3.339 \(\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{a+a \cos (c+d x)} \, dx\)
Optimal. Leaf size=174 \[ \frac{(3 A-4 B+4 C) \sin ^3(c+d x)}{3 a d}-\frac{(3 A-4 B+4 C) \sin (c+d x)}{a d}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(4 A-4 B+5 C) \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{3 (4 A-4 B+5 C) \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{3 x (4 A-4 B+5 C)}{8 a} \]
[Out]
(3*(4*A - 4*B + 5*C)*x)/(8*a) - ((3*A - 4*B + 4*C)*Sin[c + d*x])/(a*d) + (3*(4*A - 4*B + 5*C)*Cos[c + d*x]*Sin
[c + d*x])/(8*a*d) + ((4*A - 4*B + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - ((A - B + C)*Cos[c + d*x]^4*Sin
[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((3*A - 4*B + 4*C)*Sin[c + d*x]^3)/(3*a*d)
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Rubi [A] time = 0.23375, antiderivative size = 174, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 5, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.122, Rules used =
{3041, 2748, 2633, 2635, 8} \[ \frac{(3 A-4 B+4 C) \sin ^3(c+d x)}{3 a d}-\frac{(3 A-4 B+4 C) \sin (c+d x)}{a d}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(4 A-4 B+5 C) \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{3 (4 A-4 B+5 C) \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{3 x (4 A-4 B+5 C)}{8 a} \]
Antiderivative was successfully verified.
[In]
Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]
[Out]
(3*(4*A - 4*B + 5*C)*x)/(8*a) - ((3*A - 4*B + 4*C)*Sin[c + d*x])/(a*d) + (3*(4*A - 4*B + 5*C)*Cos[c + d*x]*Sin
[c + d*x])/(8*a*d) + ((4*A - 4*B + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - ((A - B + C)*Cos[c + d*x]^4*Sin
[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((3*A - 4*B + 4*C)*Sin[c + d*x]^3)/(3*a*d)
Rule 3041
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[((a*A - b*B + a*C)*Cos[e + f*x]*(
a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(f*(b*c - a*d)*(2*m + 1)), x] + Dist[1/(b*(b*c - a*d)*(2*m
+ 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) + B*(
b*c*m + a*d*(n + 1)) - C*(a*c*m + b*d*(n + 1)) + (d*(a*A - b*B)*(m + n + 2) + C*(b*c*(2*m + 1) - a*d*(m - n -
1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^
2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)]
Rule 2748
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Rule 2633
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Rule 2635
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx &=-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac{\int \cos ^3(c+d x) (-a (3 A-4 B+4 C)+a (4 A-4 B+5 C) \cos (c+d x)) \, dx}{a^2}\\ &=-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{(3 A-4 B+4 C) \int \cos ^3(c+d x) \, dx}{a}+\frac{(4 A-4 B+5 C) \int \cos ^4(c+d x) \, dx}{a}\\ &=\frac{(4 A-4 B+5 C) \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac{(3 (4 A-4 B+5 C)) \int \cos ^2(c+d x) \, dx}{4 a}+\frac{(3 A-4 B+4 C) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a d}\\ &=-\frac{(3 A-4 B+4 C) \sin (c+d x)}{a d}+\frac{3 (4 A-4 B+5 C) \cos (c+d x) \sin (c+d x)}{8 a d}+\frac{(4 A-4 B+5 C) \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac{(3 A-4 B+4 C) \sin ^3(c+d x)}{3 a d}+\frac{(3 (4 A-4 B+5 C)) \int 1 \, dx}{8 a}\\ &=\frac{3 (4 A-4 B+5 C) x}{8 a}-\frac{(3 A-4 B+4 C) \sin (c+d x)}{a d}+\frac{3 (4 A-4 B+5 C) \cos (c+d x) \sin (c+d x)}{8 a d}+\frac{(4 A-4 B+5 C) \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac{(3 A-4 B+4 C) \sin ^3(c+d x)}{3 a d}\\ \end{align*}
Mathematica [B] time = 0.762329, size = 393, normalized size = 2.26 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (72 d x (4 A-4 B+5 C) \cos \left (c+\frac{d x}{2}\right )+72 d x (4 A-4 B+5 C) \cos \left (\frac{d x}{2}\right )-96 A \sin \left (c+\frac{d x}{2}\right )-72 A \sin \left (c+\frac{3 d x}{2}\right )-72 A \sin \left (2 c+\frac{3 d x}{2}\right )+24 A \sin \left (2 c+\frac{5 d x}{2}\right )+24 A \sin \left (3 c+\frac{5 d x}{2}\right )-480 A \sin \left (\frac{d x}{2}\right )+168 B \sin \left (c+\frac{d x}{2}\right )+144 B \sin \left (c+\frac{3 d x}{2}\right )+144 B \sin \left (2 c+\frac{3 d x}{2}\right )-16 B \sin \left (2 c+\frac{5 d x}{2}\right )-16 B \sin \left (3 c+\frac{5 d x}{2}\right )+8 B \sin \left (3 c+\frac{7 d x}{2}\right )+8 B \sin \left (4 c+\frac{7 d x}{2}\right )+552 B \sin \left (\frac{d x}{2}\right )-168 C \sin \left (c+\frac{d x}{2}\right )-120 C \sin \left (c+\frac{3 d x}{2}\right )-120 C \sin \left (2 c+\frac{3 d x}{2}\right )+40 C \sin \left (2 c+\frac{5 d x}{2}\right )+40 C \sin \left (3 c+\frac{5 d x}{2}\right )-5 C \sin \left (3 c+\frac{7 d x}{2}\right )-5 C \sin \left (4 c+\frac{7 d x}{2}\right )+3 C \sin \left (4 c+\frac{9 d x}{2}\right )+3 C \sin \left (5 c+\frac{9 d x}{2}\right )-552 C \sin \left (\frac{d x}{2}\right )\right )}{192 a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
[In]
Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]
[Out]
(Cos[(c + d*x)/2]*Sec[c/2]*(72*(4*A - 4*B + 5*C)*d*x*Cos[(d*x)/2] + 72*(4*A - 4*B + 5*C)*d*x*Cos[c + (d*x)/2]
- 480*A*Sin[(d*x)/2] + 552*B*Sin[(d*x)/2] - 552*C*Sin[(d*x)/2] - 96*A*Sin[c + (d*x)/2] + 168*B*Sin[c + (d*x)/2
] - 168*C*Sin[c + (d*x)/2] - 72*A*Sin[c + (3*d*x)/2] + 144*B*Sin[c + (3*d*x)/2] - 120*C*Sin[c + (3*d*x)/2] - 7
2*A*Sin[2*c + (3*d*x)/2] + 144*B*Sin[2*c + (3*d*x)/2] - 120*C*Sin[2*c + (3*d*x)/2] + 24*A*Sin[2*c + (5*d*x)/2]
- 16*B*Sin[2*c + (5*d*x)/2] + 40*C*Sin[2*c + (5*d*x)/2] + 24*A*Sin[3*c + (5*d*x)/2] - 16*B*Sin[3*c + (5*d*x)/
2] + 40*C*Sin[3*c + (5*d*x)/2] + 8*B*Sin[3*c + (7*d*x)/2] - 5*C*Sin[3*c + (7*d*x)/2] + 8*B*Sin[4*c + (7*d*x)/2
] - 5*C*Sin[4*c + (7*d*x)/2] + 3*C*Sin[4*c + (9*d*x)/2] + 3*C*Sin[5*c + (9*d*x)/2]))/(192*a*d*(1 + Cos[c + d*x
]))
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Maple [B] time = 0.039, size = 526, normalized size = 3. \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+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c)),x)
[Out]
-1/a/d*A*tan(1/2*d*x+1/2*c)+1/a/d*B*tan(1/2*d*x+1/2*c)-1/a/d*C*tan(1/2*d*x+1/2*c)-3/a/d/(tan(1/2*d*x+1/2*c)^2+
1)^4*tan(1/2*d*x+1/2*c)^7*A-25/4/a/d/(tan(1/2*d*x+1/2*c)^2+1)^4*tan(1/2*d*x+1/2*c)^7*C+5/a/d/(tan(1/2*d*x+1/2*
c)^2+1)^4*tan(1/2*d*x+1/2*c)^7*B-7/a/d/(tan(1/2*d*x+1/2*c)^2+1)^4*tan(1/2*d*x+1/2*c)^5*A-115/12/a/d/(tan(1/2*d
*x+1/2*c)^2+1)^4*tan(1/2*d*x+1/2*c)^5*C+31/3/a/d/(tan(1/2*d*x+1/2*c)^2+1)^4*tan(1/2*d*x+1/2*c)^5*B-5/a/d/(tan(
1/2*d*x+1/2*c)^2+1)^4*tan(1/2*d*x+1/2*c)^3*A-109/12/a/d/(tan(1/2*d*x+1/2*c)^2+1)^4*tan(1/2*d*x+1/2*c)^3*C+25/3
/a/d/(tan(1/2*d*x+1/2*c)^2+1)^4*tan(1/2*d*x+1/2*c)^3*B-1/a/d/(tan(1/2*d*x+1/2*c)^2+1)^4*A*tan(1/2*d*x+1/2*c)-7
/4/a/d/(tan(1/2*d*x+1/2*c)^2+1)^4*C*tan(1/2*d*x+1/2*c)+3/a/d/(tan(1/2*d*x+1/2*c)^2+1)^4*B*tan(1/2*d*x+1/2*c)+3
/a/d*arctan(tan(1/2*d*x+1/2*c))*A-3/a/d*arctan(tan(1/2*d*x+1/2*c))*B+15/4/a/d*arctan(tan(1/2*d*x+1/2*c))*C
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Maxima [B] time = 1.54274, size = 709, normalized size = 4.07 \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+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c)),x, algorithm="maxima")
[Out]
-1/12*(C*((21*sin(d*x + c)/(cos(d*x + c) + 1) + 109*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 115*sin(d*x + c)^5/(
cos(d*x + c) + 1)^5 + 75*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/(a + 4*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6
*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a*sin(d*x + c)^8/(cos(d*x +
c) + 1)^8) - 45*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + 12*sin(d*x + c)/(a*(cos(d*x + c) + 1))) - 4*B*((9
*sin(d*x + c)/(cos(d*x + c) + 1) + 16*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 15*sin(d*x + c)^5/(cos(d*x + c) +
1)^5)/(a + 3*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + a*sin(d*x + c)^
6/(cos(d*x + c) + 1)^6) - 9*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + 3*sin(d*x + c)/(a*(cos(d*x + c) + 1)))
+ 12*A*((sin(d*x + c)/(cos(d*x + c) + 1) + 3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a + 2*a*sin(d*x + c)^2/(co
s(d*x + c) + 1)^2 + a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) - 3*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + sin
(d*x + c)/(a*(cos(d*x + c) + 1))))/d
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Fricas [A] time = 1.91629, size = 344, normalized size = 1.98 \begin{align*} \frac{9 \,{\left (4 \, A - 4 \, B + 5 \, C\right )} d x \cos \left (d x + c\right ) + 9 \,{\left (4 \, A - 4 \, B + 5 \, C\right )} d x +{\left (6 \, C \cos \left (d x + c\right )^{4} + 2 \,{\left (4 \, B - C\right )} \cos \left (d x + c\right )^{3} +{\left (12 \, A - 4 \, B + 13 \, C\right )} \cos \left (d x + c\right )^{2} -{\left (12 \, A - 28 \, B + 19 \, C\right )} \cos \left (d x + c\right ) - 48 \, A + 64 \, B - 64 \, C\right )} \sin \left (d x + c\right )}{24 \,{\left (a d \cos \left (d x + c\right ) + a d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c)),x, algorithm="fricas")
[Out]
1/24*(9*(4*A - 4*B + 5*C)*d*x*cos(d*x + c) + 9*(4*A - 4*B + 5*C)*d*x + (6*C*cos(d*x + c)^4 + 2*(4*B - C)*cos(d
*x + c)^3 + (12*A - 4*B + 13*C)*cos(d*x + c)^2 - (12*A - 28*B + 19*C)*cos(d*x + c) - 48*A + 64*B - 64*C)*sin(d
*x + c))/(a*d*cos(d*x + c) + a*d)
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Sympy [A] time = 19.3042, size = 2688, normalized size = 15.45 \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+B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+a*cos(d*x+c)),x)
[Out]
Piecewise((36*A*d*x*tan(c/2 + d*x/2)**8/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan
(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 144*A*d*x*tan(c/2 + d*x/2)**6/(24*a*d*tan(c/2 + d*x/
2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 216*
A*d*x*tan(c/2 + d*x/2)**4/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)*
*4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 144*A*d*x*tan(c/2 + d*x/2)**2/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d
*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 36*A*d*x/(24*a*d*t
an(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 2
4*a*d) - 24*A*tan(c/2 + d*x/2)**9/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 +
d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 168*A*tan(c/2 + d*x/2)**7/(24*a*d*tan(c/2 + d*x/2)**8 + 96
*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 312*A*tan(c/2
+ d*x/2)**5/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*ta
n(c/2 + d*x/2)**2 + 24*a*d) - 216*A*tan(c/2 + d*x/2)**3/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)*
*6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 48*A*tan(c/2 + d*x/2)/(24*a*d*tan(c/
2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d
) - 36*B*d*x*tan(c/2 + d*x/2)**8/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 +
d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 144*B*d*x*tan(c/2 + d*x/2)**6/(24*a*d*tan(c/2 + d*x/2)**8 +
96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 216*B*d*x*t
an(c/2 + d*x/2)**4/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96
*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 144*B*d*x*tan(c/2 + d*x/2)**2/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/
2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 36*B*d*x/(24*a*d*tan(c/2
+ d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d)
+ 24*B*tan(c/2 + d*x/2)**9/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)
**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 216*B*tan(c/2 + d*x/2)**7/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*ta
n(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 392*B*tan(c/2 + d*x/2
)**5/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 +
d*x/2)**2 + 24*a*d) + 296*B*tan(c/2 + d*x/2)**3/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 14
4*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 96*B*tan(c/2 + d*x/2)/(24*a*d*tan(c/2 + d*x
/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 45*
C*d*x*tan(c/2 + d*x/2)**8/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)*
*4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 180*C*d*x*tan(c/2 + d*x/2)**6/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d
*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 270*C*d*x*tan(c/2
+ d*x/2)**4/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*ta
n(c/2 + d*x/2)**2 + 24*a*d) + 180*C*d*x*tan(c/2 + d*x/2)**2/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x
/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 45*C*d*x/(24*a*d*tan(c/2 + d*x/2
)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 24*C*
tan(c/2 + d*x/2)**9/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 9
6*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 246*C*tan(c/2 + d*x/2)**7/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 +
d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 374*C*tan(c/2 + d*x/2)**5/(2
4*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)
**2 + 24*a*d) - 314*C*tan(c/2 + d*x/2)**3/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*t
an(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 66*C*tan(c/2 + d*x/2)/(24*a*d*tan(c/2 + d*x/2)**8
+ 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d), Ne(d, 0)),
(x*(A + B*cos(c) + C*cos(c)**2)*cos(c)**3/(a*cos(c) + a), True))
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Giac [A] time = 1.17125, size = 336, normalized size = 1.93 \begin{align*} \frac{\frac{9 \,{\left (d x + c\right )}{\left (4 \, A - 4 \, B + 5 \, C\right )}}{a} - \frac{24 \,{\left (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 ) + C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a} - \frac{2 \,{\left (36 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 60 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 75 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 84 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 124 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 115 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 60 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 100 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 109 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 12 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 36 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 21 \, C \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 )}^{4} a}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c)),x, algorithm="giac")
[Out]
1/24*(9*(d*x + c)*(4*A - 4*B + 5*C)/a - 24*(A*tan(1/2*d*x + 1/2*c) - B*tan(1/2*d*x + 1/2*c) + C*tan(1/2*d*x +
1/2*c))/a - 2*(36*A*tan(1/2*d*x + 1/2*c)^7 - 60*B*tan(1/2*d*x + 1/2*c)^7 + 75*C*tan(1/2*d*x + 1/2*c)^7 + 84*A*
tan(1/2*d*x + 1/2*c)^5 - 124*B*tan(1/2*d*x + 1/2*c)^5 + 115*C*tan(1/2*d*x + 1/2*c)^5 + 60*A*tan(1/2*d*x + 1/2*
c)^3 - 100*B*tan(1/2*d*x + 1/2*c)^3 + 109*C*tan(1/2*d*x + 1/2*c)^3 + 12*A*tan(1/2*d*x + 1/2*c) - 36*B*tan(1/2*
d*x + 1/2*c) + 21*C*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^4*a))/d