3.995 \(\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{(a+b \cos (c+d x))^3} \, dx\)

Optimal. Leaf size=314 \[ \frac{\sin (c+d x) \left (3 a^2 C-a b B+A b^2-2 b^2 C\right )}{2 b^3 d \left (a^2-b^2\right )}+\frac{\left (a^2 b^4 (A+12 C)+5 a^3 b^3 B-15 a^4 b^2 C-2 a^5 b B+6 a^6 C-6 a b^5 B+2 A b^6\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac{a \sin (c+d x) \left (a^2 b^2 (A+6 C)+a^3 b B-3 a^4 C-4 a b^3 B+2 A b^4\right )}{2 b^3 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac{x (b B-3 a C)}{b^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 2.82227, antiderivative size = 314, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146, Rules used = {3047, 3031, 3023, 2735, 2659, 205} \[ \frac{\sin (c+d x) \left (3 a^2 C-a b B+A b^2-2 b^2 C\right )}{2 b^3 d \left (a^2-b^2\right )}+\frac{\left (a^2 b^4 (A+12 C)+5 a^3 b^3 B-15 a^4 b^2 C-2 a^5 b B+6 a^6 C-6 a b^5 B+2 A b^6\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac{a \sin (c+d x) \left (a^2 b^2 (A+6 C)+a^3 b B-3 a^4 C-4 a b^3 B+2 A b^4\right )}{2 b^3 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac{x (b B-3 a C)}{b^4} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]

[Out]

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

Rule 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C - B*c*d + 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*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*
d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(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, B, 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 3031

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((b*c - a*d)*(A*b^2 - a*b*B + 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)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b
^2*d*(m + 1) - a*b*c*(m + 2)) + (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, B, C}, x] && NeQ[b*c - a*d, 0] && Ne
Q[a^2 - b^2, 0] && LtQ[m, -1]

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 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 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 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]

Rubi steps

\begin{align*} \int \frac{\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx &=-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac{\int \frac{\cos (c+d x) \left (2 \left (A b^2-a (b B-a C)\right )+2 b (b B-a (A+C)) \cos (c+d x)-\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac{a \left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}-\frac{\int \frac{-b \left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right )-\left (a^2-b^2\right ) \left (a^2 b B-2 b^3 B-3 a^3 C+a b^2 (A+4 C)\right ) \cos (c+d x)-b \left (a^2-b^2\right ) \left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )^2}\\ &=\frac{\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac{a \left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}-\frac{\int \frac{-b^2 \left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right )-2 b \left (a^2-b^2\right )^2 (b B-3 a C) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{2 b^4 \left (a^2-b^2\right )^2}\\ &=\frac{(b B-3 a C) x}{b^4}+\frac{\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac{a \left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac{\left (2 A b^6-2 a^5 b B+5 a^3 b^3 B-6 a b^5 B+6 a^6 C-15 a^4 b^2 C+a^2 b^4 (A+12 C)\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{2 b^4 \left (a^2-b^2\right )^2}\\ &=\frac{(b B-3 a C) x}{b^4}+\frac{\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac{a \left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac{\left (2 A b^6-2 a^5 b B+5 a^3 b^3 B-6 a b^5 B+6 a^6 C-15 a^4 b^2 C+a^2 b^4 (A+12 C)\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 )^2 d}\\ &=\frac{(b B-3 a C) x}{b^4}+\frac{\left (a^2 A b^4+2 A b^6-2 a^5 b B+5 a^3 b^3 B-6 a b^5 B+6 a^6 C-15 a^4 b^2 C+12 a^2 b^4 C\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{(a-b)^{5/2} b^4 (a+b)^{5/2} d}+\frac{\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac{a \left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}\\ \end{align*}

Mathematica [A]  time = 2.79907, size = 573, normalized size = 1.82 \[ \frac{\frac{a^3 A b^4 \sin (2 (c+d x))-6 a^2 A b^5 \sin (c+d x)-8 a b \left (a^2-b^2\right )^2 (c+d x) (3 a C-b B) \cos (c+d x)+2 \left (b^3-a^2 b\right )^2 (c+d x) (b B-3 a C) \cos (2 (c+d x))-4 a^5 b^2 B \sin (c+d x)-3 a^4 b^3 B \sin (2 (c+d x))+10 a^3 b^4 B \sin (c+d x)+6 a^2 b^5 B \sin (2 (c+d x))-6 a^4 b^3 B c-6 a^4 b^3 B d x+9 a^5 b^2 C \sin (2 (c+d x))-21 a^4 b^3 C \sin (c+d x)+a^4 b^3 C \sin (3 (c+d x))-16 a^3 b^4 C \sin (2 (c+d x))+2 a^2 b^5 C \sin (c+d x)-2 a^2 b^5 C \sin (3 (c+d x))+18 a^5 b^2 c C+18 a^5 b^2 C d x+4 a^6 b B c+4 a^6 b B d x+12 a^6 b C \sin (c+d x)-12 a^7 c C-12 a^7 C d x-4 a A b^6 \sin (2 (c+d x))+4 a b^6 C \sin (2 (c+d x))-6 a b^6 c C-6 a b^6 C d x+2 b^7 B c+2 b^7 B d x+b^7 C \sin (c+d x)+b^7 C \sin (3 (c+d x))}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac{4 \left (a^2 b^4 (A+12 C)+5 a^3 b^3 B-15 a^4 b^2 C-2 a^5 b B+6 a^6 C-6 a b^5 B+2 A b^6\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\left (b^2-a^2\right )^{5/2}}}{4 b^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]

[Out]

((-4*(2*A*b^6 - 2*a^5*b*B + 5*a^3*b^3*B - 6*a*b^5*B + 6*a^6*C - 15*a^4*b^2*C + a^2*b^4*(A + 12*C))*ArcTanh[((a
- b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (4*a^6*b*B*c - 6*a^4*b^3*B*c + 2*b^7*B*c - 12*
a^7*c*C + 18*a^5*b^2*c*C - 6*a*b^6*c*C + 4*a^6*b*B*d*x - 6*a^4*b^3*B*d*x + 2*b^7*B*d*x - 12*a^7*C*d*x + 18*a^5
*b^2*C*d*x - 6*a*b^6*C*d*x - 8*a*b*(a^2 - b^2)^2*(-(b*B) + 3*a*C)*(c + d*x)*Cos[c + d*x] + 2*(-(a^2*b) + b^3)^
2*(b*B - 3*a*C)*(c + d*x)*Cos[2*(c + d*x)] - 6*a^2*A*b^5*Sin[c + d*x] - 4*a^5*b^2*B*Sin[c + d*x] + 10*a^3*b^4*
B*Sin[c + d*x] + 12*a^6*b*C*Sin[c + d*x] - 21*a^4*b^3*C*Sin[c + d*x] + 2*a^2*b^5*C*Sin[c + d*x] + b^7*C*Sin[c
+ d*x] + a^3*A*b^4*Sin[2*(c + d*x)] - 4*a*A*b^6*Sin[2*(c + d*x)] - 3*a^4*b^3*B*Sin[2*(c + d*x)] + 6*a^2*b^5*B*
Sin[2*(c + d*x)] + 9*a^5*b^2*C*Sin[2*(c + d*x)] - 16*a^3*b^4*C*Sin[2*(c + d*x)] + 4*a*b^6*C*Sin[2*(c + d*x)] +
a^4*b^3*C*Sin[3*(c + d*x)] - 2*a^2*b^5*C*Sin[3*(c + d*x)] + b^7*C*Sin[3*(c + d*x)])/((a^2 - b^2)^2*(a + b*Cos
[c + d*x])^2))/(4*b^4*d)

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Maple [B]  time = 0.049, size = 1693, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x)

[Out]

-2/d*a^4/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)/(a-b)^2*tan(1/2*d*x+1/2*c)*B-1/d*a^3/
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+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B+4/d*a^5/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)/(a-b)^2*tan(1/2*d*x+1/2*c)*C-2/d*a^4/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)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+1/d*a^3/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-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+1/d/(a^4-2*a^2*b^2+b^4)/((a+
b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*a^2*A+2/d*b^2/(a^4-2*a^2*b^2+b^4)/((a+b)*
(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A+12/d/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/
2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C*a^2+2/d/b^3*B*arctan(tan(1/2*d*x+1/2*c))+1/d*a^4/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)/(a-b)^2*tan(1/2*d*x+1/2*c)*C-1/d*a^4/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)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C+2/d/b^3*C*tan(1/2*d
*x+1/2*c)/(tan(1/2*d*x+1/2*c)^2+1)-6/d/b^4*C*arctan(tan(1/2*d*x+1/2*c))*a-8/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^3/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C+6/d*a^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)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+6/d*a^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)/(a-b)^2*tan(1/2*d*x+1/2*c)*B-6/d*a*b/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arcta
n((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*B-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)/(a-b)^2*tan(1/2*d*x+1/2*c)*A+4/d*a^5/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)/(
a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*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)/(a^2
+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-8/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+b)/(a-b)^2*t
an(1/2*d*x+1/2*c)*a^3*C-1/d*a^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)/(a^2+2*a*b+b^2)*ta
n(1/2*d*x+1/2*c)^3*A-15/d/b^2/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(
a-b))^(1/2))*a^4*C+6/d/b^4/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b
))^(1/2))*a^6*C+5/d/b/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1
/2))*a^3*B-2/d/b^3/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2)
)*a^5*B+1/d*a^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)/(a-b)^2*tan(1/2*d*x+1/2*c)*A

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.1971, size = 3621, normalized size = 11.53 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(4*(3*C*a^7*b^2 - B*a^6*b^3 - 9*C*a^5*b^4 + 3*B*a^4*b^5 + 9*C*a^3*b^6 - 3*B*a^2*b^7 - 3*C*a*b^8 + B*b^9)
*d*x*cos(d*x + c)^2 + 8*(3*C*a^8*b - B*a^7*b^2 - 9*C*a^6*b^3 + 3*B*a^5*b^4 + 9*C*a^4*b^5 - 3*B*a^3*b^6 - 3*C*a
^2*b^7 + B*a*b^8)*d*x*cos(d*x + c) + 4*(3*C*a^9 - B*a^8*b - 9*C*a^7*b^2 + 3*B*a^6*b^3 + 9*C*a^5*b^4 - 3*B*a^4*
b^5 - 3*C*a^3*b^6 + B*a^2*b^7)*d*x + (6*C*a^8 - 2*B*a^7*b - 15*C*a^6*b^2 + 5*B*a^5*b^3 + (A + 12*C)*a^4*b^4 -
6*B*a^3*b^5 + 2*A*a^2*b^6 + (6*C*a^6*b^2 - 2*B*a^5*b^3 - 15*C*a^4*b^4 + 5*B*a^3*b^5 + (A + 12*C)*a^2*b^6 - 6*B
*a*b^7 + 2*A*b^8)*cos(d*x + c)^2 + 2*(6*C*a^7*b - 2*B*a^6*b^2 - 15*C*a^5*b^3 + 5*B*a^4*b^4 + (A + 12*C)*a^3*b^
5 - 6*B*a^2*b^6 + 2*A*a*b^7)*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*C*a^8*b - 2*B*a^7*b^2 - 17*C*a^6*b^3 + 7*B*a^5*b^4 - (3*A - 13*C)*a^4*b^5 - 5*B*a^3*b^6 +
(3*A - 2*C)*a^2*b^7 + 2*(C*a^6*b^3 - 3*C*a^4*b^5 + 3*C*a^2*b^7 - C*b^9)*cos(d*x + c)^2 + (9*C*a^7*b^2 - 3*B*a
^6*b^3 + (A - 25*C)*a^5*b^4 + 9*B*a^4*b^5 - 5*(A - 4*C)*a^3*b^6 - 6*B*a^2*b^7 + 4*(A - C)*a*b^8)*cos(d*x + c))
*sin(d*x + c))/((a^6*b^6 - 3*a^4*b^8 + 3*a^2*b^10 - b^12)*d*cos(d*x + c)^2 + 2*(a^7*b^5 - 3*a^5*b^7 + 3*a^3*b^
9 - a*b^11)*d*cos(d*x + c) + (a^8*b^4 - 3*a^6*b^6 + 3*a^4*b^8 - a^2*b^10)*d), -1/2*(2*(3*C*a^7*b^2 - B*a^6*b^3
- 9*C*a^5*b^4 + 3*B*a^4*b^5 + 9*C*a^3*b^6 - 3*B*a^2*b^7 - 3*C*a*b^8 + B*b^9)*d*x*cos(d*x + c)^2 + 4*(3*C*a^8*
b - B*a^7*b^2 - 9*C*a^6*b^3 + 3*B*a^5*b^4 + 9*C*a^4*b^5 - 3*B*a^3*b^6 - 3*C*a^2*b^7 + B*a*b^8)*d*x*cos(d*x + c
) + 2*(3*C*a^9 - B*a^8*b - 9*C*a^7*b^2 + 3*B*a^6*b^3 + 9*C*a^5*b^4 - 3*B*a^4*b^5 - 3*C*a^3*b^6 + B*a^2*b^7)*d*
x - (6*C*a^8 - 2*B*a^7*b - 15*C*a^6*b^2 + 5*B*a^5*b^3 + (A + 12*C)*a^4*b^4 - 6*B*a^3*b^5 + 2*A*a^2*b^6 + (6*C*
a^6*b^2 - 2*B*a^5*b^3 - 15*C*a^4*b^4 + 5*B*a^3*b^5 + (A + 12*C)*a^2*b^6 - 6*B*a*b^7 + 2*A*b^8)*cos(d*x + c)^2
+ 2*(6*C*a^7*b - 2*B*a^6*b^2 - 15*C*a^5*b^3 + 5*B*a^4*b^4 + (A + 12*C)*a^3*b^5 - 6*B*a^2*b^6 + 2*A*a*b^7)*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*C*a^8*b - 2*B*a^7*
b^2 - 17*C*a^6*b^3 + 7*B*a^5*b^4 - (3*A - 13*C)*a^4*b^5 - 5*B*a^3*b^6 + (3*A - 2*C)*a^2*b^7 + 2*(C*a^6*b^3 - 3
*C*a^4*b^5 + 3*C*a^2*b^7 - C*b^9)*cos(d*x + c)^2 + (9*C*a^7*b^2 - 3*B*a^6*b^3 + (A - 25*C)*a^5*b^4 + 9*B*a^4*b
^5 - 5*(A - 4*C)*a^3*b^6 - 6*B*a^2*b^7 + 4*(A - C)*a*b^8)*cos(d*x + c))*sin(d*x + c))/((a^6*b^6 - 3*a^4*b^8 +
3*a^2*b^10 - b^12)*d*cos(d*x + c)^2 + 2*(a^7*b^5 - 3*a^5*b^7 + 3*a^3*b^9 - a*b^11)*d*cos(d*x + c) + (a^8*b^4 -
3*a^6*b^6 + 3*a^4*b^8 - a^2*b^10)*d)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**2*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+b*cos(d*x+c))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.37204, size = 899, normalized size = 2.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((6*C*a^6 - 2*B*a^5*b - 15*C*a^4*b^2 + 5*B*a^3*b^3 + A*a^2*b^4 + 12*C*a^2*b^4 - 6*B*a*b^5 + 2*A*b^6)*(pi*floo
r(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^4*b^4 - 2*a^2*b^6 + b^8)*sqrt(a^2 - b^2)) - (4*C*a^6*tan(1/2*d*x + 1/2*c)^3 - 2*B*a^5*b*tan(1/2*
d*x + 1/2*c)^3 - 5*C*a^5*b*tan(1/2*d*x + 1/2*c)^3 + 3*B*a^4*b^2*tan(1/2*d*x + 1/2*c)^3 - 7*C*a^4*b^2*tan(1/2*d
*x + 1/2*c)^3 - A*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 + 5*B*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 + 8*C*a^3*b^3*tan(1/2*d*
x + 1/2*c)^3 - 3*A*a^2*b^4*tan(1/2*d*x + 1/2*c)^3 - 6*B*a^2*b^4*tan(1/2*d*x + 1/2*c)^3 + 4*A*a*b^5*tan(1/2*d*x
+ 1/2*c)^3 + 4*C*a^6*tan(1/2*d*x + 1/2*c) - 2*B*a^5*b*tan(1/2*d*x + 1/2*c) + 5*C*a^5*b*tan(1/2*d*x + 1/2*c) -
3*B*a^4*b^2*tan(1/2*d*x + 1/2*c) - 7*C*a^4*b^2*tan(1/2*d*x + 1/2*c) + A*a^3*b^3*tan(1/2*d*x + 1/2*c) + 5*B*a^
3*b^3*tan(1/2*d*x + 1/2*c) - 8*C*a^3*b^3*tan(1/2*d*x + 1/2*c) - 3*A*a^2*b^4*tan(1/2*d*x + 1/2*c) + 6*B*a^2*b^4
*tan(1/2*d*x + 1/2*c) - 4*A*a*b^5*tan(1/2*d*x + 1/2*c))/((a^4*b^3 - 2*a^2*b^5 + b^7)*(a*tan(1/2*d*x + 1/2*c)^2
- b*tan(1/2*d*x + 1/2*c)^2 + a + b)^2) + (3*C*a - B*b)*(d*x + c)/b^4 - 2*C*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x
+ 1/2*c)^2 + 1)*b^3))/d