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

Optimal. Leaf size=398 \[ \frac{\left (-11 a^2 b^3 B+21 a^3 b^2 C+6 a^4 b B-12 a^5 C-6 a b^4 C+2 b^5 B\right ) \sin (c+d x)}{2 b^4 d \left (a^2-b^2\right )^2}+\frac{a^2 \left (-15 a^2 b^3 B+29 a^3 b^2 C+6 a^4 b B-12 a^5 C-20 a b^4 C+12 b^5 B\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{a (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac{a \left (2 a^2 b B-4 a^3 C+7 a b^2 C-5 b^3 B\right ) \sin (c+d x) \cos ^2(c+d x)}{2 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}-\frac{\left (10 a^2 b^2 C+3 a^3 b B-6 a^4 C-6 a b^3 B-b^4 C\right ) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left (a^2-b^2\right )^2}-\frac{x \left (-12 a^2 C+6 a b B-b^2 C\right )}{2 b^5} \]

[Out]

-((6*a*b*B - 12*a^2*C - b^2*C)*x)/(2*b^5) + (a^2*(6*a^4*b*B - 15*a^2*b^3*B + 12*b^5*B - 12*a^5*C + 29*a^3*b^2*
C - 20*a*b^4*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) + ((6*
a^4*b*B - 11*a^2*b^3*B + 2*b^5*B - 12*a^5*C + 21*a^3*b^2*C - 6*a*b^4*C)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^2*d)
- ((3*a^3*b*B - 6*a*b^3*B - 6*a^4*C + 10*a^2*b^2*C - b^4*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d)
+ (a*(b*B - a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(2*a^2*b*B - 5*
b^3*B - 4*a^3*C + 7*a*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.7654, antiderivative size = 398, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3029, 2989, 3047, 3049, 3023, 2735, 2659, 205} \[ \frac{\left (-11 a^2 b^3 B+21 a^3 b^2 C+6 a^4 b B-12 a^5 C-6 a b^4 C+2 b^5 B\right ) \sin (c+d x)}{2 b^4 d \left (a^2-b^2\right )^2}+\frac{a^2 \left (-15 a^2 b^3 B+29 a^3 b^2 C+6 a^4 b B-12 a^5 C-20 a b^4 C+12 b^5 B\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{a (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac{a \left (2 a^2 b B-4 a^3 C+7 a b^2 C-5 b^3 B\right ) \sin (c+d x) \cos ^2(c+d x)}{2 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}-\frac{\left (10 a^2 b^2 C+3 a^3 b B-6 a^4 C-6 a b^3 B-b^4 C\right ) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left (a^2-b^2\right )^2}-\frac{x \left (-12 a^2 C+6 a b B-b^2 C\right )}{2 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((6*a*b*B - 12*a^2*C - b^2*C)*x)/(2*b^5) + (a^2*(6*a^4*b*B - 15*a^2*b^3*B + 12*b^5*B - 12*a^5*C + 29*a^3*b^2*
C - 20*a*b^4*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) + ((6*
a^4*b*B - 11*a^2*b^3*B + 2*b^5*B - 12*a^5*C + 21*a^3*b^2*C - 6*a*b^4*C)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^2*d)
- ((3*a^3*b*B - 6*a*b^3*B - 6*a^4*C + 10*a^2*b^2*C - b^4*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d)
+ (a*(b*B - a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(2*a^2*b*B - 5*
b^3*B - 4*a^3*C + 7*a*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))

Rule 3029

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] :> Dist[1/b^2, Int[(a + b*Sin[e + f*x])
^(m + 1)*(c + d*Sin[e + f*x])^n*(b*B - a*C + b*C*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m,
n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]

Rule 2989

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

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 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 (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx &=\int \frac{\cos ^4(c+d x) (B+C \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx\\ &=\frac{a (b B-a C) \cos ^3(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 ^2(c+d x) \left (-3 a (b B-a C)+2 b (b B-a C) \cos (c+d x)+2 \left (a b B-2 a^2 C+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{a (b B-a C) \cos ^3(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^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac{\int \frac{\cos (c+d x) \left (2 a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )+b \left (a^2 b B+2 b^3 B+a^3 C-4 a b^2 C\right ) \cos (c+d x)-2 \left (3 a^3 b B-6 a b^3 B-6 a^4 C+10 a^2 b^2 C-b^4 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (3 a^3 b B-6 a b^3 B-6 a^4 C+10 a^2 b^2 C-b^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac{a (b B-a C) \cos ^3(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^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac{\int \frac{-2 a \left (3 a^3 b B-6 a b^3 B-6 a^4 C+10 a^2 b^2 C-b^4 C\right )+2 b \left (a^3 b B-4 a b^3 B-2 a^4 C+4 a^2 b^2 C+b^4 C\right ) \cos (c+d x)+2 \left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-12 a^5 C+21 a^3 b^2 C-6 a b^4 C\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{4 b^3 \left (a^2-b^2\right )^2}\\ &=\frac{\left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-12 a^5 C+21 a^3 b^2 C-6 a b^4 C\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (3 a^3 b B-6 a b^3 B-6 a^4 C+10 a^2 b^2 C-b^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac{a (b B-a C) \cos ^3(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^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac{\int \frac{-2 a b \left (3 a^3 b B-6 a b^3 B-6 a^4 C+10 a^2 b^2 C-b^4 C\right )-2 \left (a^2-b^2\right )^2 \left (6 a b B-12 a^2 C-b^2 C\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{4 b^4 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (6 a b B-12 a^2 C-b^2 C\right ) x}{2 b^5}+\frac{\left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-12 a^5 C+21 a^3 b^2 C-6 a b^4 C\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (3 a^3 b B-6 a b^3 B-6 a^4 C+10 a^2 b^2 C-b^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac{a (b B-a C) \cos ^3(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^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac{\left (a^2 \left (6 a^4 b B-15 a^2 b^3 B+12 b^5 B-12 a^5 C+29 a^3 b^2 C-20 a b^4 C\right )\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{2 b^5 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (6 a b B-12 a^2 C-b^2 C\right ) x}{2 b^5}+\frac{\left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-12 a^5 C+21 a^3 b^2 C-6 a b^4 C\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (3 a^3 b B-6 a b^3 B-6 a^4 C+10 a^2 b^2 C-b^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac{a (b B-a C) \cos ^3(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^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac{\left (a^2 \left (6 a^4 b B-15 a^2 b^3 B+12 b^5 B-12 a^5 C+29 a^3 b^2 C-20 a b^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 )^2 d}\\ &=-\frac{\left (6 a b B-12 a^2 C-b^2 C\right ) x}{2 b^5}+\frac{a^2 \left (6 a^4 b B-15 a^2 b^3 B+12 b^5 B-12 a^5 C+29 a^3 b^2 C-20 a 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^5 (a+b)^{5/2} d}+\frac{\left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-12 a^5 C+21 a^3 b^2 C-6 a b^4 C\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (3 a^3 b B-6 a b^3 B-6 a^4 C+10 a^2 b^2 C-b^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac{a (b B-a C) \cos ^3(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^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}\\ \end{align*}

Mathematica [A]  time = 3.48729, size = 734, normalized size = 1.84 \[ \frac{\frac{16 a b \left (a^2-b^2\right )^2 (c+d x) \left (12 a^2 C-6 a b B+b^2 C\right ) \cos (c+d x)+4 \left (b^3-a^2 b\right )^2 (c+d x) \left (12 a^2 C-6 a b B+b^2 C\right ) \cos (2 (c+d x))+48 a^6 b^2 B \sin (c+d x)+36 a^5 b^3 B \sin (2 (c+d x))-84 a^4 b^4 B \sin (c+d x)+4 a^4 b^4 B \sin (3 (c+d x))-64 a^3 b^5 B \sin (2 (c+d x))+8 a^2 b^6 B \sin (c+d x)-8 a^2 b^6 B \sin (3 (c+d x))+72 a^5 b^3 B c+72 a^5 b^3 B d x-72 a^6 b^2 C \sin (2 (c+d x))+160 a^5 b^3 C \sin (c+d x)-8 a^5 b^3 C \sin (3 (c+d x))+130 a^4 b^4 C \sin (2 (c+d x))+a^4 b^4 C \sin (4 (c+d x))-32 a^3 b^5 C \sin (c+d x)+16 a^3 b^5 C \sin (3 (c+d x))-48 a^2 b^6 C \sin (2 (c+d x))-2 a^2 b^6 C \sin (4 (c+d x))-136 a^6 b^2 c C-12 a^4 b^4 c C+48 a^2 b^6 c C-136 a^6 b^2 C d x-12 a^4 b^4 C d x+48 a^2 b^6 C d x-48 a^7 b B c-48 a^7 b B d x-96 a^7 b C \sin (c+d x)+96 a^8 c C+96 a^8 C d x+16 a b^7 B \sin (2 (c+d x))-24 a b^7 B c-24 a b^7 B d x-8 a b^7 C \sin (c+d x)-8 a b^7 C \sin (3 (c+d x))+4 b^8 B \sin (c+d x)+4 b^8 B \sin (3 (c+d x))+2 b^8 C \sin (2 (c+d x))+b^8 C \sin (4 (c+d x))+4 b^8 c C+4 b^8 C d x}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac{16 a^2 \left (15 a^2 b^3 B-29 a^3 b^2 C-6 a^4 b B+12 a^5 C+20 a b^4 C-12 b^5 B\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\left (b^2-a^2\right )^{5/2}}}{16 b^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((16*a^2*(-6*a^4*b*B + 15*a^2*b^3*B - 12*b^5*B + 12*a^5*C - 29*a^3*b^2*C + 20*a*b^4*C)*ArcTanh[((a - b)*Tan[(c
+ d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (-48*a^7*b*B*c + 72*a^5*b^3*B*c - 24*a*b^7*B*c + 96*a^8*c*
C - 136*a^6*b^2*c*C - 12*a^4*b^4*c*C + 48*a^2*b^6*c*C + 4*b^8*c*C - 48*a^7*b*B*d*x + 72*a^5*b^3*B*d*x - 24*a*b
^7*B*d*x + 96*a^8*C*d*x - 136*a^6*b^2*C*d*x - 12*a^4*b^4*C*d*x + 48*a^2*b^6*C*d*x + 4*b^8*C*d*x + 16*a*b*(a^2
- b^2)^2*(-6*a*b*B + 12*a^2*C + b^2*C)*(c + d*x)*Cos[c + d*x] + 4*(-(a^2*b) + b^3)^2*(-6*a*b*B + 12*a^2*C + b^
2*C)*(c + d*x)*Cos[2*(c + d*x)] + 48*a^6*b^2*B*Sin[c + d*x] - 84*a^4*b^4*B*Sin[c + d*x] + 8*a^2*b^6*B*Sin[c +
d*x] + 4*b^8*B*Sin[c + d*x] - 96*a^7*b*C*Sin[c + d*x] + 160*a^5*b^3*C*Sin[c + d*x] - 32*a^3*b^5*C*Sin[c + d*x]
- 8*a*b^7*C*Sin[c + d*x] + 36*a^5*b^3*B*Sin[2*(c + d*x)] - 64*a^3*b^5*B*Sin[2*(c + d*x)] + 16*a*b^7*B*Sin[2*(
c + d*x)] - 72*a^6*b^2*C*Sin[2*(c + d*x)] + 130*a^4*b^4*C*Sin[2*(c + d*x)] - 48*a^2*b^6*C*Sin[2*(c + d*x)] + 2
*b^8*C*Sin[2*(c + d*x)] + 4*a^4*b^4*B*Sin[3*(c + d*x)] - 8*a^2*b^6*B*Sin[3*(c + d*x)] + 4*b^8*B*Sin[3*(c + d*x
)] - 8*a^5*b^3*C*Sin[3*(c + d*x)] + 16*a^3*b^5*C*Sin[3*(c + d*x)] - 8*a*b^7*C*Sin[3*(c + d*x)] + a^4*b^4*C*Sin
[4*(c + d*x)] - 2*a^2*b^6*C*Sin[4*(c + d*x)] + b^8*C*Sin[4*(c + d*x)])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2))
/(16*b^5*d)

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Maple [B]  time = 0.05, size = 1504, normalized size = 3.8 \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*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x)

[Out]

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)*B+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)*B-8/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-6/d*a^6/b^4/(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^5/b^3/(a*tan(1/2*d*x+1/2*c)^2-ta
n(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+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*B-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*B-8/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-6/d*a^6/b^4/(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-6/d/b^4/(tan(1/2*d*x+1/2*c)^2+1)^2*tan(1/2*d*
x+1/2*c)*a*C-6/d/b^4/(tan(1/2*d*x+1/2*c)^2+1)^2*tan(1/2*d*x+1/2*c)^3*a*C+12/d*a^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))*B+1/d/b^3*arctan(tan(1/2*d*x+1/2*c))*C+10/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+10/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+1/d/b^3/(tan
(1/2*d*x+1/2*c)^2+1)^2*tan(1/2*d*x+1/2*c)*C-6/d/b^4*arctan(tan(1/2*d*x+1/2*c))*a*B+12/d/b^5*arctan(tan(1/2*d*x
+1/2*c))*a^2*C+2/d/b^3/(tan(1/2*d*x+1/2*c)^2+1)^2*tan(1/2*d*x+1/2*c)^3*B-1/d/b^3/(tan(1/2*d*x+1/2*c)^2+1)^2*ta
n(1/2*d*x+1/2*c)^3*C+2/d/b^3/(tan(1/2*d*x+1/2*c)^2+1)^2*tan(1/2*d*x+1/2*c)*B+29/d*a^5/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))*C-20/d*a^3/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))*C-12/d*a^7/b^5/(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+1/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-15/d*a^4/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))*B+6/d*a^6/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))*B

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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*(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 = 2.92849, size = 4001, normalized size = 10.05 \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*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x, algorithm="fricas")

[Out]

[1/4*(2*(12*C*a^8*b^2 - 6*B*a^7*b^3 - 35*C*a^6*b^4 + 18*B*a^5*b^5 + 33*C*a^4*b^6 - 18*B*a^3*b^7 - 9*C*a^2*b^8
+ 6*B*a*b^9 - C*b^10)*d*x*cos(d*x + c)^2 + 4*(12*C*a^9*b - 6*B*a^8*b^2 - 35*C*a^7*b^3 + 18*B*a^6*b^4 + 33*C*a^
5*b^5 - 18*B*a^4*b^6 - 9*C*a^3*b^7 + 6*B*a^2*b^8 - C*a*b^9)*d*x*cos(d*x + c) + 2*(12*C*a^10 - 6*B*a^9*b - 35*C
*a^8*b^2 + 18*B*a^7*b^3 + 33*C*a^6*b^4 - 18*B*a^5*b^5 - 9*C*a^4*b^6 + 6*B*a^3*b^7 - C*a^2*b^8)*d*x + (12*C*a^9
- 6*B*a^8*b - 29*C*a^7*b^2 + 15*B*a^6*b^3 + 20*C*a^5*b^4 - 12*B*a^4*b^5 + (12*C*a^7*b^2 - 6*B*a^6*b^3 - 29*C*
a^5*b^4 + 15*B*a^4*b^5 + 20*C*a^3*b^6 - 12*B*a^2*b^7)*cos(d*x + c)^2 + 2*(12*C*a^8*b - 6*B*a^7*b^2 - 29*C*a^6*
b^3 + 15*B*a^5*b^4 + 20*C*a^4*b^5 - 12*B*a^3*b^6)*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^9*b - 6*B*a^8*b^2 - 33*C*a^7*b^3 + 17*B*a^6*b^4 + 27*C*a^5*b^5
- 13*B*a^4*b^6 - 6*C*a^3*b^7 + 2*B*a^2*b^8 - (C*a^6*b^4 - 3*C*a^4*b^6 + 3*C*a^2*b^8 - C*b^10)*cos(d*x + c)^3 +
2*(2*C*a^7*b^3 - B*a^6*b^4 - 6*C*a^5*b^5 + 3*B*a^4*b^6 + 6*C*a^3*b^7 - 3*B*a^2*b^8 - 2*C*a*b^9 + B*b^10)*cos(
d*x + c)^2 + (18*C*a^8*b^2 - 9*B*a^7*b^3 - 50*C*a^6*b^4 + 25*B*a^5*b^5 + 43*C*a^4*b^6 - 20*B*a^3*b^7 - 11*C*a^
2*b^8 + 4*B*a*b^9)*cos(d*x + c))*sin(d*x + c))/((a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*d*cos(d*x + c)^2 + 2
*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*d*cos(d*x + c) + (a^8*b^5 - 3*a^6*b^7 + 3*a^4*b^9 - a^2*b^11)*d),
1/2*((12*C*a^8*b^2 - 6*B*a^7*b^3 - 35*C*a^6*b^4 + 18*B*a^5*b^5 + 33*C*a^4*b^6 - 18*B*a^3*b^7 - 9*C*a^2*b^8 +
6*B*a*b^9 - C*b^10)*d*x*cos(d*x + c)^2 + 2*(12*C*a^9*b - 6*B*a^8*b^2 - 35*C*a^7*b^3 + 18*B*a^6*b^4 + 33*C*a^5*
b^5 - 18*B*a^4*b^6 - 9*C*a^3*b^7 + 6*B*a^2*b^8 - C*a*b^9)*d*x*cos(d*x + c) + (12*C*a^10 - 6*B*a^9*b - 35*C*a^8
*b^2 + 18*B*a^7*b^3 + 33*C*a^6*b^4 - 18*B*a^5*b^5 - 9*C*a^4*b^6 + 6*B*a^3*b^7 - C*a^2*b^8)*d*x - (12*C*a^9 - 6
*B*a^8*b - 29*C*a^7*b^2 + 15*B*a^6*b^3 + 20*C*a^5*b^4 - 12*B*a^4*b^5 + (12*C*a^7*b^2 - 6*B*a^6*b^3 - 29*C*a^5*
b^4 + 15*B*a^4*b^5 + 20*C*a^3*b^6 - 12*B*a^2*b^7)*cos(d*x + c)^2 + 2*(12*C*a^8*b - 6*B*a^7*b^2 - 29*C*a^6*b^3
+ 15*B*a^5*b^4 + 20*C*a^4*b^5 - 12*B*a^3*b^6)*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^9*b - 6*B*a^8*b^2 - 33*C*a^7*b^3 + 17*B*a^6*b^4 + 27*C*a^5*b^5 - 13*B*a^4
*b^6 - 6*C*a^3*b^7 + 2*B*a^2*b^8 - (C*a^6*b^4 - 3*C*a^4*b^6 + 3*C*a^2*b^8 - C*b^10)*cos(d*x + c)^3 + 2*(2*C*a^
7*b^3 - B*a^6*b^4 - 6*C*a^5*b^5 + 3*B*a^4*b^6 + 6*C*a^3*b^7 - 3*B*a^2*b^8 - 2*C*a*b^9 + B*b^10)*cos(d*x + c)^2
+ (18*C*a^8*b^2 - 9*B*a^7*b^3 - 50*C*a^6*b^4 + 25*B*a^5*b^5 + 43*C*a^4*b^6 - 20*B*a^3*b^7 - 11*C*a^2*b^8 + 4*
B*a*b^9)*cos(d*x + c))*sin(d*x + c))/((a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*d*cos(d*x + c)^2 + 2*(a^7*b^6
- 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*d*cos(d*x + c) + (a^8*b^5 - 3*a^6*b^7 + 3*a^4*b^9 - a^2*b^11)*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*(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 = 1.62264, size = 1821, normalized size = 4.58 \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*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x, algorithm="giac")

[Out]

1/2*(2*(12*C*a^7 - 6*B*a^6*b - 29*C*a^5*b^2 + 15*B*a^4*b^3 + 20*C*a^3*b^4 - 12*B*a^2*b^5)*(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^4*b^5 - 2*a^2*b^7 + b^9)*sqrt(a^2 - b^2)) - 2*(12*C*a^7*tan(1/2*d*x + 1/2*c)^7 - 6*B*a^6*b*tan(1/2*d*x + 1/2
*c)^7 - 18*C*a^6*b*tan(1/2*d*x + 1/2*c)^7 + 9*B*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 - 17*C*a^5*b^2*tan(1/2*d*x + 1/
2*c)^7 + 9*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 + 33*C*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 - 16*B*a^3*b^4*tan(1/2*d*x +
1/2*c)^7 - 2*C*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 + 2*B*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 - 13*C*a^2*b^5*tan(1/2*d*x
+ 1/2*c)^7 + 4*B*a*b^6*tan(1/2*d*x + 1/2*c)^7 + 4*C*a*b^6*tan(1/2*d*x + 1/2*c)^7 - 2*B*b^7*tan(1/2*d*x + 1/2*
c)^7 + C*b^7*tan(1/2*d*x + 1/2*c)^7 + 36*C*a^7*tan(1/2*d*x + 1/2*c)^5 - 18*B*a^6*b*tan(1/2*d*x + 1/2*c)^5 - 18
*C*a^6*b*tan(1/2*d*x + 1/2*c)^5 + 9*B*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 - 67*C*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 + 3
5*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 + 29*C*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 - 16*B*a^3*b^4*tan(1/2*d*x + 1/2*c)^5
+ 26*C*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 - 10*B*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 - 5*C*a^2*b^5*tan(1/2*d*x + 1/2*c
)^5 + 4*B*a*b^6*tan(1/2*d*x + 1/2*c)^5 - 4*C*a*b^6*tan(1/2*d*x + 1/2*c)^5 + 2*B*b^7*tan(1/2*d*x + 1/2*c)^5 - 3
*C*b^7*tan(1/2*d*x + 1/2*c)^5 + 36*C*a^7*tan(1/2*d*x + 1/2*c)^3 - 18*B*a^6*b*tan(1/2*d*x + 1/2*c)^3 + 18*C*a^6
*b*tan(1/2*d*x + 1/2*c)^3 - 9*B*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 - 67*C*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 + 35*B*a^
4*b^3*tan(1/2*d*x + 1/2*c)^3 - 29*C*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 + 16*B*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 + 26*
C*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 - 10*B*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 + 5*C*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 -
4*B*a*b^6*tan(1/2*d*x + 1/2*c)^3 - 4*C*a*b^6*tan(1/2*d*x + 1/2*c)^3 + 2*B*b^7*tan(1/2*d*x + 1/2*c)^3 + 3*C*b^7
*tan(1/2*d*x + 1/2*c)^3 + 12*C*a^7*tan(1/2*d*x + 1/2*c) - 6*B*a^6*b*tan(1/2*d*x + 1/2*c) + 18*C*a^6*b*tan(1/2*
d*x + 1/2*c) - 9*B*a^5*b^2*tan(1/2*d*x + 1/2*c) - 17*C*a^5*b^2*tan(1/2*d*x + 1/2*c) + 9*B*a^4*b^3*tan(1/2*d*x
+ 1/2*c) - 33*C*a^4*b^3*tan(1/2*d*x + 1/2*c) + 16*B*a^3*b^4*tan(1/2*d*x + 1/2*c) - 2*C*a^3*b^4*tan(1/2*d*x + 1
/2*c) + 2*B*a^2*b^5*tan(1/2*d*x + 1/2*c) + 13*C*a^2*b^5*tan(1/2*d*x + 1/2*c) - 4*B*a*b^6*tan(1/2*d*x + 1/2*c)
+ 4*C*a*b^6*tan(1/2*d*x + 1/2*c) - 2*B*b^7*tan(1/2*d*x + 1/2*c) - C*b^7*tan(1/2*d*x + 1/2*c))/((a^4*b^4 - 2*a^
2*b^6 + b^8)*(a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 + 2*a*tan(1/2*d*x + 1/2*c)^2 + a + b)^2) + (
12*C*a^2 - 6*B*a*b + C*b^2)*(d*x + c)/b^5)/d