3.355 \(\int \frac {\cos ^4(c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{(a+a \cos (c+d x))^3} \, dx\)

Optimal. Leaf size=237 \[ -\frac {4 (9 A-19 B+34 C) \sin ^3(c+d x)}{15 a^3 d}+\frac {4 (9 A-19 B+34 C) \sin (c+d x)}{5 a^3 d}-\frac {(6 A-13 B+23 C) \sin (c+d x) \cos ^3(c+d x)}{3 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac {(6 A-13 B+23 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac {x (6 A-13 B+23 C)}{2 a^3}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac {(3 A-8 B+13 C) \sin (c+d x) \cos ^4(c+d x)}{15 a d (a \cos (c+d x)+a)^2} \]

[Out]

-1/2*(6*A-13*B+23*C)*x/a^3+4/5*(9*A-19*B+34*C)*sin(d*x+c)/a^3/d-1/2*(6*A-13*B+23*C)*cos(d*x+c)*sin(d*x+c)/a^3/
d-1/5*(A-B+C)*cos(d*x+c)^5*sin(d*x+c)/d/(a+a*cos(d*x+c))^3-1/15*(3*A-8*B+13*C)*cos(d*x+c)^4*sin(d*x+c)/a/d/(a+
a*cos(d*x+c))^2-1/3*(6*A-13*B+23*C)*cos(d*x+c)^3*sin(d*x+c)/d/(a^3+a^3*cos(d*x+c))-4/15*(9*A-19*B+34*C)*sin(d*
x+c)^3/a^3/d

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Rubi [A]  time = 0.56, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ -\frac {4 (9 A-19 B+34 C) \sin ^3(c+d x)}{15 a^3 d}+\frac {4 (9 A-19 B+34 C) \sin (c+d x)}{5 a^3 d}-\frac {(6 A-13 B+23 C) \sin (c+d x) \cos ^3(c+d x)}{3 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac {(6 A-13 B+23 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac {x (6 A-13 B+23 C)}{2 a^3}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac {(3 A-8 B+13 C) \sin (c+d x) \cos ^4(c+d x)}{15 a d (a \cos (c+d x)+a)^2} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]

[Out]

-((6*A - 13*B + 23*C)*x)/(2*a^3) + (4*(9*A - 19*B + 34*C)*Sin[c + d*x])/(5*a^3*d) - ((6*A - 13*B + 23*C)*Cos[c
+ d*x]*Sin[c + d*x])/(2*a^3*d) - ((A - B + C)*Cos[c + d*x]^5*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((3
*A - 8*B + 13*C)*Cos[c + d*x]^4*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((6*A - 13*B + 23*C)*Cos[c + d
*x]^3*Sin[c + d*x])/(3*d*(a^3 + a^3*Cos[c + d*x])) - (4*(9*A - 19*B + 34*C)*Sin[c + d*x]^3)/(15*a^3*d)

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]

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 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 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 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)]

Rubi steps

\begin {align*} \int \frac {\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx &=-\frac {(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {\int \frac {\cos ^4(c+d x) (5 a (B-C)+a (3 A-3 B+8 C) \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(3 A-8 B+13 C) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {\int \frac {\cos ^3(c+d x) \left (-4 a^2 (3 A-8 B+13 C)+3 a^2 (6 A-11 B+21 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{15 a^4}\\ &=-\frac {(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(3 A-8 B+13 C) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(6 A-13 B+23 C) \cos ^3(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac {\int \cos ^2(c+d x) \left (-15 a^3 (6 A-13 B+23 C)+12 a^3 (9 A-19 B+34 C) \cos (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac {(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(3 A-8 B+13 C) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(6 A-13 B+23 C) \cos ^3(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac {(6 A-13 B+23 C) \int \cos ^2(c+d x) \, dx}{a^3}+\frac {(4 (9 A-19 B+34 C)) \int \cos ^3(c+d x) \, dx}{5 a^3}\\ &=-\frac {(6 A-13 B+23 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(3 A-8 B+13 C) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(6 A-13 B+23 C) \cos ^3(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac {(6 A-13 B+23 C) \int 1 \, dx}{2 a^3}-\frac {(4 (9 A-19 B+34 C)) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 a^3 d}\\ &=-\frac {(6 A-13 B+23 C) x}{2 a^3}+\frac {4 (9 A-19 B+34 C) \sin (c+d x)}{5 a^3 d}-\frac {(6 A-13 B+23 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(3 A-8 B+13 C) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(6 A-13 B+23 C) \cos ^3(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac {4 (9 A-19 B+34 C) \sin ^3(c+d x)}{15 a^3 d}\\ \end {align*}

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Mathematica [B]  time = 1.99, size = 663, normalized size = 2.80 \[ \frac {\sec \left (\frac {c}{2}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \left (-600 d x (6 A-13 B+23 C) \cos \left (c+\frac {d x}{2}\right )-600 d x (6 A-13 B+23 C) \cos \left (\frac {d x}{2}\right )-4500 A \sin \left (c+\frac {d x}{2}\right )+4860 A \sin \left (c+\frac {3 d x}{2}\right )-900 A \sin \left (2 c+\frac {3 d x}{2}\right )+1452 A \sin \left (2 c+\frac {5 d x}{2}\right )+300 A \sin \left (3 c+\frac {5 d x}{2}\right )+60 A \sin \left (3 c+\frac {7 d x}{2}\right )+60 A \sin \left (4 c+\frac {7 d x}{2}\right )-1800 A d x \cos \left (c+\frac {3 d x}{2}\right )-1800 A d x \cos \left (2 c+\frac {3 d x}{2}\right )-360 A d x \cos \left (2 c+\frac {5 d x}{2}\right )-360 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+7020 A \sin \left (\frac {d x}{2}\right )+7560 B \sin \left (c+\frac {d x}{2}\right )-9230 B \sin \left (c+\frac {3 d x}{2}\right )+930 B \sin \left (2 c+\frac {3 d x}{2}\right )-2782 B \sin \left (2 c+\frac {5 d x}{2}\right )-750 B \sin \left (3 c+\frac {5 d x}{2}\right )-105 B \sin \left (3 c+\frac {7 d x}{2}\right )-105 B \sin \left (4 c+\frac {7 d x}{2}\right )+15 B \sin \left (4 c+\frac {9 d x}{2}\right )+15 B \sin \left (5 c+\frac {9 d x}{2}\right )+3900 B d x \cos \left (c+\frac {3 d x}{2}\right )+3900 B d x \cos \left (2 c+\frac {3 d x}{2}\right )+780 B d x \cos \left (2 c+\frac {5 d x}{2}\right )+780 B d x \cos \left (3 c+\frac {5 d x}{2}\right )-12760 B \sin \left (\frac {d x}{2}\right )-11110 C \sin \left (c+\frac {d x}{2}\right )+15380 C \sin \left (c+\frac {3 d x}{2}\right )-380 C \sin \left (2 c+\frac {3 d x}{2}\right )+4777 C \sin \left (2 c+\frac {5 d x}{2}\right )+1625 C \sin \left (3 c+\frac {5 d x}{2}\right )+230 C \sin \left (3 c+\frac {7 d x}{2}\right )+230 C \sin \left (4 c+\frac {7 d x}{2}\right )-20 C \sin \left (4 c+\frac {9 d x}{2}\right )-20 C \sin \left (5 c+\frac {9 d x}{2}\right )+5 C \sin \left (5 c+\frac {11 d x}{2}\right )+5 C \sin \left (6 c+\frac {11 d x}{2}\right )-6900 C d x \cos \left (c+\frac {3 d x}{2}\right )-6900 C d x \cos \left (2 c+\frac {3 d x}{2}\right )-1380 C d x \cos \left (2 c+\frac {5 d x}{2}\right )-1380 C d x \cos \left (3 c+\frac {5 d x}{2}\right )+20410 C \sin \left (\frac {d x}{2}\right )\right )}{480 a^3 d (\cos (c+d x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]

[Out]

(Cos[(c + d*x)/2]*Sec[c/2]*(-600*(6*A - 13*B + 23*C)*d*x*Cos[(d*x)/2] - 600*(6*A - 13*B + 23*C)*d*x*Cos[c + (d
*x)/2] - 1800*A*d*x*Cos[c + (3*d*x)/2] + 3900*B*d*x*Cos[c + (3*d*x)/2] - 6900*C*d*x*Cos[c + (3*d*x)/2] - 1800*
A*d*x*Cos[2*c + (3*d*x)/2] + 3900*B*d*x*Cos[2*c + (3*d*x)/2] - 6900*C*d*x*Cos[2*c + (3*d*x)/2] - 360*A*d*x*Cos
[2*c + (5*d*x)/2] + 780*B*d*x*Cos[2*c + (5*d*x)/2] - 1380*C*d*x*Cos[2*c + (5*d*x)/2] - 360*A*d*x*Cos[3*c + (5*
d*x)/2] + 780*B*d*x*Cos[3*c + (5*d*x)/2] - 1380*C*d*x*Cos[3*c + (5*d*x)/2] + 7020*A*Sin[(d*x)/2] - 12760*B*Sin
[(d*x)/2] + 20410*C*Sin[(d*x)/2] - 4500*A*Sin[c + (d*x)/2] + 7560*B*Sin[c + (d*x)/2] - 11110*C*Sin[c + (d*x)/2
] + 4860*A*Sin[c + (3*d*x)/2] - 9230*B*Sin[c + (3*d*x)/2] + 15380*C*Sin[c + (3*d*x)/2] - 900*A*Sin[2*c + (3*d*
x)/2] + 930*B*Sin[2*c + (3*d*x)/2] - 380*C*Sin[2*c + (3*d*x)/2] + 1452*A*Sin[2*c + (5*d*x)/2] - 2782*B*Sin[2*c
+ (5*d*x)/2] + 4777*C*Sin[2*c + (5*d*x)/2] + 300*A*Sin[3*c + (5*d*x)/2] - 750*B*Sin[3*c + (5*d*x)/2] + 1625*C
*Sin[3*c + (5*d*x)/2] + 60*A*Sin[3*c + (7*d*x)/2] - 105*B*Sin[3*c + (7*d*x)/2] + 230*C*Sin[3*c + (7*d*x)/2] +
60*A*Sin[4*c + (7*d*x)/2] - 105*B*Sin[4*c + (7*d*x)/2] + 230*C*Sin[4*c + (7*d*x)/2] + 15*B*Sin[4*c + (9*d*x)/2
] - 20*C*Sin[4*c + (9*d*x)/2] + 15*B*Sin[5*c + (9*d*x)/2] - 20*C*Sin[5*c + (9*d*x)/2] + 5*C*Sin[5*c + (11*d*x)
/2] + 5*C*Sin[6*c + (11*d*x)/2]))/(480*a^3*d*(1 + Cos[c + d*x])^3)

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fricas [A]  time = 0.42, size = 229, normalized size = 0.97 \[ -\frac {15 \, {\left (6 \, A - 13 \, B + 23 \, C\right )} d x \cos \left (d x + c\right )^{3} + 45 \, {\left (6 \, A - 13 \, B + 23 \, C\right )} d x \cos \left (d x + c\right )^{2} + 45 \, {\left (6 \, A - 13 \, B + 23 \, C\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (6 \, A - 13 \, B + 23 \, C\right )} d x - {\left (10 \, C \cos \left (d x + c\right )^{5} + 15 \, {\left (B - C\right )} \cos \left (d x + c\right )^{4} + 5 \, {\left (6 \, A - 9 \, B + 19 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (234 \, A - 479 \, B + 869 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (114 \, A - 239 \, B + 429 \, C\right )} \cos \left (d x + c\right ) + 144 \, A - 304 \, B + 544 \, C\right )} \sin \left (d x + c\right )}{30 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/30*(15*(6*A - 13*B + 23*C)*d*x*cos(d*x + c)^3 + 45*(6*A - 13*B + 23*C)*d*x*cos(d*x + c)^2 + 45*(6*A - 13*B
+ 23*C)*d*x*cos(d*x + c) + 15*(6*A - 13*B + 23*C)*d*x - (10*C*cos(d*x + c)^5 + 15*(B - C)*cos(d*x + c)^4 + 5*(
6*A - 9*B + 19*C)*cos(d*x + c)^3 + (234*A - 479*B + 869*C)*cos(d*x + c)^2 + 3*(114*A - 239*B + 429*C)*cos(d*x
+ c) + 144*A - 304*B + 544*C)*sin(d*x + c))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x +
c) + a^3*d)

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giac [A]  time = 0.46, size = 320, normalized size = 1.35 \[ -\frac {\frac {30 \, {\left (d x + c\right )} {\left (6 \, A - 13 \, B + 23 \, C\right )}}{a^{3}} - \frac {20 \, {\left (6 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 21 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 51 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 76 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 33 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{3}} - \frac {3 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 30 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 50 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 255 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 465 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 735 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^3,x, algorithm="giac")

[Out]

-1/60*(30*(d*x + c)*(6*A - 13*B + 23*C)/a^3 - 20*(6*A*tan(1/2*d*x + 1/2*c)^5 - 21*B*tan(1/2*d*x + 1/2*c)^5 + 5
1*C*tan(1/2*d*x + 1/2*c)^5 + 12*A*tan(1/2*d*x + 1/2*c)^3 - 36*B*tan(1/2*d*x + 1/2*c)^3 + 76*C*tan(1/2*d*x + 1/
2*c)^3 + 6*A*tan(1/2*d*x + 1/2*c) - 15*B*tan(1/2*d*x + 1/2*c) + 33*C*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2
*c)^2 + 1)^3*a^3) - (3*A*a^12*tan(1/2*d*x + 1/2*c)^5 - 3*B*a^12*tan(1/2*d*x + 1/2*c)^5 + 3*C*a^12*tan(1/2*d*x
+ 1/2*c)^5 - 30*A*a^12*tan(1/2*d*x + 1/2*c)^3 + 40*B*a^12*tan(1/2*d*x + 1/2*c)^3 - 50*C*a^12*tan(1/2*d*x + 1/2
*c)^3 + 255*A*a^12*tan(1/2*d*x + 1/2*c) - 465*B*a^12*tan(1/2*d*x + 1/2*c) + 735*C*a^12*tan(1/2*d*x + 1/2*c))/a
^15)/d

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maple [B]  time = 0.14, size = 542, normalized size = 2.29 \[ \frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d \,a^{3}}-\frac {B \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d \,a^{3}}+\frac {C \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d \,a^{3}}-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{2 d \,a^{3}}+\frac {2 B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{3}}-\frac {5 C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d \,a^{3}}+\frac {17 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{3}}-\frac {31 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{3}}+\frac {49 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{3}}+\frac {2 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {7 B \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {17 C \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {12 B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {76 C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {2 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {5 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {11 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {6 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{d \,a^{3}}+\frac {13 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{d \,a^{3}}-\frac {23 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{d \,a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^3,x)

[Out]

1/20/d/a^3*A*tan(1/2*d*x+1/2*c)^5-1/20/d/a^3*B*tan(1/2*d*x+1/2*c)^5+1/20/d/a^3*C*tan(1/2*d*x+1/2*c)^5-1/2/d/a^
3*tan(1/2*d*x+1/2*c)^3*A+2/3/d/a^3*B*tan(1/2*d*x+1/2*c)^3-5/6/d/a^3*C*tan(1/2*d*x+1/2*c)^3+17/4/d/a^3*A*tan(1/
2*d*x+1/2*c)-31/4/d/a^3*B*tan(1/2*d*x+1/2*c)+49/4/d/a^3*C*tan(1/2*d*x+1/2*c)+2/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^
3*A*tan(1/2*d*x+1/2*c)^5-7/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^3*B*tan(1/2*d*x+1/2*c)^5+17/d/a^3/(1+tan(1/2*d*x+1/2
*c)^2)^3*C*tan(1/2*d*x+1/2*c)^5+4/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^3*A-12/d/a^3/(1+tan(1/2*
d*x+1/2*c)^2)^3*B*tan(1/2*d*x+1/2*c)^3+76/3/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^3*C*tan(1/2*d*x+1/2*c)^3+2/d/a^3/(1
+tan(1/2*d*x+1/2*c)^2)^3*A*tan(1/2*d*x+1/2*c)-5/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^3*B*tan(1/2*d*x+1/2*c)+11/d/a^3
/(1+tan(1/2*d*x+1/2*c)^2)^3*C*tan(1/2*d*x+1/2*c)-6/d/a^3*arctan(tan(1/2*d*x+1/2*c))*A+13/d/a^3*arctan(tan(1/2*
d*x+1/2*c))*B-23/d/a^3*arctan(tan(1/2*d*x+1/2*c))*C

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maxima [B]  time = 0.46, size = 547, normalized size = 2.31 \[ \frac {C {\left (\frac {20 \, {\left (\frac {33 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {76 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {51 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {735 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {50 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {1380 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} - B {\left (\frac {60 \, {\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{3} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {465 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {40 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {780 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} + 3 \, A {\left (\frac {40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{60 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^3,x, algorithm="maxima")

[Out]

1/60*(C*(20*(33*sin(d*x + c)/(cos(d*x + c) + 1) + 76*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 51*sin(d*x + c)^5/(
cos(d*x + c) + 1)^5)/(a^3 + 3*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1
)^4 + a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + (735*sin(d*x + c)/(cos(d*x + c) + 1) - 50*sin(d*x + c)^3/(cos
(d*x + c) + 1)^3 + 3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^3 - 1380*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a
^3) - B*(60*(5*sin(d*x + c)/(cos(d*x + c) + 1) + 7*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^3 + 2*a^3*sin(d*x +
c)^2/(cos(d*x + c) + 1)^2 + a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (465*sin(d*x + c)/(cos(d*x + c) + 1) -
40*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^3 - 780*arctan(sin(d*x + c)
/(cos(d*x + c) + 1))/a^3) + 3*A*(40*sin(d*x + c)/((a^3 + a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c
) + 1)) + (85*sin(d*x + c)/(cos(d*x + c) + 1) - 10*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + sin(d*x + c)^5/(cos(d
*x + c) + 1)^5)/a^3 - 120*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3))/d

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mupad [B]  time = 1.28, size = 259, normalized size = 1.09 \[ \frac {\left (2\,A-7\,B+17\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (4\,A-12\,B+\frac {76\,C}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A-5\,B+11\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {2\,A-4\,B+6\,C}{a^3}-\frac {A+5\,B-15\,C}{4\,a^3}+\frac {5\,\left (A-B+C\right )}{2\,a^3}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {2\,A-4\,B+6\,C}{12\,a^3}+\frac {A-B+C}{3\,a^3}\right )}{d}-\frac {x\,\left (6\,A-13\,B+23\,C\right )}{2\,a^3}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (A-B+C\right )}{20\,a^3\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + a*cos(c + d*x))^3,x)

[Out]

(tan(c/2 + (d*x)/2)*(2*A - 5*B + 11*C) + tan(c/2 + (d*x)/2)^5*(2*A - 7*B + 17*C) + tan(c/2 + (d*x)/2)^3*(4*A -
12*B + (76*C)/3))/(d*(3*a^3*tan(c/2 + (d*x)/2)^2 + 3*a^3*tan(c/2 + (d*x)/2)^4 + a^3*tan(c/2 + (d*x)/2)^6 + a^
3)) + (tan(c/2 + (d*x)/2)*((2*A - 4*B + 6*C)/a^3 - (A + 5*B - 15*C)/(4*a^3) + (5*(A - B + C))/(2*a^3)))/d - (t
an(c/2 + (d*x)/2)^3*((2*A - 4*B + 6*C)/(12*a^3) + (A - B + C)/(3*a^3)))/d - (x*(6*A - 13*B + 23*C))/(2*a^3) +
(tan(c/2 + (d*x)/2)^5*(A - B + C))/(20*a^3*d)

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sympy [A]  time = 35.74, size = 2373, normalized size = 10.01 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**4*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+a*cos(d*x+c))**3,x)

[Out]

Piecewise((-180*A*d*x*tan(c/2 + d*x/2)**6/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 18
0*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 540*A*d*x*tan(c/2 + d*x/2)**4/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180
*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 540*A*d*x*tan(c/2 + d*x/2)**2/(60*
a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 18
0*A*d*x/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*
a**3*d) + 3*A*tan(c/2 + d*x/2)**11/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*
d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 21*A*tan(c/2 + d*x/2)**9/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(
c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 174*A*tan(c/2 + d*x/2)**7/(60*a**3*d*tan(c/2 +
d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 798*A*tan(c/2 + d*
x/2)**5/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*
a**3*d) + 975*A*tan(c/2 + d*x/2)**3/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3
*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 375*A*tan(c/2 + d*x/2)/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c
/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 390*B*d*x*tan(c/2 + d*x/2)**6/(60*a**3*d*tan(c/
2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 1170*B*d*x*tan(
c/2 + d*x/2)**4/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)*
*2 + 60*a**3*d) + 1170*B*d*x*tan(c/2 + d*x/2)**2/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)*
*4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 390*B*d*x/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c
/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 3*B*tan(c/2 + d*x/2)**11/(60*a**3*d*tan(c/2 + d
*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 31*B*tan(c/2 + d*x/2
)**9/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**
3*d) - 354*B*tan(c/2 + d*x/2)**7/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*
tan(c/2 + d*x/2)**2 + 60*a**3*d) - 1698*B*tan(c/2 + d*x/2)**5/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(
c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 2075*B*tan(c/2 + d*x/2)**3/(60*a**3*d*tan(c/2
+ d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 765*B*tan(c/2 + d
*x/2)/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a*
*3*d) - 690*C*d*x*tan(c/2 + d*x/2)**6/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a*
*3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 2070*C*d*x*tan(c/2 + d*x/2)**4/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a*
*3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 2070*C*d*x*tan(c/2 + d*x/2)**2/(60*a*
*3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 690*
C*d*x/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a*
*3*d) + 3*C*tan(c/2 + d*x/2)**11/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*
tan(c/2 + d*x/2)**2 + 60*a**3*d) - 41*C*tan(c/2 + d*x/2)**9/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/
2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 594*C*tan(c/2 + d*x/2)**7/(60*a**3*d*tan(c/2 + d
*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 3078*C*tan(c/2 + d*x
/2)**5/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a
**3*d) + 3675*C*tan(c/2 + d*x/2)**3/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3
*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 1395*C*tan(c/2 + d*x/2)/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(
c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d), Ne(d, 0)), (x*(A + B*cos(c) + C*cos(c)**2)*cos(
c)**4/(a*cos(c) + a)**3, True))

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