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

Optimal. Leaf size=332 \[ \frac{\left (a^2 b^2 (6 A-7 C)+12 a^4 C-b^4 (3 A+2 C)\right ) \sin (c+d x)}{3 b^4 d \left (a^2-b^2\right )}+\frac{2 a^2 \left (2 a^2 A b^2-5 a^2 b^2 C+4 a^4 C-3 A b^4\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac{\left (4 a^2 C+3 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d \left (a^2-b^2\right )}-\frac{a \left (2 a^2 C+A b^2-b^2 C\right ) \sin (c+d x) \cos (c+d x)}{b^3 d \left (a^2-b^2\right )}-\frac{a x \left (C \left (4 a^2+b^2\right )+2 A b^2\right )}{b^5} \]

[Out]

-((a*(2*A*b^2 + (4*a^2 + b^2)*C)*x)/b^5) + (2*a^2*(2*a^2*A*b^2 - 3*A*b^4 + 4*a^4*C - 5*a^2*b^2*C)*ArcTan[(Sqrt
[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^5*(a + b)^(3/2)*d) + ((a^2*b^2*(6*A - 7*C) + 12*a^4*C
- b^4*(3*A + 2*C))*Sin[c + d*x])/(3*b^4*(a^2 - b^2)*d) - (a*(A*b^2 + 2*a^2*C - b^2*C)*Cos[c + d*x]*Sin[c + d*
x])/(b^3*(a^2 - b^2)*d) + ((3*A*b^2 + 4*a^2*C - b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - ((
A*b^2 + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))

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Rubi [A]  time = 1.11783, antiderivative size = 332, normalized size of antiderivative = 1., 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} \[ \frac{\left (a^2 b^2 (6 A-7 C)+12 a^4 C-b^4 (3 A+2 C)\right ) \sin (c+d x)}{3 b^4 d \left (a^2-b^2\right )}+\frac{2 a^2 \left (2 a^2 A b^2-5 a^2 b^2 C+4 a^4 C-3 A b^4\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac{\left (4 a^2 C+3 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d \left (a^2-b^2\right )}-\frac{a \left (2 a^2 C+A b^2-b^2 C\right ) \sin (c+d x) \cos (c+d x)}{b^3 d \left (a^2-b^2\right )}-\frac{a x \left (C \left (4 a^2+b^2\right )+2 A b^2\right )}{b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*(2*A*b^2 + (4*a^2 + b^2)*C)*x)/b^5) + (2*a^2*(2*a^2*A*b^2 - 3*A*b^4 + 4*a^4*C - 5*a^2*b^2*C)*ArcTan[(Sqrt
[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^5*(a + b)^(3/2)*d) + ((a^2*b^2*(6*A - 7*C) + 12*a^4*C
- b^4*(3*A + 2*C))*Sin[c + d*x])/(3*b^4*(a^2 - b^2)*d) - (a*(A*b^2 + 2*a^2*C - b^2*C)*Cos[c + d*x]*Sin[c + d*
x])/(b^3*(a^2 - b^2)*d) + ((3*A*b^2 + 4*a^2*C - b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - ((
A*b^2 + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))

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 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 ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx &=-\frac{\left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac{\int \frac{\cos ^2(c+d x) \left (3 \left (A b^2+a^2 C\right )-a b (A+C) \cos (c+d x)-\left (3 A b^2+4 a^2 C-b^2 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{b \left (a^2-b^2\right )}\\ &=\frac{\left (3 A b^2+4 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac{\left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac{\int \frac{\cos (c+d x) \left (-2 a \left (3 A b^2+\left (4 a^2-b^2\right ) C\right )+b \left (3 A b^2+\left (a^2+2 b^2\right ) C\right ) \cos (c+d x)+6 a \left (A b^2+2 a^2 C-b^2 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{3 b^2 \left (a^2-b^2\right )}\\ &=-\frac{a \left (A b^2+2 a^2 C-b^2 C\right ) \cos (c+d x) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (3 A b^2+4 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac{\left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac{\int \frac{6 a^2 \left (A b^2+2 a^2 C-b^2 C\right )-2 a b \left (3 A b^2+\left (2 a^2+b^2\right ) C\right ) \cos (c+d x)-2 \left (a^2 b^2 (6 A-7 C)+12 a^4 C-b^4 (3 A+2 C)\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^3 \left (a^2-b^2\right )}\\ &=\frac{\left (a^2 b^2 (6 A-7 C)+12 a^4 C-b^4 (3 A+2 C)\right ) \sin (c+d x)}{3 b^4 \left (a^2-b^2\right ) d}-\frac{a \left (A b^2+2 a^2 C-b^2 C\right ) \cos (c+d x) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (3 A b^2+4 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac{\left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac{\int \frac{6 a^2 b \left (A b^2+2 a^2 C-b^2 C\right )+6 a \left (a^2-b^2\right ) \left (2 A b^2+4 a^2 C+b^2 C\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^4 \left (a^2-b^2\right )}\\ &=-\frac{a \left (2 A b^2+\left (4 a^2+b^2\right ) C\right ) x}{b^5}+\frac{\left (a^2 b^2 (6 A-7 C)+12 a^4 C-b^4 (3 A+2 C)\right ) \sin (c+d x)}{3 b^4 \left (a^2-b^2\right ) d}-\frac{a \left (A b^2+2 a^2 C-b^2 C\right ) \cos (c+d x) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (3 A b^2+4 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac{\left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac{\left (a^2 \left (3 A b^4-a^2 b^2 (2 A-5 C)-4 a^4 C\right )\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{b^5 \left (a^2-b^2\right )}\\ &=-\frac{a \left (2 A b^2+\left (4 a^2+b^2\right ) C\right ) x}{b^5}+\frac{\left (a^2 b^2 (6 A-7 C)+12 a^4 C-b^4 (3 A+2 C)\right ) \sin (c+d x)}{3 b^4 \left (a^2-b^2\right ) d}-\frac{a \left (A b^2+2 a^2 C-b^2 C\right ) \cos (c+d x) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (3 A b^2+4 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac{\left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac{\left (2 a^2 \left (3 A b^4-a^2 b^2 (2 A-5 C)-4 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 )}{b^5 \left (a^2-b^2\right ) d}\\ &=-\frac{a \left (2 A b^2+\left (4 a^2+b^2\right ) C\right ) x}{b^5}+\frac{2 a^2 \left (2 a^2 A b^2-3 A b^4+4 a^4 C-5 a^2 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)^{3/2} b^5 (a+b)^{3/2} d}+\frac{\left (a^2 b^2 (6 A-7 C)+12 a^4 C-b^4 (3 A+2 C)\right ) \sin (c+d x)}{3 b^4 \left (a^2-b^2\right ) d}-\frac{a \left (A b^2+2 a^2 C-b^2 C\right ) \cos (c+d x) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (3 A b^2+4 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac{\left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}\\ \end{align*}

Mathematica [A]  time = 1.11899, size = 215, normalized size = 0.65 \[ \frac{-12 a (c+d x) \left (C \left (4 a^2+b^2\right )+2 A b^2\right )+3 b \left (3 C \left (4 a^2+b^2\right )+4 A b^2\right ) \sin (c+d x)+\frac{12 a^3 b \left (a^2 C+A b^2\right ) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}+\frac{24 a^2 \left (a^2 b^2 (2 A-5 C)+4 a^4 C-3 A b^4\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\left (b^2-a^2\right )^{3/2}}-6 a b^2 C \sin (2 (c+d x))+b^3 C \sin (3 (c+d x))}{12 b^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-12*a*(2*A*b^2 + (4*a^2 + b^2)*C)*(c + d*x) + (24*a^2*(-3*A*b^4 + a^2*b^2*(2*A - 5*C) + 4*a^4*C)*ArcTanh[((a
- b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + 3*b*(4*A*b^2 + 3*(4*a^2 + b^2)*C)*Sin[c + d*x]
+ (12*a^3*b*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])) - 6*a*b^2*C*Sin[2*(c + d*x)]
+ b^3*C*Sin[3*(c + d*x)])/(12*b^5*d)

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Maple [B]  time = 0.043, size = 828, normalized size = 2.5 \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)^3*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^2,x)

[Out]

2/d/b^2/(tan(1/2*d*x+1/2*c)^2+1)^3*A*tan(1/2*d*x+1/2*c)^5+6/d/b^4/(tan(1/2*d*x+1/2*c)^2+1)^3*C*tan(1/2*d*x+1/2
*c)^5*a^2+2/d/b^3/(tan(1/2*d*x+1/2*c)^2+1)^3*C*tan(1/2*d*x+1/2*c)^5*a+2/d/b^2/(tan(1/2*d*x+1/2*c)^2+1)^3*C*tan
(1/2*d*x+1/2*c)^5+4/d/b^2/(tan(1/2*d*x+1/2*c)^2+1)^3*tan(1/2*d*x+1/2*c)^3*A+12/d/b^4/(tan(1/2*d*x+1/2*c)^2+1)^
3*tan(1/2*d*x+1/2*c)^3*a^2*C+4/3/d/b^2/(tan(1/2*d*x+1/2*c)^2+1)^3*tan(1/2*d*x+1/2*c)^3*C+2/d/b^2/(tan(1/2*d*x+
1/2*c)^2+1)^3*A*tan(1/2*d*x+1/2*c)+6/d/b^4/(tan(1/2*d*x+1/2*c)^2+1)^3*C*tan(1/2*d*x+1/2*c)*a^2-2/d/b^3/(tan(1/
2*d*x+1/2*c)^2+1)^3*C*tan(1/2*d*x+1/2*c)*a+2/d/b^2/(tan(1/2*d*x+1/2*c)^2+1)^3*C*tan(1/2*d*x+1/2*c)-4/d/b^3*A*a
rctan(tan(1/2*d*x+1/2*c))*a-8/d/b^5*C*arctan(tan(1/2*d*x+1/2*c))*a^3-2/d/b^3*C*arctan(tan(1/2*d*x+1/2*c))*a+2/
d*a^3/b^2/(a^2-b^2)*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)*A+2/d*a^5/b^4/(a^2-
b^2)*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)*C+4/d*a^4/b^3/(a+b)/(a-b)/((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/d*a^2/b/(a+b)/(a-b)/((a+b)*(a-b))^(1/2)*a
rctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A+8/d*a^6/b^5/(a+b)/(a-b)/((a+b)*(a-b))^(1/2)*arctan((a-b)
*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C-10/d*a^4/b^3/(a+b)/(a-b)/((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(cos(d*x+c)^3*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.08631, size = 2171, normalized size = 6.54 \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)^3*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^2,x, algorithm="fricas")

[Out]

[-1/6*(6*(4*C*a^7*b + (2*A - 7*C)*a^5*b^3 - 2*(2*A - C)*a^3*b^5 + (2*A + C)*a*b^7)*d*x*cos(d*x + c) + 6*(4*C*a
^8 + (2*A - 7*C)*a^6*b^2 - 2*(2*A - C)*a^4*b^4 + (2*A + C)*a^2*b^6)*d*x + 3*(4*C*a^7 + (2*A - 5*C)*a^5*b^2 - 3
*A*a^3*b^4 + (4*C*a^6*b + (2*A - 5*C)*a^4*b^3 - 3*A*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*C*a^7*b + (6*A - 19*C)*a^5*b^3 - (9*A - 5*C)*a^3*b^5 + (
3*A + 2*C)*a*b^7 + (C*a^4*b^4 - 2*C*a^2*b^6 + C*b^8)*cos(d*x + c)^3 - 2*(C*a^5*b^3 - 2*C*a^3*b^5 + C*a*b^7)*co
s(d*x + c)^2 + (6*C*a^6*b^2 + (3*A - 10*C)*a^4*b^4 - 2*(3*A - C)*a^2*b^6 + (3*A + 2*C)*b^8)*cos(d*x + c))*sin(
d*x + c))/((a^4*b^6 - 2*a^2*b^8 + b^10)*d*cos(d*x + c) + (a^5*b^5 - 2*a^3*b^7 + a*b^9)*d), -1/3*(3*(4*C*a^7*b
+ (2*A - 7*C)*a^5*b^3 - 2*(2*A - C)*a^3*b^5 + (2*A + C)*a*b^7)*d*x*cos(d*x + c) + 3*(4*C*a^8 + (2*A - 7*C)*a^6
*b^2 - 2*(2*A - C)*a^4*b^4 + (2*A + C)*a^2*b^6)*d*x - 3*(4*C*a^7 + (2*A - 5*C)*a^5*b^2 - 3*A*a^3*b^4 + (4*C*a^
6*b + (2*A - 5*C)*a^4*b^3 - 3*A*a^2*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))) - (12*C*a^7*b + (6*A - 19*C)*a^5*b^3 - (9*A - 5*C)*a^3*b^5 + (3*A + 2*C)*a*b^7 + (C*a^4*
b^4 - 2*C*a^2*b^6 + C*b^8)*cos(d*x + c)^3 - 2*(C*a^5*b^3 - 2*C*a^3*b^5 + C*a*b^7)*cos(d*x + c)^2 + (6*C*a^6*b^
2 + (3*A - 10*C)*a^4*b^4 - 2*(3*A - C)*a^2*b^6 + (3*A + 2*C)*b^8)*cos(d*x + c))*sin(d*x + c))/((a^4*b^6 - 2*a^
2*b^8 + b^10)*d*cos(d*x + c) + (a^5*b^5 - 2*a^3*b^7 + a*b^9)*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)**3*(A+C*cos(d*x+c)**2)/(a+b*cos(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.46857, size = 593, normalized size = 1.79 \begin{align*} -\frac{\frac{6 \,{\left (4 \, C a^{6} + 2 \, A a^{4} b^{2} - 5 \, C a^{4} b^{2} - 3 \, A a^{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^{5} - b^{7}\right )} \sqrt{a^{2} - b^{2}}} - \frac{6 \,{\left (C a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + A a^{3} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (a^{2} b^{4} - b^{6}\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 )}} + \frac{3 \,{\left (4 \, C a^{3} + 2 \, A a b^{2} + C a b^{2}\right )}{\left (d x + c\right )}}{b^{5}} - \frac{2 \,{\left (9 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 18 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, C b^{2} \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 )}^{3} b^{4}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(6*(4*C*a^6 + 2*A*a^4*b^2 - 5*C*a^4*b^2 - 3*A*a^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^5 - b^7)*sqrt(a^2 - b^2)
) - 6*(C*a^5*tan(1/2*d*x + 1/2*c) + A*a^3*b^2*tan(1/2*d*x + 1/2*c))/((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*(4*C*a^3 + 2*A*a*b^2 + C*a*b^2)*(d*x + c)/b^5 - 2*(9*C*a^2*tan(1/2*d
*x + 1/2*c)^5 + 3*C*a*b*tan(1/2*d*x + 1/2*c)^5 + 3*A*b^2*tan(1/2*d*x + 1/2*c)^5 + 3*C*b^2*tan(1/2*d*x + 1/2*c)
^5 + 18*C*a^2*tan(1/2*d*x + 1/2*c)^3 + 6*A*b^2*tan(1/2*d*x + 1/2*c)^3 + 2*C*b^2*tan(1/2*d*x + 1/2*c)^3 + 9*C*a
^2*tan(1/2*d*x + 1/2*c) - 3*C*a*b*tan(1/2*d*x + 1/2*c) + 3*A*b^2*tan(1/2*d*x + 1/2*c) + 3*C*b^2*tan(1/2*d*x +
1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*b^4))/d