3.589 \(\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx\)

Optimal. Leaf size=252 \[ \frac{a \left (a^2 (2 A+C)+b^2 (3 A+4 C)\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\left (-a^2 b^2 (11 A+10 C)+a^4 C-2 b^4 (2 A+3 C)\right ) \sin (c+d x)}{6 b d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac{a \left (a^2 (-C)+5 A b^2+6 b^2 C\right ) \sin (c+d x)}{6 b d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac{\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3} \]

[Out]

(a*(a^2*(2*A + C) + b^2*(3*A + 4*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b
)^(7/2)*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a*(5*A*b^2 - a^2*C +
6*b^2*C)*Sin[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((a^4*C - 2*b^4*(2*A + 3*C) - a^2*b^2*(
11*A + 10*C))*Sin[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))

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Rubi [A]  time = 0.46314, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3022, 2754, 12, 2659, 205} \[ \frac{a \left (a^2 (2 A+C)+b^2 (3 A+4 C)\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\left (-a^2 b^2 (11 A+10 C)+a^4 C-2 b^4 (2 A+3 C)\right ) \sin (c+d x)}{6 b d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac{a \left (a^2 (-C)+5 A b^2+6 b^2 C\right ) \sin (c+d x)}{6 b d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac{\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(a^2*(2*A + C) + b^2*(3*A + 4*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b
)^(7/2)*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a*(5*A*b^2 - a^2*C +
6*b^2*C)*Sin[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((a^4*C - 2*b^4*(2*A + 3*C) - a^2*b^2*(
11*A + 10*C))*Sin[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))

Rule 3022

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

Rule 2754

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

Rule 12

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

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{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx &=-\frac{\left (A b^2+a^2 C\right ) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\int \frac{-3 a b (A+C)+\left (2 A b^2-a^2 C+3 b^2 C\right ) \cos (c+d x)}{(a+b \cos (c+d x))^3} \, dx}{3 b \left (a^2-b^2\right )}\\ &=-\frac{\left (A b^2+a^2 C\right ) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{a \left (5 A b^2-a^2 C+6 b^2 C\right ) \sin (c+d x)}{6 b \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{\int \frac{2 \left (b^3 (2 A+3 C)+\frac{1}{2} a^2 (6 A b+4 b C)\right )-a \left (5 A b^2-\left (a^2-6 b^2\right ) C\right ) \cos (c+d x)}{(a+b \cos (c+d x))^2} \, dx}{6 b \left (a^2-b^2\right )^2}\\ &=-\frac{\left (A b^2+a^2 C\right ) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{a \left (5 A b^2-a^2 C+6 b^2 C\right ) \sin (c+d x)}{6 b \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{\left (a^4 C-2 b^4 (2 A+3 C)-a^2 b^2 (11 A+10 C)\right ) \sin (c+d x)}{6 b \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac{\int -\frac{3 a b \left (a^2 (2 A+C)+b^2 (3 A+4 C)\right )}{a+b \cos (c+d x)} \, dx}{6 b \left (a^2-b^2\right )^3}\\ &=-\frac{\left (A b^2+a^2 C\right ) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{a \left (5 A b^2-a^2 C+6 b^2 C\right ) \sin (c+d x)}{6 b \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{\left (a^4 C-2 b^4 (2 A+3 C)-a^2 b^2 (11 A+10 C)\right ) \sin (c+d x)}{6 b \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\left (a \left (a^2 (2 A+C)+b^2 (3 A+4 C)\right )\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{2 \left (a^2-b^2\right )^3}\\ &=-\frac{\left (A b^2+a^2 C\right ) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{a \left (5 A b^2-a^2 C+6 b^2 C\right ) \sin (c+d x)}{6 b \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{\left (a^4 C-2 b^4 (2 A+3 C)-a^2 b^2 (11 A+10 C)\right ) \sin (c+d x)}{6 b \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\left (a \left (a^2 (2 A+C)+b^2 (3 A+4 C)\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 )}{\left (a^2-b^2\right )^3 d}\\ &=\frac{a \left (2 a^2 A+3 A b^2+a^2 C+4 b^2 C\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} d}-\frac{\left (A b^2+a^2 C\right ) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{a \left (5 A b^2-a^2 C+6 b^2 C\right ) \sin (c+d x)}{6 b \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{\left (a^4 C-2 b^4 (2 A+3 C)-a^2 b^2 (11 A+10 C)\right ) \sin (c+d x)}{6 b \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}\\ \end{align*}

Mathematica [A]  time = 1.08064, size = 224, normalized size = 0.89 \[ \frac{\frac{2 \sin (c+d x) \left (6 a \left (-9 a^2 b^2 (A+C)+a^4 C-b^4 (A+2 C)\right ) \cos (c+d x)-b \left (\left (a^2 b^2 (11 A+10 C)+a^4 (-C)+2 b^4 (2 A+3 C)\right ) \cos (2 (c+d x))+a^2 b^2 (A+14 C)+a^4 (36 A+25 C)+2 b^4 (4 A+3 C)\right )\right )}{(a+b \cos (c+d x))^3}-\frac{24 a \left (a^2 (2 A+C)+b^2 (3 A+4 C)\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}}}{24 d \left (a^2-b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-24*a*(a^2*(2*A + C) + b^2*(3*A + 4*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^
2] + (2*(6*a*(a^4*C - 9*a^2*b^2*(A + C) - b^4*(A + 2*C))*Cos[c + d*x] - b*(2*b^4*(4*A + 3*C) + a^2*b^2*(A + 14
*C) + a^4*(36*A + 25*C) + (-(a^4*C) + 2*b^4*(2*A + 3*C) + a^2*b^2*(11*A + 10*C))*Cos[2*(c + d*x)]))*Sin[c + d*
x])/(a + b*Cos[c + d*x])^3)/(24*(a^2 - b^2)^3*d)

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Maple [B]  time = 0.036, size = 1726, normalized size = 6.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/d*a^2*b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1
/2*c)^5*A-3/d*b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*a/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(
1/2*d*x+1/2*c)^5*A-2/d*b^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^
3)*tan(1/2*d*x+1/2*c)^5*A-1/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*a^3/(a-b)/(a^3+3*a^2*b+3*a
*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C-6/d*b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*
b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C*a^2-2/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3
+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C*a*b^2-2/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(
a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C*b^3-12/d*a^2*b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c
)^2*b+a+b)^3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-4/3/d*b^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*
d*x+1/2*c)^2*b+a+b)^3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-28/3/d*b/(a*tan(1/2*d*x+1/2*c)^2-
tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C*a^2-4/d/(a*tan(1/2*d*x+1/
2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C-6/d*a^2*b/(a*t
an(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A+3/d*b^2
/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*a/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A-
2/d*b^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*
c)*A+1/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*a^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x
+1/2*c)*C-6/d*b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*
d*x+1/2*c)*C*a^2+2/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan
(1/2*d*x+1/2*c)*C*a*b^2-2/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b
^3)*tan(1/2*d*x+1/2*c)*C*b^3+2/d*a^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((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/d*a*b^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1
/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A+1/d*a^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-b)*ta
n(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C+4/d*b^2*a/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-
b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C

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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((A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.39121, size = 2456, normalized size = 9.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(3*((2*A + C)*a^6 + (3*A + 4*C)*a^4*b^2 + ((2*A + C)*a^3*b^3 + (3*A + 4*C)*a*b^5)*cos(d*x + c)^3 + 3*((2
*A + C)*a^4*b^2 + (3*A + 4*C)*a^2*b^4)*cos(d*x + c)^2 + 3*((2*A + C)*a^5*b + (3*A + 4*C)*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)) - 2*((18*A + 13*C)*a^6*b -
(23*A + 11*C)*a^4*b^3 + (7*A - 2*C)*a^2*b^5 - 2*A*b^7 - (C*a^6*b - 11*(A + C)*a^4*b^3 + (7*A + 4*C)*a^2*b^5 +
2*(2*A + 3*C)*b^7)*cos(d*x + c)^2 - 3*(C*a^7 - (9*A + 10*C)*a^5*b^2 + (8*A + 7*C)*a^3*b^4 + (A + 2*C)*a*b^6)*c
os(d*x + c))*sin(d*x + c))/((a^8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11)*d*cos(d*x + c)^3 + 3*(a^9*b^2
- 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10)*d*cos(d*x + c)^2 + 3*(a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^
7 + a^2*b^9)*d*cos(d*x + c) + (a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*d), 1/6*(3*((2*A + C)*a^6 +
(3*A + 4*C)*a^4*b^2 + ((2*A + C)*a^3*b^3 + (3*A + 4*C)*a*b^5)*cos(d*x + c)^3 + 3*((2*A + C)*a^4*b^2 + (3*A +
4*C)*a^2*b^4)*cos(d*x + c)^2 + 3*((2*A + C)*a^5*b + (3*A + 4*C)*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))) - ((18*A + 13*C)*a^6*b - (23*A + 11*C)*a^4*b^3 + (7*A -
2*C)*a^2*b^5 - 2*A*b^7 - (C*a^6*b - 11*(A + C)*a^4*b^3 + (7*A + 4*C)*a^2*b^5 + 2*(2*A + 3*C)*b^7)*cos(d*x + c)
^2 - 3*(C*a^7 - (9*A + 10*C)*a^5*b^2 + (8*A + 7*C)*a^3*b^4 + (A + 2*C)*a*b^6)*cos(d*x + c))*sin(d*x + c))/((a^
8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11)*d*cos(d*x + c)^3 + 3*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^
3*b^8 + a*b^10)*d*cos(d*x + c)^2 + 3*(a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d*cos(d*x + c) + (
a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*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((A+C*cos(d*x+c)**2)/(a+b*cos(d*x+c))**4,x)

[Out]

Timed out

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Giac [B]  time = 1.39153, size = 930, normalized size = 3.69 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*(2*A*a^3 + C*a^3 + 3*A*a*b^2 + 4*C*a*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)))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sqrt(
a^2 - b^2)) + (3*C*a^5*tan(1/2*d*x + 1/2*c)^5 + 18*A*a^4*b*tan(1/2*d*x + 1/2*c)^5 + 12*C*a^4*b*tan(1/2*d*x + 1
/2*c)^5 - 27*A*a^3*b^2*tan(1/2*d*x + 1/2*c)^5 - 27*C*a^3*b^2*tan(1/2*d*x + 1/2*c)^5 + 6*A*a^2*b^3*tan(1/2*d*x
+ 1/2*c)^5 + 12*C*a^2*b^3*tan(1/2*d*x + 1/2*c)^5 - 3*A*a*b^4*tan(1/2*d*x + 1/2*c)^5 - 6*C*a*b^4*tan(1/2*d*x +
1/2*c)^5 + 6*A*b^5*tan(1/2*d*x + 1/2*c)^5 + 6*C*b^5*tan(1/2*d*x + 1/2*c)^5 + 36*A*a^4*b*tan(1/2*d*x + 1/2*c)^3
+ 28*C*a^4*b*tan(1/2*d*x + 1/2*c)^3 - 32*A*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 - 16*C*a^2*b^3*tan(1/2*d*x + 1/2*c)
^3 - 4*A*b^5*tan(1/2*d*x + 1/2*c)^3 - 12*C*b^5*tan(1/2*d*x + 1/2*c)^3 - 3*C*a^5*tan(1/2*d*x + 1/2*c) + 18*A*a^
4*b*tan(1/2*d*x + 1/2*c) + 12*C*a^4*b*tan(1/2*d*x + 1/2*c) + 27*A*a^3*b^2*tan(1/2*d*x + 1/2*c) + 27*C*a^3*b^2*
tan(1/2*d*x + 1/2*c) + 6*A*a^2*b^3*tan(1/2*d*x + 1/2*c) + 12*C*a^2*b^3*tan(1/2*d*x + 1/2*c) + 3*A*a*b^4*tan(1/
2*d*x + 1/2*c) + 6*C*a*b^4*tan(1/2*d*x + 1/2*c) + 6*A*b^5*tan(1/2*d*x + 1/2*c) + 6*C*b^5*tan(1/2*d*x + 1/2*c))
/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 + a + b)^3))/d