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

Optimal. Leaf size=182 \[ -\frac {a \left (3 a^2-2 b^2\right ) \sin (c+d x)}{2 b^3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\left (6 a^4-9 a^2 b^2+2 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac {3 a x}{b^4}+\frac {\sin (c+d x) \cos ^2(c+d x)}{2 b d (a+b \cos (c+d x))^2}-\frac {3 \sin (c+d x)}{2 b^3 d} \]

[Out]

3*a*x/b^4-(6*a^4-9*a^2*b^2+2*b^4)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(3/2)/b^4/(a+b)^(3/
2)/d-3/2*sin(d*x+c)/b^3/d+1/2*cos(d*x+c)^2*sin(d*x+c)/b/d/(a+b*cos(d*x+c))^2-1/2*a*(3*a^2-2*b^2)*sin(d*x+c)/b^
3/(a^2-b^2)/d/(a+b*cos(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 0.46, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3048, 3032, 3023, 2735, 2659, 205} \[ -\frac {\left (-9 a^2 b^2+6 a^4+2 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}-\frac {a \left (3 a^2-2 b^2\right ) \sin (c+d x)}{2 b^3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {3 a x}{b^4}+\frac {\sin (c+d x) \cos ^2(c+d x)}{2 b d (a+b \cos (c+d x))^2}-\frac {3 \sin (c+d x)}{2 b^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a*x)/b^4 - ((6*a^4 - 9*a^2*b^2 + 2*b^4)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*
b^4*(a + b)^(3/2)*d) - (3*Sin[c + d*x])/(2*b^3*d) + (Cos[c + d*x]^2*Sin[c + d*x])/(2*b*d*(a + b*Cos[c + d*x])^
2) - (a*(3*a^2 - 2*b^2)*Sin[c + d*x])/(2*b^3*(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 3032

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (C_.)*sin[(e
_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((b*c - a*d)*(A*b^2 + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1
))/(b^2*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(b^2*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b
*(m + 1)*(a*C*(b*c - a*d) + A*b*(a*c - b*d)) - ((b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f
*x] + b*C*d*(m + 1)*(a^2 - b^2)*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] && 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]

Rubi steps

\begin {align*} \int \frac {\cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx &=\frac {\cos ^2(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}-\frac {\int \frac {\cos (c+d x) \left (-2 \left (a^2-b^2\right )+3 \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 ^2(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}-\frac {a \left (3 a^2-2 b^2\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {b \left (3 a^4-5 a^2 b^2+2 b^4\right )+3 a \left (a^2-b^2\right )^2 \cos (c+d x)-3 b \left (a^2-b^2\right )^2 \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {3 \sin (c+d x)}{2 b^3 d}+\frac {\cos ^2(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}-\frac {a \left (3 a^2-2 b^2\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {b^2 \left (3 a^4-5 a^2 b^2+2 b^4\right )+6 a b \left (a^2-b^2\right )^2 \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{2 b^4 \left (a^2-b^2\right )^2}\\ &=\frac {3 a x}{b^4}-\frac {3 \sin (c+d x)}{2 b^3 d}+\frac {\cos ^2(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}-\frac {a \left (3 a^2-2 b^2\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\left (6 a^4-9 a^2 b^2+2 b^4\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{2 b^4 \left (a^2-b^2\right )}\\ &=\frac {3 a x}{b^4}-\frac {3 \sin (c+d x)}{2 b^3 d}+\frac {\cos ^2(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}-\frac {a \left (3 a^2-2 b^2\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\left (6 a^4-9 a^2 b^2+2 b^4\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^4 \left (a^2-b^2\right ) d}\\ &=\frac {3 a x}{b^4}-\frac {\left (6 a^4-9 a^2 b^2+2 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^4 (a+b)^{3/2} d}-\frac {3 \sin (c+d x)}{2 b^3 d}+\frac {\cos ^2(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}-\frac {a \left (3 a^2-2 b^2\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.07, size = 159, normalized size = 0.87 \[ \frac {\frac {a b \left (4 b^2-5 a^2\right ) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}+\frac {a^2 b \sin (c+d x)}{(a+b \cos (c+d x))^2}-\frac {2 \left (6 a^4-9 a^2 b^2+2 b^4\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\left (b^2-a^2\right )^{3/2}}+6 a (c+d x)-2 b \sin (c+d x)}{2 b^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*a*(c + d*x) - (2*(6*a^4 - 9*a^2*b^2 + 2*b^4)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 +
b^2)^(3/2) - 2*b*Sin[c + d*x] + (a^2*b*Sin[c + d*x])/(a + b*Cos[c + d*x])^2 + (a*b*(-5*a^2 + 4*b^2)*Sin[c + d*
x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/(2*b^4*d)

________________________________________________________________________________________

fricas [B]  time = 0.57, size = 856, normalized size = 4.70 \[ \left [\frac {12 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} d x \cos \left (d x + c\right )^{2} + 24 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d x \cos \left (d x + c\right ) + 12 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d x + {\left (6 \, a^{6} - 9 \, a^{4} b^{2} + 2 \, a^{2} b^{4} + {\left (6 \, a^{4} b^{2} - 9 \, a^{2} b^{4} + 2 \, b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{5} b - 9 \, a^{3} b^{3} + 2 \, a 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 (6 \, a^{6} b - 11 \, a^{4} b^{3} + 5 \, a^{2} b^{5} + 2 \, {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (9 \, a^{5} b^{2} - 17 \, a^{3} b^{4} + 8 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{4} b^{6} - 2 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{5} - 2 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b^{4} - 2 \, a^{4} b^{6} + a^{2} b^{8}\right )} d\right )}}, \frac {6 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} d x \cos \left (d x + c\right )^{2} + 12 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d x \cos \left (d x + c\right ) + 6 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d x - {\left (6 \, a^{6} - 9 \, a^{4} b^{2} + 2 \, a^{2} b^{4} + {\left (6 \, a^{4} b^{2} - 9 \, a^{2} b^{4} + 2 \, b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{5} b - 9 \, a^{3} b^{3} + 2 \, a 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 (6 \, a^{6} b - 11 \, a^{4} b^{3} + 5 \, a^{2} b^{5} + 2 \, {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (9 \, a^{5} b^{2} - 17 \, a^{3} b^{4} + 8 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b^{6} - 2 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{5} - 2 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b^{4} - 2 \, a^{4} b^{6} + a^{2} b^{8}\right )} d\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(12*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*d*x*cos(d*x + c)^2 + 24*(a^6*b - 2*a^4*b^3 + a^2*b^5)*d*x*cos(d*x + c)
+ 12*(a^7 - 2*a^5*b^2 + a^3*b^4)*d*x + (6*a^6 - 9*a^4*b^2 + 2*a^2*b^4 + (6*a^4*b^2 - 9*a^2*b^4 + 2*b^6)*cos(d*
x + c)^2 + 2*(6*a^5*b - 9*a^3*b^3 + 2*a*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*(6*a^6*b - 11*a^4*b^3 + 5*a^2*b^5 + 2*(a^4*b^3 - 2*a^2*b^5 + b^7)*cos(d*x +
c)^2 + (9*a^5*b^2 - 17*a^3*b^4 + 8*a*b^6)*cos(d*x + c))*sin(d*x + c))/((a^4*b^6 - 2*a^2*b^8 + b^10)*d*cos(d*x
+ c)^2 + 2*(a^5*b^5 - 2*a^3*b^7 + a*b^9)*d*cos(d*x + c) + (a^6*b^4 - 2*a^4*b^6 + a^2*b^8)*d), 1/2*(6*(a^5*b^2
- 2*a^3*b^4 + a*b^6)*d*x*cos(d*x + c)^2 + 12*(a^6*b - 2*a^4*b^3 + a^2*b^5)*d*x*cos(d*x + c) + 6*(a^7 - 2*a^5*b
^2 + a^3*b^4)*d*x - (6*a^6 - 9*a^4*b^2 + 2*a^2*b^4 + (6*a^4*b^2 - 9*a^2*b^4 + 2*b^6)*cos(d*x + c)^2 + 2*(6*a^5
*b - 9*a^3*b^3 + 2*a*b^5)*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))) - (6*a^6*b - 11*a^4*b^3 + 5*a^2*b^5 + 2*(a^4*b^3 - 2*a^2*b^5 + b^7)*cos(d*x + c)^2 + (9*a^5*b^2 - 17*a^
3*b^4 + 8*a*b^6)*cos(d*x + c))*sin(d*x + c))/((a^4*b^6 - 2*a^2*b^8 + b^10)*d*cos(d*x + c)^2 + 2*(a^5*b^5 - 2*a
^3*b^7 + a*b^9)*d*cos(d*x + c) + (a^6*b^4 - 2*a^4*b^6 + a^2*b^8)*d)]

________________________________________________________________________________________

giac [A]  time = 0.76, size = 333, normalized size = 1.83 \[ \frac {\frac {{\left (6 \, a^{4} - 9 \, a^{2} b^{2} + 2 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{2} b^{4} - b^{6}\right )} \sqrt {a^{2} - b^{2}}} - \frac {4 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{2} b^{3} - b^{5}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )}^{2}} + \frac {3 \, {\left (d x + c\right )} a}{b^{4}} - \frac {2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} b^{3}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((6*a^4 - 9*a^2*b^2 + 2*b^4)*(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)))/((a^2*b^4 - b^6)*sqrt(a^2 - b^2)) - (4*a^4*tan(1/2*d*x + 1/2*c)
^3 - 5*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 3*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 4*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 4*a^
4*tan(1/2*d*x + 1/2*c) + 5*a^3*b*tan(1/2*d*x + 1/2*c) - 3*a^2*b^2*tan(1/2*d*x + 1/2*c) - 4*a*b^3*tan(1/2*d*x +
1/2*c))/((a^2*b^3 - b^5)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 + a + b)^2) + 3*(d*x + c)*a/b^4
- 2*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 + 1)*b^3))/d

________________________________________________________________________________________

maple [B]  time = 0.10, size = 576, normalized size = 3.16 \[ -\frac {4 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 {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 {4 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d b \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 {4 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 {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 {4 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d b \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} \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} \left (a^{2}-b^{2}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {9 a^{2} \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^{2} \left (a^{2}-b^{2}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {2 \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 \left (a^{2}-b^{2}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {2 \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 )}+\frac {6 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{d \,b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/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)^3+1/d*a^2/b^2/(a*t
an(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+4/d/b/(a*tan(1/2*d*x+1/2*c)^2-tan
(1/2*d*x+1/2*c)^2*b+a+b)^2*a/(a+b)*tan(1/2*d*x+1/2*c)^3-4/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)-1/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)*t
an(1/2*d*x+1/2*c)+4/d/b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2*a/(a-b)*tan(1/2*d*x+1/2*c)-6/d*a
^4/b^4/(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))+9/d*a^2/b^2/(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))-2/d/(a^2-b^2)/((a-b)*(a+b))^(1/2)*a
rctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))-2/d/b^3*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)+6/d/b^
4*arctan(tan(1/2*d*x+1/2*c))*a

________________________________________________________________________________________

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)^2*(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?

________________________________________________________________________________________

mupad [B]  time = 8.23, size = 3380, normalized size = 18.57 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((tan(c/2 + (d*x)/2)*(6*a*b^2 - 3*a^2*b - 6*a^3 + 2*b^3))/(a*b^3 - b^4) + (tan(c/2 + (d*x)/2)^5*(6*a*b^2 + 3*a
^2*b - 6*a^3 - 2*b^3))/(b^3*(a + b)) - (2*tan(c/2 + (d*x)/2)^3*(6*a^4 + 2*b^4 - 7*a^2*b^2))/(b^3*(a + b)*(a -
b)))/(d*(2*a*b + tan(c/2 + (d*x)/2)^2*(2*a*b + 3*a^2 - b^2) + tan(c/2 + (d*x)/2)^6*(a^2 - 2*a*b + b^2) + a^2 +
b^2 - tan(c/2 + (d*x)/2)^4*(2*a*b - 3*a^2 + b^2))) + (6*a*atan(((3*a*((8*tan(c/2 + (d*x)/2)*(72*a^8 - 72*a^7*
b + 4*b^8 - 72*a^3*b^5 + 69*a^4*b^4 + 144*a^5*b^3 - 144*a^6*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) + (a*((8*(
8*a*b^13 + 4*b^14 - 22*a^2*b^12 - 14*a^3*b^11 + 30*a^4*b^10 + 6*a^5*b^9 - 12*a^6*b^8))/(a*b^11 + b^12 - a^2*b^
10 - a^3*b^9) - (a*tan(c/2 + (d*x)/2)*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8*a^6*b
^8)*24i)/(b^4*(a*b^8 + b^9 - a^2*b^7 - a^3*b^6)))*3i)/b^4))/b^4 + (3*a*((8*tan(c/2 + (d*x)/2)*(72*a^8 - 72*a^7
*b + 4*b^8 - 72*a^3*b^5 + 69*a^4*b^4 + 144*a^5*b^3 - 144*a^6*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) - (a*((8*
(8*a*b^13 + 4*b^14 - 22*a^2*b^12 - 14*a^3*b^11 + 30*a^4*b^10 + 6*a^5*b^9 - 12*a^6*b^8))/(a*b^11 + b^12 - a^2*b
^10 - a^3*b^9) + (a*tan(c/2 + (d*x)/2)*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8*a^6*
b^8)*24i)/(b^4*(a*b^8 + b^9 - a^2*b^7 - a^3*b^6)))*3i)/b^4))/b^4)/((16*(12*a*b^7 - 54*a^7*b + 108*a^8 - 36*a^2
*b^6 - 72*a^3*b^5 + 198*a^4*b^4 + 117*a^5*b^3 - 270*a^6*b^2))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) - (a*((8*ta
n(c/2 + (d*x)/2)*(72*a^8 - 72*a^7*b + 4*b^8 - 72*a^3*b^5 + 69*a^4*b^4 + 144*a^5*b^3 - 144*a^6*b^2))/(a*b^8 + b
^9 - a^2*b^7 - a^3*b^6) + (a*((8*(8*a*b^13 + 4*b^14 - 22*a^2*b^12 - 14*a^3*b^11 + 30*a^4*b^10 + 6*a^5*b^9 - 12
*a^6*b^8))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) - (a*tan(c/2 + (d*x)/2)*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 +
16*a^4*b^10 + 8*a^5*b^9 - 8*a^6*b^8)*24i)/(b^4*(a*b^8 + b^9 - a^2*b^7 - a^3*b^6)))*3i)/b^4)*3i)/b^4 + (a*((8*
tan(c/2 + (d*x)/2)*(72*a^8 - 72*a^7*b + 4*b^8 - 72*a^3*b^5 + 69*a^4*b^4 + 144*a^5*b^3 - 144*a^6*b^2))/(a*b^8 +
b^9 - a^2*b^7 - a^3*b^6) - (a*((8*(8*a*b^13 + 4*b^14 - 22*a^2*b^12 - 14*a^3*b^11 + 30*a^4*b^10 + 6*a^5*b^9 -
12*a^6*b^8))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) + (a*tan(c/2 + (d*x)/2)*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11
+ 16*a^4*b^10 + 8*a^5*b^9 - 8*a^6*b^8)*24i)/(b^4*(a*b^8 + b^9 - a^2*b^7 - a^3*b^6)))*3i)/b^4)*3i)/b^4)))/(b^4
*d) + (atan(((((8*tan(c/2 + (d*x)/2)*(72*a^8 - 72*a^7*b + 4*b^8 - 72*a^3*b^5 + 69*a^4*b^4 + 144*a^5*b^3 - 144*
a^6*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) + (((8*(8*a*b^13 + 4*b^14 - 22*a^2*b^12 - 14*a^3*b^11 + 30*a^4*b^1
0 + 6*a^5*b^9 - 12*a^6*b^8))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) - (8*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a - b)^
3)^(1/2)*(3*a^4 + b^4 - (9*a^2*b^2)/2)*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8*a^6*
b^8))/((a*b^8 + b^9 - a^2*b^7 - a^3*b^6)*(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4)))*(-(a + b)^3*(a - b)^3)^(1/
2)*(3*a^4 + b^4 - (9*a^2*b^2)/2))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4))*(-(a + b)^3*(a - b)^3)^(1/2)*(3*a^
4 + b^4 - (9*a^2*b^2)/2)*1i)/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4) + (((8*tan(c/2 + (d*x)/2)*(72*a^8 - 72*a
^7*b + 4*b^8 - 72*a^3*b^5 + 69*a^4*b^4 + 144*a^5*b^3 - 144*a^6*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) - (((8*
(8*a*b^13 + 4*b^14 - 22*a^2*b^12 - 14*a^3*b^11 + 30*a^4*b^10 + 6*a^5*b^9 - 12*a^6*b^8))/(a*b^11 + b^12 - a^2*b
^10 - a^3*b^9) + (8*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a - b)^3)^(1/2)*(3*a^4 + b^4 - (9*a^2*b^2)/2)*(8*a*b^13 -
8*a^2*b^12 - 16*a^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8*a^6*b^8))/((a*b^8 + b^9 - a^2*b^7 - a^3*b^6)*(b^10 - 3*
a^2*b^8 + 3*a^4*b^6 - a^6*b^4)))*(-(a + b)^3*(a - b)^3)^(1/2)*(3*a^4 + b^4 - (9*a^2*b^2)/2))/(b^10 - 3*a^2*b^8
+ 3*a^4*b^6 - a^6*b^4))*(-(a + b)^3*(a - b)^3)^(1/2)*(3*a^4 + b^4 - (9*a^2*b^2)/2)*1i)/(b^10 - 3*a^2*b^8 + 3*
a^4*b^6 - a^6*b^4))/((16*(12*a*b^7 - 54*a^7*b + 108*a^8 - 36*a^2*b^6 - 72*a^3*b^5 + 198*a^4*b^4 + 117*a^5*b^3
- 270*a^6*b^2))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) - (((8*tan(c/2 + (d*x)/2)*(72*a^8 - 72*a^7*b + 4*b^8 - 72
*a^3*b^5 + 69*a^4*b^4 + 144*a^5*b^3 - 144*a^6*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) + (((8*(8*a*b^13 + 4*b^1
4 - 22*a^2*b^12 - 14*a^3*b^11 + 30*a^4*b^10 + 6*a^5*b^9 - 12*a^6*b^8))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) -
(8*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a - b)^3)^(1/2)*(3*a^4 + b^4 - (9*a^2*b^2)/2)*(8*a*b^13 - 8*a^2*b^12 - 16*a
^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8*a^6*b^8))/((a*b^8 + b^9 - a^2*b^7 - a^3*b^6)*(b^10 - 3*a^2*b^8 + 3*a^4*b
^6 - a^6*b^4)))*(-(a + b)^3*(a - b)^3)^(1/2)*(3*a^4 + b^4 - (9*a^2*b^2)/2))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^
6*b^4))*(-(a + b)^3*(a - b)^3)^(1/2)*(3*a^4 + b^4 - (9*a^2*b^2)/2))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4) +
(((8*tan(c/2 + (d*x)/2)*(72*a^8 - 72*a^7*b + 4*b^8 - 72*a^3*b^5 + 69*a^4*b^4 + 144*a^5*b^3 - 144*a^6*b^2))/(a
*b^8 + b^9 - a^2*b^7 - a^3*b^6) - (((8*(8*a*b^13 + 4*b^14 - 22*a^2*b^12 - 14*a^3*b^11 + 30*a^4*b^10 + 6*a^5*b^
9 - 12*a^6*b^8))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) + (8*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a - b)^3)^(1/2)*(3*
a^4 + b^4 - (9*a^2*b^2)/2)*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8*a^6*b^8))/((a*b^
8 + b^9 - a^2*b^7 - a^3*b^6)*(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4)))*(-(a + b)^3*(a - b)^3)^(1/2)*(3*a^4 +
b^4 - (9*a^2*b^2)/2))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4))*(-(a + b)^3*(a - b)^3)^(1/2)*(3*a^4 + b^4 - (9
*a^2*b^2)/2))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4)))*(-(a + b)^3*(a - b)^3)^(1/2)*(3*a^4 + b^4 - (9*a^2*b^
2)/2)*2i)/(d*(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4))

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________