3.1007 \(\int \frac{(A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx\)

Optimal. Leaf size=480 \[ -\frac{\left (-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6-8 a^5 b^3 B+7 a^3 b^5 B+8 a^7 b B-2 a^8 C-2 a b^7 B+8 A b^8\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\tan (c+d x) \left (-a^4 b^2 (65 A+4 C)+68 a^2 A b^4+a^6 (6 A-11 C)-17 a^3 b^3 B+26 a^5 b B+6 a b^5 B-24 A b^6\right )}{6 a^4 d \left (a^2-b^2\right )^3}-\frac{\tan (c+d x) \left (-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-2 a^3 b^3 B+6 a^5 b B-2 a^6 C+a b^5 B-4 A b^6\right )}{2 a^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac{\tan (c+d x) \left (-a^2 b^2 (9 A+2 C)+6 a^3 b B-3 a^4 C-a b^3 B+4 A b^4\right )}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac{\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac{(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d} \]

[Out]

-(((35*a^4*A*b^4 - 28*a^2*A*b^6 + 8*A*b^8 + 8*a^7*b*B - 8*a^5*b^3*B + 7*a^3*b^5*B - 2*a*b^7*B - 2*a^8*C - a^6*
b^2*(20*A + 3*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d)) - (
(4*A*b - a*B)*ArcTanh[Sin[c + d*x]])/(a^5*d) + ((68*a^2*A*b^4 - 24*A*b^6 + 26*a^5*b*B - 17*a^3*b^3*B + 6*a*b^5
*B + a^6*(6*A - 11*C) - a^4*b^2*(65*A + 4*C))*Tan[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + ((A*b^2 - a*(b*B - a*C))
*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((4*A*b^4 + 6*a^3*b*B - a*b^3*B - 3*a^4*C - a^2*b^
2*(9*A + 2*C))*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((11*a^2*A*b^4 - 4*A*b^6 + 6*a^5
*b*B - 2*a^3*b^3*B + a*b^5*B - 2*a^6*C - 3*a^4*b^2*(4*A + C))*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Cos[
c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 10.5378, antiderivative size = 480, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.122, Rules used = {3055, 3001, 3770, 2659, 205} \[ -\frac{\left (-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6-8 a^5 b^3 B+7 a^3 b^5 B+8 a^7 b B-2 a^8 C-2 a b^7 B+8 A b^8\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\tan (c+d x) \left (-a^4 b^2 (65 A+4 C)+68 a^2 A b^4+a^6 (6 A-11 C)-17 a^3 b^3 B+26 a^5 b B+6 a b^5 B-24 A b^6\right )}{6 a^4 d \left (a^2-b^2\right )^3}-\frac{\tan (c+d x) \left (-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-2 a^3 b^3 B+6 a^5 b B-2 a^6 C+a b^5 B-4 A b^6\right )}{2 a^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac{\tan (c+d x) \left (-a^2 b^2 (9 A+2 C)+6 a^3 b B-3 a^4 C-a b^3 B+4 A b^4\right )}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac{\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac{(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((35*a^4*A*b^4 - 28*a^2*A*b^6 + 8*A*b^8 + 8*a^7*b*B - 8*a^5*b^3*B + 7*a^3*b^5*B - 2*a*b^7*B - 2*a^8*C - a^6*
b^2*(20*A + 3*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d)) - (
(4*A*b - a*B)*ArcTanh[Sin[c + d*x]])/(a^5*d) + ((68*a^2*A*b^4 - 24*A*b^6 + 26*a^5*b*B - 17*a^3*b^3*B + 6*a*b^5
*B + a^6*(6*A - 11*C) - a^4*b^2*(65*A + 4*C))*Tan[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + ((A*b^2 - a*(b*B - a*C))
*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((4*A*b^4 + 6*a^3*b*B - a*b^3*B - 3*a^4*C - a^2*b^
2*(9*A + 2*C))*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((11*a^2*A*b^4 - 4*A*b^6 + 6*a^5
*b*B - 2*a^3*b^3*B + a*b^5*B - 2*a^6*C - 3*a^4*b^2*(4*A + C))*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Cos[
c + d*x]))

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 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 3770

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

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx &=\frac{\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{\int \frac{\left (-4 A b^2+a b B+a^2 (3 A-C)-3 a (A b-a B+b C) \cos (c+d x)+3 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac{\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{\int \frac{\left (-23 a^2 A b^2+12 A b^4+8 a^3 b B-3 a b^3 B+a^4 (6 A-5 C)+2 a \left (A b^3+3 a^3 B+2 a b^2 B-a^2 b (6 A+5 C)\right ) \cos (c+d x)-2 \left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx}{6 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac{\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\int \frac{\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)-a \left (4 A b^5-6 a^5 B-8 a^3 b^2 B-a b^4 B-a^2 b^3 (7 A-4 C)+a^4 b (18 A+11 C)\right ) \cos (c+d x)-3 \left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^3}\\ &=\frac{\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac{\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac{\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\int \frac{\left (-6 \left (a^2-b^2\right )^3 (4 A b-a B)-3 a \left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{6 a^4 \left (a^2-b^2\right )^3}\\ &=\frac{\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac{\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac{\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac{(4 A b-a B) \int \sec (c+d x) \, dx}{a^5}-\frac{\left (35 a^4 A b^4-28 a^2 A b^6+8 A b^8+8 a^7 b B-8 a^5 b^3 B+7 a^3 b^5 B-2 a b^7 B-2 a^8 C-a^6 b^2 (20 A+3 C)\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )^3}\\ &=-\frac{(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac{\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac{\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac{\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac{\left (35 a^4 A b^4-28 a^2 A b^6+8 A b^8+8 a^7 b B-8 a^5 b^3 B+7 a^3 b^5 B-2 a b^7 B-2 a^8 C-a^6 b^2 (20 A+3 C)\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^5 \left (a^2-b^2\right )^3 d}\\ &=\frac{\left (20 a^6 A b^2-35 a^4 A b^4+28 a^2 A b^6-8 A b^8-8 a^7 b B+8 a^5 b^3 B-7 a^3 b^5 B+2 a b^7 B+2 a^8 C+3 a^6 b^2 C\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 (a-b)^{7/2} (a+b)^{7/2} d}-\frac{(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac{\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac{\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac{\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}\\ \end{align*}

Mathematica [A]  time = 4.45901, size = 709, normalized size = 1.48 \[ \frac{\cos (c+d x) \left (A \sec ^2(c+d x)+B \sec (c+d x)+C\right ) \left (\frac{2 a \sin (c+d x) \left (6 a b^2 \cos (2 (c+d x)) \left (-a^4 b^2 (53 A+C)+57 a^2 A b^4+a^6 (6 A-9 C)-15 a^3 b^3 B+20 a^5 b B+5 a b^5 B-20 A b^6\right )-b \cos (c+d x) \left (a^6 b^2 (438 A+13 C)-5 a^4 b^4 (61 A-4 C)-28 a^2 A b^6-72 a^8 (A-C)+50 a^5 b^3 B+7 a^3 b^5 B-144 a^7 b B-18 a b^7 B+72 A b^8\right )+6 a^6 A b^3 \cos (3 (c+d x))-65 a^4 A b^5 \cos (3 (c+d x))+68 a^2 A b^7 \cos (3 (c+d x))-36 a^7 A b^2-246 a^5 A b^4+318 a^3 A b^6+24 a^9 A+26 a^5 b^4 B \cos (3 (c+d x))-17 a^3 b^6 B \cos (3 (c+d x))+120 a^6 b^3 B-90 a^4 b^5 B+30 a^2 b^7 B-11 a^6 b^3 C \cos (3 (c+d x))-4 a^4 b^5 C \cos (3 (c+d x))-54 a^7 b^2 C-6 a^5 b^4 C-120 a A b^8+6 a b^8 B \cos (3 (c+d x))-24 A b^9 \cos (3 (c+d x))\right )}{\left (a^2-b^2\right )^3 (a+b \cos (c+d x))^3}+\frac{48 \cos (c+d x) \left (a^6 b^2 (20 A+3 C)-35 a^4 A b^4+28 a^2 A b^6+8 a^5 b^3 B-7 a^3 b^5 B-8 a^7 b B+2 a^8 C+2 a b^7 B-8 A b^8\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\left (b^2-a^2\right )^{7/2}}+48 (4 A b-a B) \cos (c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+48 (a B-4 A b) \cos (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{24 a^5 d (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Cos[c + d*x]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((48*(-35*a^4*A*b^4 + 28*a^2*A*b^6 - 8*A*b^8 - 8*a^7*b*B
+ 8*a^5*b^3*B - 7*a^3*b^5*B + 2*a*b^7*B + 2*a^8*C + a^6*b^2*(20*A + 3*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/
Sqrt[-a^2 + b^2]]*Cos[c + d*x])/(-a^2 + b^2)^(7/2) + 48*(4*A*b - a*B)*Cos[c + d*x]*Log[Cos[(c + d*x)/2] - Sin[
(c + d*x)/2]] + 48*(-4*A*b + a*B)*Cos[c + d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*a*(24*a^9*A - 36*
a^7*A*b^2 - 246*a^5*A*b^4 + 318*a^3*A*b^6 - 120*a*A*b^8 + 120*a^6*b^3*B - 90*a^4*b^5*B + 30*a^2*b^7*B - 54*a^7
*b^2*C - 6*a^5*b^4*C - b*(-28*a^2*A*b^6 + 72*A*b^8 - 144*a^7*b*B + 50*a^5*b^3*B + 7*a^3*b^5*B - 18*a*b^7*B - 5
*a^4*b^4*(61*A - 4*C) - 72*a^8*(A - C) + a^6*b^2*(438*A + 13*C))*Cos[c + d*x] + 6*a*b^2*(57*a^2*A*b^4 - 20*A*b
^6 + 20*a^5*b*B - 15*a^3*b^3*B + 5*a*b^5*B + a^6*(6*A - 9*C) - a^4*b^2*(53*A + C))*Cos[2*(c + d*x)] + 6*a^6*A*
b^3*Cos[3*(c + d*x)] - 65*a^4*A*b^5*Cos[3*(c + d*x)] + 68*a^2*A*b^7*Cos[3*(c + d*x)] - 24*A*b^9*Cos[3*(c + d*x
)] + 26*a^5*b^4*B*Cos[3*(c + d*x)] - 17*a^3*b^6*B*Cos[3*(c + d*x)] + 6*a*b^8*B*Cos[3*(c + d*x)] - 11*a^6*b^3*C
*Cos[3*(c + d*x)] - 4*a^4*b^5*C*Cos[3*(c + d*x)])*Sin[c + d*x])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3)))/(24*a
^5*d*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)]))

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Maple [B]  time = 0.115, size = 3628, normalized size = 7.6 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/d/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^4/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1
/2*c)^5*B-2/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*
x+1/2*c)^5*C*b^3-4/3/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^
2)*tan(1/2*d*x+1/2*c)^3*C-2/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2
-b^3)*tan(1/2*d*x+1/2*c)*C*b^3+2/d*a^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*
d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C-1/d/a^4*ln(tan(1/2*d*x+1/2*c)-1)*B+1/d/a^4*ln(tan(1/2*d*x+1/2*c)+1)*B+4/d/a^
3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^6/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)
^3*B-1/d/a^4*A/(tan(1/2*d*x+1/2*c)-1)-1/d/a^4*A/(tan(1/2*d*x+1/2*c)+1)-4/d*A*b/a^5*ln(tan(1/2*d*x+1/2*c)+1)+4/
d*A*b/a^5*ln(tan(1/2*d*x+1/2*c)-1)-3/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*
b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C*a*b^2+8/d*b^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((
a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*B+3/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)
/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C*a*b^2-1/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)^3*b^5/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B-44/3/d/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+
1/2*c)^2*b+a+b)^3*b^4/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+2/d/a^3/(a*tan(1/2*d*x+1/2*c)^2-t
an(1/2*d*x+1/2*c)^2*b+a+b)^3*b^6/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B+24/d*b^2/(a*tan(1/2*d*
x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*a*B+12/d*b^2/(a*
tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*a*B-20/d
*b^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*
A-20/d*b^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1
/2*c)^5*A+3/d*b^2*a/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(
a-b))^(1/2))*C-40/d*b^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*
tan(1/2*d*x+1/2*c)^3*A+12/d*b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b
^2+b^3)*tan(1/2*d*x+1/2*c)^5*a*B+20/d*a*b^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan
(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A-12/d*b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a^2+2*a*b
+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C*a^2-6/d*b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(
a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C*a^2-6/d*b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a
+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C*a^2-35/d/a/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(
a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A*b^4+28/d/a^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/
((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A*b^6-8/d/a^5/(a^6-3*a^4*b^2+3*a^2*b^
4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A*b^8+2/d/a^3/(a*tan(1/2*d*x+1
/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A*b^6-6/d/a^4/(a*ta
n(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^7/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A-6/d/
a^4/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^7/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*
c)^5*A+116/3/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)^3*b^5/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*t
an(1/2*d*x+1/2*c)^3*A-12/d/a^4/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^7/(a^2+2*a*b+b^2)/(a^2-
2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+5/d/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2
*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A*b^4+18/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)^3/(a+b)/
(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A*b^5-2/d/a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)
^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A*b^6-5/d/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2
*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A*b^4+18/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)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A*b^5+4/d/(a*tan(1/2*d*x+1/2*c)^2-t
an(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*b^3*B-4/d/(a*tan(1/2*d*x+1/2
*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*b^3*B+2/d/a^4/(a^6-3*a^
4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*b^7*B-7/d/a^2/(a
^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*b^5*B-8/d
*a^2*b/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*
B+1/d/a^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^5/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*
x+1/2*c)*B+2/d/a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^6/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*t
an(1/2*d*x+1/2*c)*B-6/d/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^4/(a+b)/(a^3-3*a^2*b+3*a*b^2
-b^3)*tan(1/2*d*x+1/2*c)*B

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.36395, size = 1694, normalized size = 3.53 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*(2*C*a^8 - 8*B*a^7*b + 20*A*a^6*b^2 + 3*C*a^6*b^2 + 8*B*a^5*b^3 - 35*A*a^4*b^4 - 7*B*a^3*b^5 + 28*A*a^
2*b^6 + 2*B*a*b^7 - 8*A*b^8)*(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^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*sqrt(a^2 - b^2)) + (1
8*C*a^8*b*tan(1/2*d*x + 1/2*c)^5 - 36*B*a^7*b^2*tan(1/2*d*x + 1/2*c)^5 - 27*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^5 +
60*A*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 60*B*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^6*b^3*tan(1/2*d*x + 1/2*c)^
5 - 105*A*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 - 3*C*a^5*b^4*tan(1/2*d*x + 1/2*
c)^5 - 24*A*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 - 45*B*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^4*b^5*tan(1/2*d*x + 1
/2*c)^5 + 117*A*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 - 24*A*a^2*b^7*tan(1/2*d*x
+ 1/2*c)^5 + 15*B*a^2*b^7*tan(1/2*d*x + 1/2*c)^5 - 42*A*a*b^8*tan(1/2*d*x + 1/2*c)^5 - 6*B*a*b^8*tan(1/2*d*x
+ 1/2*c)^5 + 18*A*b^9*tan(1/2*d*x + 1/2*c)^5 + 36*C*a^8*b*tan(1/2*d*x + 1/2*c)^3 - 72*B*a^7*b^2*tan(1/2*d*x +
1/2*c)^3 + 120*A*a^6*b^3*tan(1/2*d*x + 1/2*c)^3 - 32*C*a^6*b^3*tan(1/2*d*x + 1/2*c)^3 + 116*B*a^5*b^4*tan(1/2*
d*x + 1/2*c)^3 - 236*A*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 - 4*C*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 - 56*B*a^3*b^6*tan(
1/2*d*x + 1/2*c)^3 + 152*A*a^2*b^7*tan(1/2*d*x + 1/2*c)^3 + 12*B*a*b^8*tan(1/2*d*x + 1/2*c)^3 - 36*A*b^9*tan(1
/2*d*x + 1/2*c)^3 + 18*C*a^8*b*tan(1/2*d*x + 1/2*c) - 36*B*a^7*b^2*tan(1/2*d*x + 1/2*c) + 27*C*a^7*b^2*tan(1/2
*d*x + 1/2*c) + 60*A*a^6*b^3*tan(1/2*d*x + 1/2*c) - 60*B*a^6*b^3*tan(1/2*d*x + 1/2*c) + 6*C*a^6*b^3*tan(1/2*d*
x + 1/2*c) + 105*A*a^5*b^4*tan(1/2*d*x + 1/2*c) + 6*B*a^5*b^4*tan(1/2*d*x + 1/2*c) + 3*C*a^5*b^4*tan(1/2*d*x +
1/2*c) - 24*A*a^4*b^5*tan(1/2*d*x + 1/2*c) + 45*B*a^4*b^5*tan(1/2*d*x + 1/2*c) + 6*C*a^4*b^5*tan(1/2*d*x + 1/
2*c) - 117*A*a^3*b^6*tan(1/2*d*x + 1/2*c) + 6*B*a^3*b^6*tan(1/2*d*x + 1/2*c) - 24*A*a^2*b^7*tan(1/2*d*x + 1/2*
c) - 15*B*a^2*b^7*tan(1/2*d*x + 1/2*c) + 42*A*a*b^8*tan(1/2*d*x + 1/2*c) - 6*B*a*b^8*tan(1/2*d*x + 1/2*c) + 18
*A*b^9*tan(1/2*d*x + 1/2*c))/((a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d
*x + 1/2*c)^2 + a + b)^3) - 3*(B*a - 4*A*b)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^5 + 3*(B*a - 4*A*b)*log(abs(t
an(1/2*d*x + 1/2*c) - 1))/a^5 + 6*A*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 - 1)*a^4))/d