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

Optimal. Leaf size=299 \[ -\frac {(3 b B-a C) \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac {b (b B-a C) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {b \left (-4 a^3 C+6 a^2 b B+a b^2 C-3 b^3 B\right ) \tan (c+d x)}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {\left (2 a^4 B+5 a^3 b C-11 a^2 b^2 B-2 a b^3 C+6 b^4 B\right ) \tan (c+d x)}{2 a^3 d \left (a^2-b^2\right )^2}+\frac {b \left (-6 a^5 C+12 a^4 b B+5 a^3 b^2 C-15 a^2 b^3 B-2 a b^4 C+6 b^5 B\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d (a-b)^{5/2} (a+b)^{5/2}} \]

[Out]

b*(12*B*a^4*b-15*B*a^2*b^3+6*B*b^5-6*C*a^5+5*C*a^3*b^2-2*C*a*b^4)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^
(1/2))/a^4/(a-b)^(5/2)/(a+b)^(5/2)/d-(3*B*b-C*a)*arctanh(sin(d*x+c))/a^4/d+1/2*(2*B*a^4-11*B*a^2*b^2+6*B*b^4+5
*C*a^3*b-2*C*a*b^3)*tan(d*x+c)/a^3/(a^2-b^2)^2/d+1/2*b*(B*b-C*a)*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*cos(d*x+c))^2+1
/2*b*(6*B*a^2*b-3*B*b^3-4*C*a^3+C*a*b^2)*tan(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))

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Rubi [A]  time = 1.80, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {3029, 3000, 3055, 3001, 3770, 2659, 205} \[ \frac {b \left (-15 a^2 b^3 B+5 a^3 b^2 C+12 a^4 b B-6 a^5 C-2 a b^4 C+6 b^5 B\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac {\left (-11 a^2 b^2 B+5 a^3 b C+2 a^4 B-2 a b^3 C+6 b^4 B\right ) \tan (c+d x)}{2 a^3 d \left (a^2-b^2\right )^2}+\frac {b \left (6 a^2 b B-4 a^3 C+a b^2 C-3 b^3 B\right ) \tan (c+d x)}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {b (b B-a C) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {(3 b B-a C) \tanh ^{-1}(\sin (c+d x))}{a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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 3000

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^2 - a*b*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*
Sin[e + f*x])^(1 + n))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int
[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(a*A - b*B)*(b*c - a*d)*(m + 1) + b*d*(A*b - a*B)*(m
+ n + 2) + (A*b - a*B)*(a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(A*b - a*B)*(m + n + 3)*Sin[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^
2, 0] && RationalQ[m] && m < -1 && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n,
-1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3001

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*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]

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 3055

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*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*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] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

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Mathematica [A]  time = 5.89, size = 352, normalized size = 1.18 \[ \frac {\frac {a^2 b^2 (a C-b B) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+\frac {a b^2 \left (5 a^3 C-7 a^2 b B-2 a b^2 C+4 b^3 B\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}-\frac {2 b \left (-6 a^5 C+12 a^4 b B+5 a^3 b^2 C-15 a^2 b^3 B-2 a b^4 C+6 b^5 B\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\left (b^2-a^2\right )^{5/2}}+2 (3 b B-a C) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 (a C-3 b B) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 a B \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {2 a B \sin \left (\frac {1}{2} (c+d x)\right )}{\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )}}{2 a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*b*(12*a^4*b*B - 15*a^2*b^3*B + 6*b^5*B - 6*a^5*C + 5*a^3*b^2*C - 2*a*b^4*C)*ArcTanh[((a - b)*Tan[(c + d*x
)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + 2*(3*b*B - a*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(-3
*b*B + a*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*a*B*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d
*x)/2]) + (2*a*B*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + (a^2*b^2*(-(b*B) + a*C)*Sin[c + d*x
])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (a*b^2*(-7*a^2*b*B + 4*b^3*B + 5*a^3*C - 2*a*b^2*C)*Sin[c + d*x]
)/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])))/(2*a^4*d)

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fricas [B]  time = 47.69, size = 2100, normalized size = 7.02 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(((6*C*a^5*b^3 - 12*B*a^4*b^4 - 5*C*a^3*b^5 + 15*B*a^2*b^6 + 2*C*a*b^7 - 6*B*b^8)*cos(d*x + c)^3 + 2*(6*C
*a^6*b^2 - 12*B*a^5*b^3 - 5*C*a^4*b^4 + 15*B*a^3*b^5 + 2*C*a^2*b^6 - 6*B*a*b^7)*cos(d*x + c)^2 + (6*C*a^7*b -
12*B*a^6*b^2 - 5*C*a^5*b^3 + 15*B*a^4*b^4 + 2*C*a^3*b^5 - 6*B*a^2*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*((C*a^7*b^2 - 3*B*a^6*b^3 - 3*C*a^5*b^4 + 9*B*a^4*
b^5 + 3*C*a^3*b^6 - 9*B*a^2*b^7 - C*a*b^8 + 3*B*b^9)*cos(d*x + c)^3 + 2*(C*a^8*b - 3*B*a^7*b^2 - 3*C*a^6*b^3 +
9*B*a^5*b^4 + 3*C*a^4*b^5 - 9*B*a^3*b^6 - C*a^2*b^7 + 3*B*a*b^8)*cos(d*x + c)^2 + (C*a^9 - 3*B*a^8*b - 3*C*a^
7*b^2 + 9*B*a^6*b^3 + 3*C*a^5*b^4 - 9*B*a^4*b^5 - C*a^3*b^6 + 3*B*a^2*b^7)*cos(d*x + c))*log(sin(d*x + c) + 1)
- 2*((C*a^7*b^2 - 3*B*a^6*b^3 - 3*C*a^5*b^4 + 9*B*a^4*b^5 + 3*C*a^3*b^6 - 9*B*a^2*b^7 - C*a*b^8 + 3*B*b^9)*co
s(d*x + c)^3 + 2*(C*a^8*b - 3*B*a^7*b^2 - 3*C*a^6*b^3 + 9*B*a^5*b^4 + 3*C*a^4*b^5 - 9*B*a^3*b^6 - C*a^2*b^7 +
3*B*a*b^8)*cos(d*x + c)^2 + (C*a^9 - 3*B*a^8*b - 3*C*a^7*b^2 + 9*B*a^6*b^3 + 3*C*a^5*b^4 - 9*B*a^4*b^5 - C*a^3
*b^6 + 3*B*a^2*b^7)*cos(d*x + c))*log(-sin(d*x + c) + 1) + 2*(2*B*a^9 - 6*B*a^7*b^2 + 6*B*a^5*b^4 - 2*B*a^3*b^
6 + (2*B*a^7*b^2 + 5*C*a^6*b^3 - 13*B*a^5*b^4 - 7*C*a^4*b^5 + 17*B*a^3*b^6 + 2*C*a^2*b^7 - 6*B*a*b^8)*cos(d*x
+ c)^2 + (4*B*a^8*b + 6*C*a^7*b^2 - 20*B*a^6*b^3 - 9*C*a^5*b^4 + 25*B*a^4*b^5 + 3*C*a^3*b^6 - 9*B*a^2*b^7)*cos
(d*x + c))*sin(d*x + c))/((a^10*b^2 - 3*a^8*b^4 + 3*a^6*b^6 - a^4*b^8)*d*cos(d*x + c)^3 + 2*(a^11*b - 3*a^9*b^
3 + 3*a^7*b^5 - a^5*b^7)*d*cos(d*x + c)^2 + (a^12 - 3*a^10*b^2 + 3*a^8*b^4 - a^6*b^6)*d*cos(d*x + c)), -1/2*((
(6*C*a^5*b^3 - 12*B*a^4*b^4 - 5*C*a^3*b^5 + 15*B*a^2*b^6 + 2*C*a*b^7 - 6*B*b^8)*cos(d*x + c)^3 + 2*(6*C*a^6*b^
2 - 12*B*a^5*b^3 - 5*C*a^4*b^4 + 15*B*a^3*b^5 + 2*C*a^2*b^6 - 6*B*a*b^7)*cos(d*x + c)^2 + (6*C*a^7*b - 12*B*a^
6*b^2 - 5*C*a^5*b^3 + 15*B*a^4*b^4 + 2*C*a^3*b^5 - 6*B*a^2*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))) - ((C*a^7*b^2 - 3*B*a^6*b^3 - 3*C*a^5*b^4 + 9*B*a^4*b^5 + 3*C*a^3
*b^6 - 9*B*a^2*b^7 - C*a*b^8 + 3*B*b^9)*cos(d*x + c)^3 + 2*(C*a^8*b - 3*B*a^7*b^2 - 3*C*a^6*b^3 + 9*B*a^5*b^4
+ 3*C*a^4*b^5 - 9*B*a^3*b^6 - C*a^2*b^7 + 3*B*a*b^8)*cos(d*x + c)^2 + (C*a^9 - 3*B*a^8*b - 3*C*a^7*b^2 + 9*B*a
^6*b^3 + 3*C*a^5*b^4 - 9*B*a^4*b^5 - C*a^3*b^6 + 3*B*a^2*b^7)*cos(d*x + c))*log(sin(d*x + c) + 1) + ((C*a^7*b^
2 - 3*B*a^6*b^3 - 3*C*a^5*b^4 + 9*B*a^4*b^5 + 3*C*a^3*b^6 - 9*B*a^2*b^7 - C*a*b^8 + 3*B*b^9)*cos(d*x + c)^3 +
2*(C*a^8*b - 3*B*a^7*b^2 - 3*C*a^6*b^3 + 9*B*a^5*b^4 + 3*C*a^4*b^5 - 9*B*a^3*b^6 - C*a^2*b^7 + 3*B*a*b^8)*cos(
d*x + c)^2 + (C*a^9 - 3*B*a^8*b - 3*C*a^7*b^2 + 9*B*a^6*b^3 + 3*C*a^5*b^4 - 9*B*a^4*b^5 - C*a^3*b^6 + 3*B*a^2*
b^7)*cos(d*x + c))*log(-sin(d*x + c) + 1) - (2*B*a^9 - 6*B*a^7*b^2 + 6*B*a^5*b^4 - 2*B*a^3*b^6 + (2*B*a^7*b^2
+ 5*C*a^6*b^3 - 13*B*a^5*b^4 - 7*C*a^4*b^5 + 17*B*a^3*b^6 + 2*C*a^2*b^7 - 6*B*a*b^8)*cos(d*x + c)^2 + (4*B*a^8
*b + 6*C*a^7*b^2 - 20*B*a^6*b^3 - 9*C*a^5*b^4 + 25*B*a^4*b^5 + 3*C*a^3*b^6 - 9*B*a^2*b^7)*cos(d*x + c))*sin(d*
x + c))/((a^10*b^2 - 3*a^8*b^4 + 3*a^6*b^6 - a^4*b^8)*d*cos(d*x + c)^3 + 2*(a^11*b - 3*a^9*b^3 + 3*a^7*b^5 - a
^5*b^7)*d*cos(d*x + c)^2 + (a^12 - 3*a^10*b^2 + 3*a^8*b^4 - a^6*b^6)*d*cos(d*x + c))]

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giac [B]  time = 0.36, size = 574, normalized size = 1.92 \[ \frac {\frac {{\left (6 \, C a^{5} b - 12 \, B a^{4} b^{2} - 5 \, C a^{3} b^{3} + 15 \, B a^{2} b^{4} + 2 \, C a b^{5} - 6 \, B b^{6}\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^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} \sqrt {a^{2} - b^{2}}} + \frac {6 \, C a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, B a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, C a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7 \, B a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, C a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, B a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, C a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, B b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, C a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, B a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, C a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 7 \, B a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, C a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, B a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, C a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, B b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\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 )}^{2}} + \frac {{\left (C a - 3 \, B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {{\left (C a - 3 \, B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} - \frac {2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a^{3}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((6*C*a^5*b - 12*B*a^4*b^2 - 5*C*a^3*b^3 + 15*B*a^2*b^4 + 2*C*a*b^5 - 6*B*b^6)*(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^8 - 2*a^6
*b^2 + a^4*b^4)*sqrt(a^2 - b^2)) + (6*C*a^4*b^2*tan(1/2*d*x + 1/2*c)^3 - 8*B*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 -
5*C*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 + 7*B*a^2*b^4*tan(1/2*d*x + 1/2*c)^3 - 3*C*a^2*b^4*tan(1/2*d*x + 1/2*c)^3 +
5*B*a*b^5*tan(1/2*d*x + 1/2*c)^3 + 2*C*a*b^5*tan(1/2*d*x + 1/2*c)^3 - 4*B*b^6*tan(1/2*d*x + 1/2*c)^3 + 6*C*a^
4*b^2*tan(1/2*d*x + 1/2*c) - 8*B*a^3*b^3*tan(1/2*d*x + 1/2*c) + 5*C*a^3*b^3*tan(1/2*d*x + 1/2*c) - 7*B*a^2*b^4
*tan(1/2*d*x + 1/2*c) - 3*C*a^2*b^4*tan(1/2*d*x + 1/2*c) + 5*B*a*b^5*tan(1/2*d*x + 1/2*c) - 2*C*a*b^5*tan(1/2*
d*x + 1/2*c) + 4*B*b^6*tan(1/2*d*x + 1/2*c))/((a^7 - 2*a^5*b^2 + a^3*b^4)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/
2*d*x + 1/2*c)^2 + a + b)^2) + (C*a - 3*B*b)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^4 - (C*a - 3*B*b)*log(abs(ta
n(1/2*d*x + 1/2*c) - 1))/a^4 - 2*B*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 - 1)*a^3))/d

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maple [B]  time = 0.27, size = 1358, normalized size = 4.54 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/d/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2*b^3/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-
1/d/a^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2*b^4/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B
+4/d*b^5/a^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+6/d/(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^2*C+1
/d*b^3/a/(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-2/
d*b^4/a^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-8
/d/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2*b^3/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B+1/d/a^2/(a*t
an(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2*b^4/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B+4/d*b^5/a^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+6/d/(a*tan(1/2*d*x+1/2*c)^2-ta
n(1/2*d*x+1/2*c)^2*b+a+b)^2*b^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*C-1/d*b^3/a/(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-2/d*b^4/a^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+12/d*b^2/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2
*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B-15/d*b^4/a^2/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d
*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B+6/d*b^6/a^4/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+
1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B-6/d*b/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a
-b)/((a-b)*(a+b))^(1/2))*C*a+5/d*b^3/a/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)
/((a-b)*(a+b))^(1/2))*C-2/d*b^5/a^3/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((
a-b)*(a+b))^(1/2))*C-1/d*B/a^3/(tan(1/2*d*x+1/2*c)-1)+3/d/a^4*ln(tan(1/2*d*x+1/2*c)-1)*B*b-1/d/a^3*ln(tan(1/2*
d*x+1/2*c)-1)*C-1/d*B/a^3/(tan(1/2*d*x+1/2*c)+1)-3/d/a^4*ln(tan(1/2*d*x+1/2*c)+1)*B*b+1/d/a^3*ln(tan(1/2*d*x+1
/2*c)+1)*C

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?

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mupad [B]  time = 14.32, size = 9312, normalized size = 31.14 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((tan(c/2 + (d*x)/2)^5*(6*B*b^5 - 2*B*a^5 - 12*B*a^2*b^3 + 4*B*a^3*b^2 + C*a^2*b^3 + 6*C*a^3*b^2 - 3*B*a*b^4 +
2*B*a^4*b - 2*C*a*b^4))/((a^3*b - a^4)*(a + b)^2) + (tan(c/2 + (d*x)/2)*(2*B*a^5 + 6*B*b^5 - 12*B*a^2*b^3 - 4
*B*a^3*b^2 - C*a^2*b^3 + 6*C*a^3*b^2 + 3*B*a*b^4 + 2*B*a^4*b - 2*C*a*b^4))/((a + b)*(a^5 - 2*a^4*b + a^3*b^2))
- (2*tan(c/2 + (d*x)/2)^3*(2*B*a^6 - 6*B*b^6 + 13*B*a^2*b^4 - 6*B*a^4*b^2 - 5*C*a^3*b^3 + 2*C*a*b^5))/(a*(a^2
*b - a^3)*(a + b)^2*(a - b)))/(d*(2*a*b - tan(c/2 + (d*x)/2)^2*(2*a*b - a^2 + 3*b^2) - tan(c/2 + (d*x)/2)^6*(a
^2 - 2*a*b + b^2) + a^2 + b^2 - tan(c/2 + (d*x)/2)^4*(2*a*b + a^2 - 3*b^2))) + (atan((((3*B*b - C*a)*((8*tan(c
/2 + (d*x)/2)*(72*B^2*b^12 + 4*C^2*a^12 - 72*B^2*a*b^11 - 8*C^2*a^11*b - 288*B^2*a^2*b^10 + 288*B^2*a^3*b^9 +
441*B^2*a^4*b^8 - 432*B^2*a^5*b^7 - 288*B^2*a^6*b^6 + 288*B^2*a^7*b^5 + 36*B^2*a^8*b^4 - 72*B^2*a^9*b^3 + 36*B
^2*a^10*b^2 + 8*C^2*a^2*b^10 - 8*C^2*a^3*b^9 - 32*C^2*a^4*b^8 + 32*C^2*a^5*b^7 + 57*C^2*a^6*b^6 - 48*C^2*a^7*b
^5 - 52*C^2*a^8*b^4 + 32*C^2*a^9*b^3 + 24*C^2*a^10*b^2 - 48*B*C*a*b^11 - 24*B*C*a^11*b + 48*B*C*a^2*b^10 + 192
*B*C*a^3*b^9 - 192*B*C*a^4*b^8 - 318*B*C*a^5*b^7 + 288*B*C*a^6*b^6 + 252*B*C*a^7*b^5 - 192*B*C*a^8*b^4 - 72*B*
C*a^9*b^3 + 48*B*C*a^10*b^2))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11
*b^2) + (((8*(4*C*a^18 + 12*B*a^8*b^10 - 6*B*a^9*b^9 - 54*B*a^10*b^8 + 24*B*a^11*b^7 + 96*B*a^12*b^6 - 42*B*a^
13*b^5 - 78*B*a^14*b^4 + 36*B*a^15*b^3 + 24*B*a^16*b^2 - 4*C*a^9*b^9 + 2*C*a^10*b^8 + 18*C*a^11*b^7 - 4*C*a^12
*b^6 - 36*C*a^13*b^5 + 6*C*a^14*b^4 + 34*C*a^15*b^3 - 8*C*a^16*b^2 - 12*B*a^17*b - 12*C*a^17*b))/(a^15*b + a^1
6 - a^9*b^7 - a^10*b^6 + 3*a^11*b^5 + 3*a^12*b^4 - 3*a^13*b^3 - 3*a^14*b^2) + (8*tan(c/2 + (d*x)/2)*(3*B*b - C
*a)*(8*a^17*b - 8*a^8*b^10 + 8*a^9*b^9 + 32*a^10*b^8 - 32*a^11*b^7 - 48*a^12*b^6 + 48*a^13*b^5 + 32*a^14*b^4 -
32*a^15*b^3 - 8*a^16*b^2))/(a^4*(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a
^11*b^2)))*(3*B*b - C*a))/a^4)*1i)/a^4 + ((3*B*b - C*a)*((8*tan(c/2 + (d*x)/2)*(72*B^2*b^12 + 4*C^2*a^12 - 72*
B^2*a*b^11 - 8*C^2*a^11*b - 288*B^2*a^2*b^10 + 288*B^2*a^3*b^9 + 441*B^2*a^4*b^8 - 432*B^2*a^5*b^7 - 288*B^2*a
^6*b^6 + 288*B^2*a^7*b^5 + 36*B^2*a^8*b^4 - 72*B^2*a^9*b^3 + 36*B^2*a^10*b^2 + 8*C^2*a^2*b^10 - 8*C^2*a^3*b^9
- 32*C^2*a^4*b^8 + 32*C^2*a^5*b^7 + 57*C^2*a^6*b^6 - 48*C^2*a^7*b^5 - 52*C^2*a^8*b^4 + 32*C^2*a^9*b^3 + 24*C^2
*a^10*b^2 - 48*B*C*a*b^11 - 24*B*C*a^11*b + 48*B*C*a^2*b^10 + 192*B*C*a^3*b^9 - 192*B*C*a^4*b^8 - 318*B*C*a^5*
b^7 + 288*B*C*a^6*b^6 + 252*B*C*a^7*b^5 - 192*B*C*a^8*b^4 - 72*B*C*a^9*b^3 + 48*B*C*a^10*b^2))/(a^12*b + a^13
- a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) - (((8*(4*C*a^18 + 12*B*a^8*b^10 - 6*B*
a^9*b^9 - 54*B*a^10*b^8 + 24*B*a^11*b^7 + 96*B*a^12*b^6 - 42*B*a^13*b^5 - 78*B*a^14*b^4 + 36*B*a^15*b^3 + 24*B
*a^16*b^2 - 4*C*a^9*b^9 + 2*C*a^10*b^8 + 18*C*a^11*b^7 - 4*C*a^12*b^6 - 36*C*a^13*b^5 + 6*C*a^14*b^4 + 34*C*a^
15*b^3 - 8*C*a^16*b^2 - 12*B*a^17*b - 12*C*a^17*b))/(a^15*b + a^16 - a^9*b^7 - a^10*b^6 + 3*a^11*b^5 + 3*a^12*
b^4 - 3*a^13*b^3 - 3*a^14*b^2) - (8*tan(c/2 + (d*x)/2)*(3*B*b - C*a)*(8*a^17*b - 8*a^8*b^10 + 8*a^9*b^9 + 32*a
^10*b^8 - 32*a^11*b^7 - 48*a^12*b^6 + 48*a^13*b^5 + 32*a^14*b^4 - 32*a^15*b^3 - 8*a^16*b^2))/(a^4*(a^12*b + a^
13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2)))*(3*B*b - C*a))/a^4)*1i)/a^4)/((16*
(108*B^3*b^12 - 54*B^3*a*b^11 - 12*C^3*a^11*b - 486*B^3*a^2*b^10 + 243*B^3*a^3*b^9 + 864*B^3*a^4*b^8 - 378*B^3
*a^5*b^7 - 702*B^3*a^6*b^6 + 216*B^3*a^7*b^5 + 216*B^3*a^8*b^4 - 4*C^3*a^3*b^9 + 2*C^3*a^4*b^8 + 18*C^3*a^5*b^
7 - 13*C^3*a^6*b^6 - 36*C^3*a^7*b^5 + 26*C^3*a^8*b^4 + 34*C^3*a^9*b^3 - 24*C^3*a^10*b^2 - 108*B^2*C*a*b^11 + 3
6*B*C^2*a^2*b^10 - 18*B*C^2*a^3*b^9 - 162*B*C^2*a^4*b^8 + 105*B*C^2*a^5*b^7 + 312*B*C^2*a^6*b^6 - 198*B*C^2*a^
7*b^5 - 282*B*C^2*a^8*b^4 + 156*B*C^2*a^9*b^3 + 96*B*C^2*a^10*b^2 + 54*B^2*C*a^2*b^10 + 486*B^2*C*a^3*b^9 - 27
9*B^2*C*a^4*b^8 - 900*B^2*C*a^5*b^7 + 486*B^2*C*a^6*b^6 + 774*B^2*C*a^7*b^5 - 324*B^2*C*a^8*b^4 - 252*B^2*C*a^
9*b^3))/(a^15*b + a^16 - a^9*b^7 - a^10*b^6 + 3*a^11*b^5 + 3*a^12*b^4 - 3*a^13*b^3 - 3*a^14*b^2) - ((3*B*b - C
*a)*((8*tan(c/2 + (d*x)/2)*(72*B^2*b^12 + 4*C^2*a^12 - 72*B^2*a*b^11 - 8*C^2*a^11*b - 288*B^2*a^2*b^10 + 288*B
^2*a^3*b^9 + 441*B^2*a^4*b^8 - 432*B^2*a^5*b^7 - 288*B^2*a^6*b^6 + 288*B^2*a^7*b^5 + 36*B^2*a^8*b^4 - 72*B^2*a
^9*b^3 + 36*B^2*a^10*b^2 + 8*C^2*a^2*b^10 - 8*C^2*a^3*b^9 - 32*C^2*a^4*b^8 + 32*C^2*a^5*b^7 + 57*C^2*a^6*b^6 -
48*C^2*a^7*b^5 - 52*C^2*a^8*b^4 + 32*C^2*a^9*b^3 + 24*C^2*a^10*b^2 - 48*B*C*a*b^11 - 24*B*C*a^11*b + 48*B*C*a
^2*b^10 + 192*B*C*a^3*b^9 - 192*B*C*a^4*b^8 - 318*B*C*a^5*b^7 + 288*B*C*a^6*b^6 + 252*B*C*a^7*b^5 - 192*B*C*a^
8*b^4 - 72*B*C*a^9*b^3 + 48*B*C*a^10*b^2))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10
*b^3 - 3*a^11*b^2) + (((8*(4*C*a^18 + 12*B*a^8*b^10 - 6*B*a^9*b^9 - 54*B*a^10*b^8 + 24*B*a^11*b^7 + 96*B*a^12*
b^6 - 42*B*a^13*b^5 - 78*B*a^14*b^4 + 36*B*a^15*b^3 + 24*B*a^16*b^2 - 4*C*a^9*b^9 + 2*C*a^10*b^8 + 18*C*a^11*b
^7 - 4*C*a^12*b^6 - 36*C*a^13*b^5 + 6*C*a^14*b^4 + 34*C*a^15*b^3 - 8*C*a^16*b^2 - 12*B*a^17*b - 12*C*a^17*b))/
(a^15*b + a^16 - a^9*b^7 - a^10*b^6 + 3*a^11*b^5 + 3*a^12*b^4 - 3*a^13*b^3 - 3*a^14*b^2) + (8*tan(c/2 + (d*x)/
2)*(3*B*b - C*a)*(8*a^17*b - 8*a^8*b^10 + 8*a^9*b^9 + 32*a^10*b^8 - 32*a^11*b^7 - 48*a^12*b^6 + 48*a^13*b^5 +
32*a^14*b^4 - 32*a^15*b^3 - 8*a^16*b^2))/(a^4*(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a
^10*b^3 - 3*a^11*b^2)))*(3*B*b - C*a))/a^4))/a^4 + ((3*B*b - C*a)*((8*tan(c/2 + (d*x)/2)*(72*B^2*b^12 + 4*C^2*
a^12 - 72*B^2*a*b^11 - 8*C^2*a^11*b - 288*B^2*a^2*b^10 + 288*B^2*a^3*b^9 + 441*B^2*a^4*b^8 - 432*B^2*a^5*b^7 -
288*B^2*a^6*b^6 + 288*B^2*a^7*b^5 + 36*B^2*a^8*b^4 - 72*B^2*a^9*b^3 + 36*B^2*a^10*b^2 + 8*C^2*a^2*b^10 - 8*C^
2*a^3*b^9 - 32*C^2*a^4*b^8 + 32*C^2*a^5*b^7 + 57*C^2*a^6*b^6 - 48*C^2*a^7*b^5 - 52*C^2*a^8*b^4 + 32*C^2*a^9*b^
3 + 24*C^2*a^10*b^2 - 48*B*C*a*b^11 - 24*B*C*a^11*b + 48*B*C*a^2*b^10 + 192*B*C*a^3*b^9 - 192*B*C*a^4*b^8 - 31
8*B*C*a^5*b^7 + 288*B*C*a^6*b^6 + 252*B*C*a^7*b^5 - 192*B*C*a^8*b^4 - 72*B*C*a^9*b^3 + 48*B*C*a^10*b^2))/(a^12
*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) - (((8*(4*C*a^18 + 12*B*a^8*b
^10 - 6*B*a^9*b^9 - 54*B*a^10*b^8 + 24*B*a^11*b^7 + 96*B*a^12*b^6 - 42*B*a^13*b^5 - 78*B*a^14*b^4 + 36*B*a^15*
b^3 + 24*B*a^16*b^2 - 4*C*a^9*b^9 + 2*C*a^10*b^8 + 18*C*a^11*b^7 - 4*C*a^12*b^6 - 36*C*a^13*b^5 + 6*C*a^14*b^4
+ 34*C*a^15*b^3 - 8*C*a^16*b^2 - 12*B*a^17*b - 12*C*a^17*b))/(a^15*b + a^16 - a^9*b^7 - a^10*b^6 + 3*a^11*b^5
+ 3*a^12*b^4 - 3*a^13*b^3 - 3*a^14*b^2) - (8*tan(c/2 + (d*x)/2)*(3*B*b - C*a)*(8*a^17*b - 8*a^8*b^10 + 8*a^9*
b^9 + 32*a^10*b^8 - 32*a^11*b^7 - 48*a^12*b^6 + 48*a^13*b^5 + 32*a^14*b^4 - 32*a^15*b^3 - 8*a^16*b^2))/(a^4*(a
^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2)))*(3*B*b - C*a))/a^4))/a^4
))*(3*B*b - C*a)*2i)/(a^4*d) + (b*atan(((b*(-(a + b)^5*(a - b)^5)^(1/2)*((8*tan(c/2 + (d*x)/2)*(72*B^2*b^12 +
4*C^2*a^12 - 72*B^2*a*b^11 - 8*C^2*a^11*b - 288*B^2*a^2*b^10 + 288*B^2*a^3*b^9 + 441*B^2*a^4*b^8 - 432*B^2*a^5
*b^7 - 288*B^2*a^6*b^6 + 288*B^2*a^7*b^5 + 36*B^2*a^8*b^4 - 72*B^2*a^9*b^3 + 36*B^2*a^10*b^2 + 8*C^2*a^2*b^10
- 8*C^2*a^3*b^9 - 32*C^2*a^4*b^8 + 32*C^2*a^5*b^7 + 57*C^2*a^6*b^6 - 48*C^2*a^7*b^5 - 52*C^2*a^8*b^4 + 32*C^2*
a^9*b^3 + 24*C^2*a^10*b^2 - 48*B*C*a*b^11 - 24*B*C*a^11*b + 48*B*C*a^2*b^10 + 192*B*C*a^3*b^9 - 192*B*C*a^4*b^
8 - 318*B*C*a^5*b^7 + 288*B*C*a^6*b^6 + 252*B*C*a^7*b^5 - 192*B*C*a^8*b^4 - 72*B*C*a^9*b^3 + 48*B*C*a^10*b^2))
/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) - (b*((8*(4*C*a^18 + 12
*B*a^8*b^10 - 6*B*a^9*b^9 - 54*B*a^10*b^8 + 24*B*a^11*b^7 + 96*B*a^12*b^6 - 42*B*a^13*b^5 - 78*B*a^14*b^4 + 36
*B*a^15*b^3 + 24*B*a^16*b^2 - 4*C*a^9*b^9 + 2*C*a^10*b^8 + 18*C*a^11*b^7 - 4*C*a^12*b^6 - 36*C*a^13*b^5 + 6*C*
a^14*b^4 + 34*C*a^15*b^3 - 8*C*a^16*b^2 - 12*B*a^17*b - 12*C*a^17*b))/(a^15*b + a^16 - a^9*b^7 - a^10*b^6 + 3*
a^11*b^5 + 3*a^12*b^4 - 3*a^13*b^3 - 3*a^14*b^2) - (4*b*tan(c/2 + (d*x)/2)*(-(a + b)^5*(a - b)^5)^(1/2)*(6*B*b
^5 - 6*C*a^5 - 15*B*a^2*b^3 + 5*C*a^3*b^2 + 12*B*a^4*b - 2*C*a*b^4)*(8*a^17*b - 8*a^8*b^10 + 8*a^9*b^9 + 32*a^
10*b^8 - 32*a^11*b^7 - 48*a^12*b^6 + 48*a^13*b^5 + 32*a^14*b^4 - 32*a^15*b^3 - 8*a^16*b^2))/((a^14 - a^4*b^10
+ 5*a^6*b^8 - 10*a^8*b^6 + 10*a^10*b^4 - 5*a^12*b^2)*(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^
4 - 3*a^10*b^3 - 3*a^11*b^2)))*(-(a + b)^5*(a - b)^5)^(1/2)*(6*B*b^5 - 6*C*a^5 - 15*B*a^2*b^3 + 5*C*a^3*b^2 +
12*B*a^4*b - 2*C*a*b^4))/(2*(a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b^6 + 10*a^10*b^4 - 5*a^12*b^2)))*(6*B*b^5 -
6*C*a^5 - 15*B*a^2*b^3 + 5*C*a^3*b^2 + 12*B*a^4*b - 2*C*a*b^4)*1i)/(2*(a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b
^6 + 10*a^10*b^4 - 5*a^12*b^2)) + (b*(-(a + b)^5*(a - b)^5)^(1/2)*((8*tan(c/2 + (d*x)/2)*(72*B^2*b^12 + 4*C^2*
a^12 - 72*B^2*a*b^11 - 8*C^2*a^11*b - 288*B^2*a^2*b^10 + 288*B^2*a^3*b^9 + 441*B^2*a^4*b^8 - 432*B^2*a^5*b^7 -
288*B^2*a^6*b^6 + 288*B^2*a^7*b^5 + 36*B^2*a^8*b^4 - 72*B^2*a^9*b^3 + 36*B^2*a^10*b^2 + 8*C^2*a^2*b^10 - 8*C^
2*a^3*b^9 - 32*C^2*a^4*b^8 + 32*C^2*a^5*b^7 + 57*C^2*a^6*b^6 - 48*C^2*a^7*b^5 - 52*C^2*a^8*b^4 + 32*C^2*a^9*b^
3 + 24*C^2*a^10*b^2 - 48*B*C*a*b^11 - 24*B*C*a^11*b + 48*B*C*a^2*b^10 + 192*B*C*a^3*b^9 - 192*B*C*a^4*b^8 - 31
8*B*C*a^5*b^7 + 288*B*C*a^6*b^6 + 252*B*C*a^7*b^5 - 192*B*C*a^8*b^4 - 72*B*C*a^9*b^3 + 48*B*C*a^10*b^2))/(a^12
*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) + (b*((8*(4*C*a^18 + 12*B*a^8
*b^10 - 6*B*a^9*b^9 - 54*B*a^10*b^8 + 24*B*a^11*b^7 + 96*B*a^12*b^6 - 42*B*a^13*b^5 - 78*B*a^14*b^4 + 36*B*a^1
5*b^3 + 24*B*a^16*b^2 - 4*C*a^9*b^9 + 2*C*a^10*b^8 + 18*C*a^11*b^7 - 4*C*a^12*b^6 - 36*C*a^13*b^5 + 6*C*a^14*b
^4 + 34*C*a^15*b^3 - 8*C*a^16*b^2 - 12*B*a^17*b - 12*C*a^17*b))/(a^15*b + a^16 - a^9*b^7 - a^10*b^6 + 3*a^11*b
^5 + 3*a^12*b^4 - 3*a^13*b^3 - 3*a^14*b^2) + (4*b*tan(c/2 + (d*x)/2)*(-(a + b)^5*(a - b)^5)^(1/2)*(6*B*b^5 - 6
*C*a^5 - 15*B*a^2*b^3 + 5*C*a^3*b^2 + 12*B*a^4*b - 2*C*a*b^4)*(8*a^17*b - 8*a^8*b^10 + 8*a^9*b^9 + 32*a^10*b^8
- 32*a^11*b^7 - 48*a^12*b^6 + 48*a^13*b^5 + 32*a^14*b^4 - 32*a^15*b^3 - 8*a^16*b^2))/((a^14 - a^4*b^10 + 5*a^
6*b^8 - 10*a^8*b^6 + 10*a^10*b^4 - 5*a^12*b^2)*(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*
a^10*b^3 - 3*a^11*b^2)))*(-(a + b)^5*(a - b)^5)^(1/2)*(6*B*b^5 - 6*C*a^5 - 15*B*a^2*b^3 + 5*C*a^3*b^2 + 12*B*a
^4*b - 2*C*a*b^4))/(2*(a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b^6 + 10*a^10*b^4 - 5*a^12*b^2)))*(6*B*b^5 - 6*C*a
^5 - 15*B*a^2*b^3 + 5*C*a^3*b^2 + 12*B*a^4*b - 2*C*a*b^4)*1i)/(2*(a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b^6 + 1
0*a^10*b^4 - 5*a^12*b^2)))/((16*(108*B^3*b^12 - 54*B^3*a*b^11 - 12*C^3*a^11*b - 486*B^3*a^2*b^10 + 243*B^3*a^3
*b^9 + 864*B^3*a^4*b^8 - 378*B^3*a^5*b^7 - 702*B^3*a^6*b^6 + 216*B^3*a^7*b^5 + 216*B^3*a^8*b^4 - 4*C^3*a^3*b^9
+ 2*C^3*a^4*b^8 + 18*C^3*a^5*b^7 - 13*C^3*a^6*b^6 - 36*C^3*a^7*b^5 + 26*C^3*a^8*b^4 + 34*C^3*a^9*b^3 - 24*C^3
*a^10*b^2 - 108*B^2*C*a*b^11 + 36*B*C^2*a^2*b^10 - 18*B*C^2*a^3*b^9 - 162*B*C^2*a^4*b^8 + 105*B*C^2*a^5*b^7 +
312*B*C^2*a^6*b^6 - 198*B*C^2*a^7*b^5 - 282*B*C^2*a^8*b^4 + 156*B*C^2*a^9*b^3 + 96*B*C^2*a^10*b^2 + 54*B^2*C*a
^2*b^10 + 486*B^2*C*a^3*b^9 - 279*B^2*C*a^4*b^8 - 900*B^2*C*a^5*b^7 + 486*B^2*C*a^6*b^6 + 774*B^2*C*a^7*b^5 -
324*B^2*C*a^8*b^4 - 252*B^2*C*a^9*b^3))/(a^15*b + a^16 - a^9*b^7 - a^10*b^6 + 3*a^11*b^5 + 3*a^12*b^4 - 3*a^13
*b^3 - 3*a^14*b^2) + (b*(-(a + b)^5*(a - b)^5)^(1/2)*((8*tan(c/2 + (d*x)/2)*(72*B^2*b^12 + 4*C^2*a^12 - 72*B^2
*a*b^11 - 8*C^2*a^11*b - 288*B^2*a^2*b^10 + 288*B^2*a^3*b^9 + 441*B^2*a^4*b^8 - 432*B^2*a^5*b^7 - 288*B^2*a^6*
b^6 + 288*B^2*a^7*b^5 + 36*B^2*a^8*b^4 - 72*B^2*a^9*b^3 + 36*B^2*a^10*b^2 + 8*C^2*a^2*b^10 - 8*C^2*a^3*b^9 - 3
2*C^2*a^4*b^8 + 32*C^2*a^5*b^7 + 57*C^2*a^6*b^6 - 48*C^2*a^7*b^5 - 52*C^2*a^8*b^4 + 32*C^2*a^9*b^3 + 24*C^2*a^
10*b^2 - 48*B*C*a*b^11 - 24*B*C*a^11*b + 48*B*C*a^2*b^10 + 192*B*C*a^3*b^9 - 192*B*C*a^4*b^8 - 318*B*C*a^5*b^7
+ 288*B*C*a^6*b^6 + 252*B*C*a^7*b^5 - 192*B*C*a^8*b^4 - 72*B*C*a^9*b^3 + 48*B*C*a^10*b^2))/(a^12*b + a^13 - a
^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) - (b*((8*(4*C*a^18 + 12*B*a^8*b^10 - 6*B*a
^9*b^9 - 54*B*a^10*b^8 + 24*B*a^11*b^7 + 96*B*a^12*b^6 - 42*B*a^13*b^5 - 78*B*a^14*b^4 + 36*B*a^15*b^3 + 24*B*
a^16*b^2 - 4*C*a^9*b^9 + 2*C*a^10*b^8 + 18*C*a^11*b^7 - 4*C*a^12*b^6 - 36*C*a^13*b^5 + 6*C*a^14*b^4 + 34*C*a^1
5*b^3 - 8*C*a^16*b^2 - 12*B*a^17*b - 12*C*a^17*b))/(a^15*b + a^16 - a^9*b^7 - a^10*b^6 + 3*a^11*b^5 + 3*a^12*b
^4 - 3*a^13*b^3 - 3*a^14*b^2) - (4*b*tan(c/2 + (d*x)/2)*(-(a + b)^5*(a - b)^5)^(1/2)*(6*B*b^5 - 6*C*a^5 - 15*B
*a^2*b^3 + 5*C*a^3*b^2 + 12*B*a^4*b - 2*C*a*b^4)*(8*a^17*b - 8*a^8*b^10 + 8*a^9*b^9 + 32*a^10*b^8 - 32*a^11*b^
7 - 48*a^12*b^6 + 48*a^13*b^5 + 32*a^14*b^4 - 32*a^15*b^3 - 8*a^16*b^2))/((a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^
8*b^6 + 10*a^10*b^4 - 5*a^12*b^2)*(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*
a^11*b^2)))*(-(a + b)^5*(a - b)^5)^(1/2)*(6*B*b^5 - 6*C*a^5 - 15*B*a^2*b^3 + 5*C*a^3*b^2 + 12*B*a^4*b - 2*C*a*
b^4))/(2*(a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b^6 + 10*a^10*b^4 - 5*a^12*b^2)))*(6*B*b^5 - 6*C*a^5 - 15*B*a^2
*b^3 + 5*C*a^3*b^2 + 12*B*a^4*b - 2*C*a*b^4))/(2*(a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b^6 + 10*a^10*b^4 - 5*a
^12*b^2)) - (b*(-(a + b)^5*(a - b)^5)^(1/2)*((8*tan(c/2 + (d*x)/2)*(72*B^2*b^12 + 4*C^2*a^12 - 72*B^2*a*b^11 -
8*C^2*a^11*b - 288*B^2*a^2*b^10 + 288*B^2*a^3*b^9 + 441*B^2*a^4*b^8 - 432*B^2*a^5*b^7 - 288*B^2*a^6*b^6 + 288
*B^2*a^7*b^5 + 36*B^2*a^8*b^4 - 72*B^2*a^9*b^3 + 36*B^2*a^10*b^2 + 8*C^2*a^2*b^10 - 8*C^2*a^3*b^9 - 32*C^2*a^4
*b^8 + 32*C^2*a^5*b^7 + 57*C^2*a^6*b^6 - 48*C^2*a^7*b^5 - 52*C^2*a^8*b^4 + 32*C^2*a^9*b^3 + 24*C^2*a^10*b^2 -
48*B*C*a*b^11 - 24*B*C*a^11*b + 48*B*C*a^2*b^10 + 192*B*C*a^3*b^9 - 192*B*C*a^4*b^8 - 318*B*C*a^5*b^7 + 288*B*
C*a^6*b^6 + 252*B*C*a^7*b^5 - 192*B*C*a^8*b^4 - 72*B*C*a^9*b^3 + 48*B*C*a^10*b^2))/(a^12*b + a^13 - a^6*b^7 -
a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) + (b*((8*(4*C*a^18 + 12*B*a^8*b^10 - 6*B*a^9*b^9 -
54*B*a^10*b^8 + 24*B*a^11*b^7 + 96*B*a^12*b^6 - 42*B*a^13*b^5 - 78*B*a^14*b^4 + 36*B*a^15*b^3 + 24*B*a^16*b^2
- 4*C*a^9*b^9 + 2*C*a^10*b^8 + 18*C*a^11*b^7 - 4*C*a^12*b^6 - 36*C*a^13*b^5 + 6*C*a^14*b^4 + 34*C*a^15*b^3 - 8
*C*a^16*b^2 - 12*B*a^17*b - 12*C*a^17*b))/(a^15*b + a^16 - a^9*b^7 - a^10*b^6 + 3*a^11*b^5 + 3*a^12*b^4 - 3*a^
13*b^3 - 3*a^14*b^2) + (4*b*tan(c/2 + (d*x)/2)*(-(a + b)^5*(a - b)^5)^(1/2)*(6*B*b^5 - 6*C*a^5 - 15*B*a^2*b^3
+ 5*C*a^3*b^2 + 12*B*a^4*b - 2*C*a*b^4)*(8*a^17*b - 8*a^8*b^10 + 8*a^9*b^9 + 32*a^10*b^8 - 32*a^11*b^7 - 48*a^
12*b^6 + 48*a^13*b^5 + 32*a^14*b^4 - 32*a^15*b^3 - 8*a^16*b^2))/((a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b^6 + 1
0*a^10*b^4 - 5*a^12*b^2)*(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2)
))*(-(a + b)^5*(a - b)^5)^(1/2)*(6*B*b^5 - 6*C*a^5 - 15*B*a^2*b^3 + 5*C*a^3*b^2 + 12*B*a^4*b - 2*C*a*b^4))/(2*
(a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b^6 + 10*a^10*b^4 - 5*a^12*b^2)))*(6*B*b^5 - 6*C*a^5 - 15*B*a^2*b^3 + 5*
C*a^3*b^2 + 12*B*a^4*b - 2*C*a*b^4))/(2*(a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b^6 + 10*a^10*b^4 - 5*a^12*b^2))
))*(-(a + b)^5*(a - b)^5)^(1/2)*(6*B*b^5 - 6*C*a^5 - 15*B*a^2*b^3 + 5*C*a^3*b^2 + 12*B*a^4*b - 2*C*a*b^4)*1i)/
(d*(a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b^6 + 10*a^10*b^4 - 5*a^12*b^2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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