3.347 \(\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{(a+a \cos (c+d x))^2} \, dx\)
Optimal. Leaf size=185 \[ -\frac{(5 A-8 B+12 C) \sin ^3(c+d x)}{3 a^2 d}+\frac{(5 A-8 B+12 C) \sin (c+d x)}{a^2 d}-\frac{(4 A-7 B+10 C) \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(4 A-7 B+10 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{x (4 A-7 B+10 C)}{2 a^2}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
[Out]
-((4*A - 7*B + 10*C)*x)/(2*a^2) + ((5*A - 8*B + 12*C)*Sin[c + d*x])/(a^2*d) - ((4*A - 7*B + 10*C)*Cos[c + d*x]
*Sin[c + d*x])/(2*a^2*d) - ((4*A - 7*B + 10*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A
- B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((5*A - 8*B + 12*C)*Sin[c + d*x]^3)/(3*a
^2*d)
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Rubi [A] time = 0.379138, antiderivative size = 185, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146, Rules used =
{3041, 2977, 2748, 2635, 8, 2633} \[ -\frac{(5 A-8 B+12 C) \sin ^3(c+d x)}{3 a^2 d}+\frac{(5 A-8 B+12 C) \sin (c+d x)}{a^2 d}-\frac{(4 A-7 B+10 C) \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(4 A-7 B+10 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{x (4 A-7 B+10 C)}{2 a^2}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
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])^2,x]
[Out]
-((4*A - 7*B + 10*C)*x)/(2*a^2) + ((5*A - 8*B + 12*C)*Sin[c + d*x])/(a^2*d) - ((4*A - 7*B + 10*C)*Cos[c + d*x]
*Sin[c + d*x])/(2*a^2*d) - ((4*A - 7*B + 10*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A
- B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((5*A - 8*B + 12*C)*Sin[c + d*x]^3)/(3*a
^2*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 2977
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]
)^n)/(a*f*(2*m + 1)), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n -
1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ
[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
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 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]
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]
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))^2} \, dx &=-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int \frac{\cos ^3(c+d x) (-a (A-4 B+4 C)+3 a (A-B+2 C) \cos (c+d x))}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=-\frac{(4 A-7 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int \cos ^2(c+d x) \left (-3 a^2 (4 A-7 B+10 C)+3 a^2 (5 A-8 B+12 C) \cos (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac{(4 A-7 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{(4 A-7 B+10 C) \int \cos ^2(c+d x) \, dx}{a^2}+\frac{(5 A-8 B+12 C) \int \cos ^3(c+d x) \, dx}{a^2}\\ &=-\frac{(4 A-7 B+10 C) \cos (c+d x) \sin (c+d x)}{2 a^2 d}-\frac{(4 A-7 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{(4 A-7 B+10 C) \int 1 \, dx}{2 a^2}-\frac{(5 A-8 B+12 C) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a^2 d}\\ &=-\frac{(4 A-7 B+10 C) x}{2 a^2}+\frac{(5 A-8 B+12 C) \sin (c+d x)}{a^2 d}-\frac{(4 A-7 B+10 C) \cos (c+d x) \sin (c+d x)}{2 a^2 d}-\frac{(4 A-7 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{(5 A-8 B+12 C) \sin ^3(c+d x)}{3 a^2 d}\\ \end{align*}
Mathematica [B] time = 0.915014, size = 481, normalized size = 2.6 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (-36 d x (4 A-7 B+10 C) \cos \left (c+\frac{d x}{2}\right )-36 d x (4 A-7 B+10 C) \cos \left (\frac{d x}{2}\right )-120 A \sin \left (c+\frac{d x}{2}\right )+164 A \sin \left (c+\frac{3 d x}{2}\right )+36 A \sin \left (2 c+\frac{3 d x}{2}\right )+12 A \sin \left (2 c+\frac{5 d x}{2}\right )+12 A \sin \left (3 c+\frac{5 d x}{2}\right )-48 A d x \cos \left (c+\frac{3 d x}{2}\right )-48 A d x \cos \left (2 c+\frac{3 d x}{2}\right )+264 A \sin \left (\frac{d x}{2}\right )+147 B \sin \left (c+\frac{d x}{2}\right )-239 B \sin \left (c+\frac{3 d x}{2}\right )-63 B \sin \left (2 c+\frac{3 d x}{2}\right )-15 B \sin \left (2 c+\frac{5 d x}{2}\right )-15 B \sin \left (3 c+\frac{5 d x}{2}\right )+3 B \sin \left (3 c+\frac{7 d x}{2}\right )+3 B \sin \left (4 c+\frac{7 d x}{2}\right )+84 B d x \cos \left (c+\frac{3 d x}{2}\right )+84 B d x \cos \left (2 c+\frac{3 d x}{2}\right )-381 B \sin \left (\frac{d x}{2}\right )-156 C \sin \left (c+\frac{d x}{2}\right )+342 C \sin \left (c+\frac{3 d x}{2}\right )+118 C \sin \left (2 c+\frac{3 d x}{2}\right )+30 C \sin \left (2 c+\frac{5 d x}{2}\right )+30 C \sin \left (3 c+\frac{5 d x}{2}\right )-3 C \sin \left (3 c+\frac{7 d x}{2}\right )-3 C \sin \left (4 c+\frac{7 d x}{2}\right )+C \sin \left (4 c+\frac{9 d x}{2}\right )+C \sin \left (5 c+\frac{9 d x}{2}\right )-120 C d x \cos \left (c+\frac{3 d x}{2}\right )-120 C d x \cos \left (2 c+\frac{3 d x}{2}\right )+516 C \sin \left (\frac{d x}{2}\right )\right )}{48 a^2 d (\cos (c+d x)+1)^2} \]
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])^2,x]
[Out]
(Cos[(c + d*x)/2]*Sec[c/2]*(-36*(4*A - 7*B + 10*C)*d*x*Cos[(d*x)/2] - 36*(4*A - 7*B + 10*C)*d*x*Cos[c + (d*x)/
2] - 48*A*d*x*Cos[c + (3*d*x)/2] + 84*B*d*x*Cos[c + (3*d*x)/2] - 120*C*d*x*Cos[c + (3*d*x)/2] - 48*A*d*x*Cos[2
*c + (3*d*x)/2] + 84*B*d*x*Cos[2*c + (3*d*x)/2] - 120*C*d*x*Cos[2*c + (3*d*x)/2] + 264*A*Sin[(d*x)/2] - 381*B*
Sin[(d*x)/2] + 516*C*Sin[(d*x)/2] - 120*A*Sin[c + (d*x)/2] + 147*B*Sin[c + (d*x)/2] - 156*C*Sin[c + (d*x)/2] +
164*A*Sin[c + (3*d*x)/2] - 239*B*Sin[c + (3*d*x)/2] + 342*C*Sin[c + (3*d*x)/2] + 36*A*Sin[2*c + (3*d*x)/2] -
63*B*Sin[2*c + (3*d*x)/2] + 118*C*Sin[2*c + (3*d*x)/2] + 12*A*Sin[2*c + (5*d*x)/2] - 15*B*Sin[2*c + (5*d*x)/2]
+ 30*C*Sin[2*c + (5*d*x)/2] + 12*A*Sin[3*c + (5*d*x)/2] - 15*B*Sin[3*c + (5*d*x)/2] + 30*C*Sin[3*c + (5*d*x)/
2] + 3*B*Sin[3*c + (7*d*x)/2] - 3*C*Sin[3*c + (7*d*x)/2] + 3*B*Sin[4*c + (7*d*x)/2] - 3*C*Sin[4*c + (7*d*x)/2]
+ C*Sin[4*c + (9*d*x)/2] + C*Sin[5*c + (9*d*x)/2]))/(48*a^2*d*(1 + Cos[c + d*x])^2)
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Maple [B] time = 0.036, size = 482, normalized size = 2.6 \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))^2,x)
[Out]
-1/6/d/a^2*tan(1/2*d*x+1/2*c)^3*A+1/6/d/a^2*tan(1/2*d*x+1/2*c)^3*B-1/6/d/a^2*C*tan(1/2*d*x+1/2*c)^3+5/2/d/a^2*
A*tan(1/2*d*x+1/2*c)-7/2/d/a^2*B*tan(1/2*d*x+1/2*c)+9/2/d/a^2*C*tan(1/2*d*x+1/2*c)+2/d/a^2/(tan(1/2*d*x+1/2*c)
^2+1)^3*tan(1/2*d*x+1/2*c)^5*A-5/d/a^2/(tan(1/2*d*x+1/2*c)^2+1)^3*tan(1/2*d*x+1/2*c)^5*B+10/d/a^2/(tan(1/2*d*x
+1/2*c)^2+1)^3*tan(1/2*d*x+1/2*c)^5*C+4/d/a^2/(tan(1/2*d*x+1/2*c)^2+1)^3*tan(1/2*d*x+1/2*c)^3*A-8/d/a^2/(tan(1
/2*d*x+1/2*c)^2+1)^3*tan(1/2*d*x+1/2*c)^3*B+40/3/d/a^2/(tan(1/2*d*x+1/2*c)^2+1)^3*C*tan(1/2*d*x+1/2*c)^3+2/d/a
^2/(tan(1/2*d*x+1/2*c)^2+1)^3*A*tan(1/2*d*x+1/2*c)-3/d/a^2/(tan(1/2*d*x+1/2*c)^2+1)^3*B*tan(1/2*d*x+1/2*c)+6/d
/a^2/(tan(1/2*d*x+1/2*c)^2+1)^3*C*tan(1/2*d*x+1/2*c)-4/d/a^2*arctan(tan(1/2*d*x+1/2*c))*A+7/d/a^2*arctan(tan(1
/2*d*x+1/2*c))*B-10/d/a^2*arctan(tan(1/2*d*x+1/2*c))*C
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Maxima [B] time = 1.47181, size = 657, normalized size = 3.55 \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))^2,x, algorithm="maxima")
[Out]
1/6*(C*(4*(9*sin(d*x + c)/(cos(d*x + c) + 1) + 20*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 15*sin(d*x + c)^5/(cos
(d*x + c) + 1)^5)/(a^2 + 3*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4
+ a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + (27*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d*x + c)^3/(cos(d*x +
c) + 1)^3)/a^2 - 60*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2) - B*(6*(3*sin(d*x + c)/(cos(d*x + c) + 1) + 5
*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^2 + 2*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^2*sin(d*x + c)^4/(c
os(d*x + c) + 1)^4) + (21*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/a^2 - 42*arct
an(sin(d*x + c)/(cos(d*x + c) + 1))/a^2) + A*((15*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d*x + c)^3/(cos(d*x +
c) + 1)^3)/a^2 - 24*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 + 12*sin(d*x + c)/((a^2 + a^2*sin(d*x + c)^2/(
cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1))))/d
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Fricas [A] time = 2.01982, size = 437, normalized size = 2.36 \begin{align*} -\frac{3 \,{\left (4 \, A - 7 \, B + 10 \, C\right )} d x \cos \left (d x + c\right )^{2} + 6 \,{\left (4 \, A - 7 \, B + 10 \, C\right )} d x \cos \left (d x + c\right ) + 3 \,{\left (4 \, A - 7 \, B + 10 \, C\right )} d x -{\left (2 \, C \cos \left (d x + c\right )^{4} +{\left (3 \, B - 2 \, C\right )} \cos \left (d x + c\right )^{3} + 6 \,{\left (A - B + 2 \, C\right )} \cos \left (d x + c\right )^{2} +{\left (28 \, A - 43 \, B + 66 \, C\right )} \cos \left (d x + c\right ) + 20 \, A - 32 \, B + 48 \, C\right )} \sin \left (d x + c\right )}{6 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} 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))^2,x, algorithm="fricas")
[Out]
-1/6*(3*(4*A - 7*B + 10*C)*d*x*cos(d*x + c)^2 + 6*(4*A - 7*B + 10*C)*d*x*cos(d*x + c) + 3*(4*A - 7*B + 10*C)*d
*x - (2*C*cos(d*x + c)^4 + (3*B - 2*C)*cos(d*x + c)^3 + 6*(A - B + 2*C)*cos(d*x + c)^2 + (28*A - 43*B + 66*C)*
cos(d*x + c) + 20*A - 32*B + 48*C)*sin(d*x + c))/(a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)
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Sympy [A] time = 30.9776, size = 2134, normalized size = 11.54 \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+B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+a*cos(d*x+c))**2,x)
[Out]
Piecewise((-12*A*d*x*tan(c/2 + d*x/2)**6/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a*
*2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) - 36*A*d*x*tan(c/2 + d*x/2)**4/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*
tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) - 36*A*d*x*tan(c/2 + d*x/2)**2/(6*a**2*d*tan(c
/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) - 12*A*d*x/(6*a**2*
d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) - A*tan(c/2
+ d*x/2)**9/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*
a**2*d) + 12*A*tan(c/2 + d*x/2)**7/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*t
an(c/2 + d*x/2)**2 + 6*a**2*d) + 54*A*tan(c/2 + d*x/2)**5/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 +
d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) + 68*A*tan(c/2 + d*x/2)**3/(6*a**2*d*tan(c/2 + d*x/2)**6
+ 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) + 27*A*tan(c/2 + d*x/2)/(6*a**2*d
*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) + 21*B*d*x*ta
n(c/2 + d*x/2)**6/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**
2 + 6*a**2*d) + 63*B*d*x*tan(c/2 + d*x/2)**4/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 1
8*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) + 63*B*d*x*tan(c/2 + d*x/2)**2/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**
2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) + 21*B*d*x/(6*a**2*d*tan(c/2 + d*x/2)**6 +
18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) + B*tan(c/2 + d*x/2)**9/(6*a**2*d*t
an(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) - 18*B*tan(c/2
+ d*x/2)**7/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*
a**2*d) - 90*B*tan(c/2 + d*x/2)**5/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*t
an(c/2 + d*x/2)**2 + 6*a**2*d) - 110*B*tan(c/2 + d*x/2)**3/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 +
d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) - 39*B*tan(c/2 + d*x/2)/(6*a**2*d*tan(c/2 + d*x/2)**6 +
18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) - 30*C*d*x*tan(c/2 + d*x/2)**6/(6*a
**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) - 90*C*d
*x*tan(c/2 + d*x/2)**4/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x
/2)**2 + 6*a**2*d) - 90*C*d*x*tan(c/2 + d*x/2)**2/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**
4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) - 30*C*d*x/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d
*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) - C*tan(c/2 + d*x/2)**9/(6*a**2*d*tan(c/2 + d*x/2)**6 + 1
8*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) + 24*C*tan(c/2 + d*x/2)**7/(6*a**2*d*
tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) + 138*C*tan(c/
2 + d*x/2)**5/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 +
6*a**2*d) + 160*C*tan(c/2 + d*x/2)**3/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*
d*tan(c/2 + d*x/2)**2 + 6*a**2*d) + 63*C*tan(c/2 + d*x/2)/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 +
d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d), Ne(d, 0)), (x*(A + B*cos(c) + C*cos(c)**2)*cos(c)**3/(a
*cos(c) + a)**2, True))
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Giac [A] time = 1.15369, size = 359, normalized size = 1.94 \begin{align*} -\frac{\frac{3 \,{\left (d x + c\right )}{\left (4 \, A - 7 \, B + 10 \, C\right )}}{a^{2}} - \frac{2 \,{\left (6 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 15 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 30 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 12 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 24 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 40 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 9 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 18 \, 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 )}^{3} a^{2}} + \frac{A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 21 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 27 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{6}}}{6 \, 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))^2,x, algorithm="giac")
[Out]
-1/6*(3*(d*x + c)*(4*A - 7*B + 10*C)/a^2 - 2*(6*A*tan(1/2*d*x + 1/2*c)^5 - 15*B*tan(1/2*d*x + 1/2*c)^5 + 30*C*
tan(1/2*d*x + 1/2*c)^5 + 12*A*tan(1/2*d*x + 1/2*c)^3 - 24*B*tan(1/2*d*x + 1/2*c)^3 + 40*C*tan(1/2*d*x + 1/2*c)
^3 + 6*A*tan(1/2*d*x + 1/2*c) - 9*B*tan(1/2*d*x + 1/2*c) + 18*C*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2
+ 1)^3*a^2) + (A*a^4*tan(1/2*d*x + 1/2*c)^3 - B*a^4*tan(1/2*d*x + 1/2*c)^3 + C*a^4*tan(1/2*d*x + 1/2*c)^3 - 1
5*A*a^4*tan(1/2*d*x + 1/2*c) + 21*B*a^4*tan(1/2*d*x + 1/2*c) - 27*C*a^4*tan(1/2*d*x + 1/2*c))/a^6)/d