4 + 12*b^4 - 32*a^2*b^2))/(6*b^5) - (tan(c/2 + (d*x)/2)^7*(23*a*b
^3 - 60*a^3*b + 240*a^4 + 12*b^4 - 32*a^2*b^2))/(6*b^5))/(d*(a + b + tan(c/2 + (d*x)/2)^10*(a - b) + tan(c/2 +
(d*x)/2)^2*(5*a + 3*b) + tan(c/2 + (d*x)/2)^4*(10*a + 2*b) + tan(c/2 + (d*x)/2)^8*(5*a - 3*b) + tan(c/2 + (d*
x)/2)^6*(10*a - 2*b))) + (a^3*atan(((a^3*(-(a + b)*(a - b))^(1/2)*((tan(c/2 + (d*x)/2)*(3*a*b^10 - 6400*a^10*b
+ 3200*a^11 - b^11 - 27*a^2*b^9 + 73*a^3*b^8 - 136*a^4*b^7 + 216*a^5*b^6 - 256*a^6*b^5 - 1792*a^7*b^4 + 3840*
a^8*b^3 + 1280*a^9*b^2))/(2*b^10) + (a^3*(-(a + b)*(a - b))^(1/2)*(5*a^2 - 4*b^2)*((4*b^18 - 4*a*b^17 + 44*a^2
*b^16 - 172*a^3*b^15 + 48*a^4*b^14 + 240*a^5*b^13 - 160*a^6*b^12)/b^15 + (a^3*tan(c/2 + (d*x)/2)*(-(a + b)*(a
- b))^(1/2)*(5*a^2 - 4*b^2)*(128*a*b^14 - 256*a^2*b^13 + 128*a^3*b^12))/(2*b^10*(b^8 - a^2*b^6))))/(b^8 - a^2*
b^6))*(5*a^2 - 4*b^2)*1i)/(b^8 - a^2*b^6) + (a^3*(-(a + b)*(a - b))^(1/2)*((tan(c/2 + (d*x)/2)*(3*a*b^10 - 640
0*a^10*b + 3200*a^11 - b^11 - 27*a^2*b^9 + 73*a^3*b^8 - 136*a^4*b^7 + 216*a^5*b^6 - 256*a^6*b^5 - 1792*a^7*b^4
+ 3840*a^8*b^3 + 1280*a^9*b^2))/(2*b^10) - (a^3*(-(a + b)*(a - b))^(1/2)*(5*a^2 - 4*b^2)*((4*b^18 - 4*a*b^17
+ 44*a^2*b^16 - 172*a^3*b^15 + 48*a^4*b^14 + 240*a^5*b^13 - 160*a^6*b^12)/b^15 - (a^3*tan(c/2 + (d*x)/2)*(-(a
+ b)*(a - b))^(1/2)*(5*a^2 - 4*b^2)*(128*a*b^14 - 256*a^2*b^13 + 128*a^3*b^12))/(2*b^10*(b^8 - a^2*b^6))))/(b^
8 - a^2*b^6))*(5*a^2 - 4*b^2)*1i)/(b^8 - a^2*b^6))/((12000*a^13*b - 8000*a^14 - 4*a^3*b^11 + 8*a^4*b^10 - 95*a
^5*b^9 + 54*a^6*b^8 - 99*a^7*b^7 - 944*a^8*b^6 + 5240*a^9*b^5 + 440*a^10*b^4 - 15800*a^11*b^3 + 7200*a^12*b^2)
/b^15 + (a^3*(-(a + b)*(a - b))^(1/2)*((tan(c/2 + (d*x)/2)*(3*a*b^10 - 6400*a^10*b + 3200*a^11 - b^11 - 27*a^2
*b^9 + 73*a^3*b^8 - 136*a^4*b^7 + 216*a^5*b^6 - 256*a^6*b^5 - 1792*a^7*b^4 + 3840*a^8*b^3 + 1280*a^9*b^2))/(2*
b^10) + (a^3*(-(a + b)*(a - b))^(1/2)*(5*a^2 - 4*b^2)*((4*b^18 - 4*a*b^17 + 44*a^2*b^16 - 172*a^3*b^15 + 48*a^
4*b^14 + 240*a^5*b^13 - 160*a^6*b^12)/b^15 + (a^3*tan(c/2 + (d*x)/2)*(-(a + b)*(a - b))^(1/2)*(5*a^2 - 4*b^2)*
(128*a*b^14 - 256*a^2*b^13 + 128*a^3*b^12))/(2*b^10*(b^8 - a^2*b^6))))/(b^8 - a^2*b^6))*(5*a^2 - 4*b^2))/(b^8
- a^2*b^6) - (a^3*(-(a + b)*(a - b))^(1/2)*((tan(c/2 + (d*x)/2)*(3*a*b^10 - 6400*a^10*b + 3200*a^11 - b^11 - 2
7*a^2*b^9 + 73*a^3*b^8 - 136*a^4*b^7 + 216*a^5*b^6 - 256*a^6*b^5 - 1792*a^7*b^4 + 3840*a^8*b^3 + 1280*a^9*b^2)
)/(2*b^10) - (a^3*(-(a + b)*(a - b))^(1/2)*(5*a^2 - 4*b^2)*((4*b^18 - 4*a*b^17 + 44*a^2*b^16 - 172*a^3*b^15 +
48*a^4*b^14 + 240*a^5*b^13 - 160*a^6*b^12)/b^15 - (a^3*tan(c/2 + (d*x)/2)*(-(a + b)*(a - b))^(1/2)*(5*a^2 - 4*
b^2)*(128*a*b^14 - 256*a^2*b^13 + 128*a^3*b^12))/(2*b^10*(b^8 - a^2*b^6))))/(b^8 - a^2*b^6))*(5*a^2 - 4*b^2))/
(b^8 - a^2*b^6)))*(-(a + b)*(a - b))^(1/2)*(5*a^2 - 4*b^2)*2i)/(d*(b^8 - a^2*b^6))

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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)**4*(1-cos(d*x+c)**2)/(a+b*cos(d*x+c))**2,x)

[Out]

Timed out

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