∫ cos2 (c+dx)(A+B sec(c+dx)+C sec2(c+dx))______________________________

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

Optimal. Leaf size=453 \[ \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {x \left (a^2 (A+2 C)-6 a b B+12 A b^2\right )}{2 a^5}+\frac {\sin (c+d x) \cos (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{2 a^3 d \left (a^2-b^2\right )^2}+\frac {\sin (c+d x) \cos (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac {\sin (c+d x) \left (-2 a^5 B+a^4 b (6 A-5 C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right )}{2 a^4 d \left (a^2-b^2\right )^2}-\frac {b \left (6 a^6 C-12 a^5 b B+5 a^4 b^2 (4 A-C)+15 a^3 b^3 B-a^2 b^4 (29 A-2 C)-6 a b^5 B+12 A b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d (a-b)^{5/2} (a+b)^{5/2}} \]

[Out]

1/2*(12*A*b^2-6*a*b*B+a^2*(A+2*C))*x/a^5-b*(12*A*b^6-12*a^5*b*B+15*a^3*b^3*B-6*a*b^5*B-a^2*b^4*(29*A-2*C)+5*a^

4*b^2*(4*A-C)+6*a^6*C)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/(a-b)^(5/2)/(a+b)^(5/2)/d-1/2*(

12*A*b^5-2*a^5*B+11*a^3*b^2*B-6*a*b^4*B+a^4*b*(6*A-5*C)-a^2*b^3*(21*A-2*C))*sin(d*x+c)/a^4/(a^2-b^2)^2/d+1/2*(

6*A*b^4+6*a^3*b*B-3*a*b^3*B+a^4*(A-4*C)-a^2*b^2*(10*A-C))*cos(d*x+c)*sin(d*x+c)/a^3/(a^2-b^2)^2/d+1/2*(A*b^2-a

*(B*b-C*a))*cos(d*x+c)*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))^2+1/2*(7*A*a^2*b^2-4*A*b^4-5*B*a^3*b+2*B*a*b^

3+3*C*a^4)*cos(d*x+c)*sin(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))

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Rubi [A] time = 4.82, antiderivative size = 453, 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 = {4100, 4104, 3919, 3831, 2659, 208} \[ -\frac {\sin (c+d x) \left (-a^2 b^3 (21 A-2 C)+a^4 b (6 A-5 C)+11 a^3 b^2 B-2 a^5 B-6 a b^4 B+12 A b^5\right )}{2 a^4 d \left (a^2-b^2\right )^2}+\frac {\sin (c+d x) \cos (c+d x) \left (-a^2 b^2 (10 A-C)+a^4 (A-4 C)+6 a^3 b B-3 a b^3 B+6 A b^4\right )}{2 a^3 d \left (a^2-b^2\right )^2}-\frac {b \left (5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+15 a^3 b^3 B-12 a^5 b B+6 a^6 C-6 a b^5 B+12 A b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d (a-b)^{5/2} (a+b)^{5/2}}+\frac {\sin (c+d x) \cos (c+d x) \left (7 a^2 A b^2-5 a^3 b B+3 a^4 C+2 a b^3 B-4 A b^4\right )}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {x \left (a^2 (A+2 C)-6 a b B+12 A b^2\right )}{2 a^5} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((12*A*b^2 - 6*a*b*B + a^2*(A + 2*C))*x)/(2*a^5) - (b*(12*A*b^6 - 12*a^5*b*B + 15*a^3*b^3*B - 6*a*b^5*B - a^2*

b^4*(29*A - 2*C) + 5*a^4*b^2*(4*A - C) + 6*a^6*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a

- b)^(5/2)*(a + b)^(5/2)*d) - ((12*A*b^5 - 2*a^5*B + 11*a^3*b^2*B - 6*a*b^4*B + a^4*b*(6*A - 5*C) - a^2*b^3*(

21*A - 2*C))*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((6*A*b^4 + 6*a^3*b*B - 3*a*b^3*B + a^4*(A - 4*C) - a^2*b

^2*(10*A - C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]*Sin[

c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((7*a^2*A*b^2 - 4*A*b^4 - 5*a^3*b*B + 2*a*b^3*B + 3*a^4

*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))

Rule 208

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

b}, x] && NegQ[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 3831

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e

+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3919

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,

x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[

b*c - a*d, 0]

Rule 4100

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^

(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[((A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a +

b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n)/(a*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), I

nt[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*

(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x

], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && !(ILtQ[m + 1/2, 0] &

& ILtQ[n, 0])

Rule 4104

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^

(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m +

1)*(d*Csc[e + f*x])^n)/(a*f*n), x] + Dist[1/(a*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[

a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ

[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]

Rubi steps

\begin {align*} \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx &=\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\int \frac {\cos ^2(c+d x) \left (2 \left (2 A b^2-a b B-a^2 (A-C)\right )+2 a (A b-a B+b C) \sec (c+d x)-3 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {\cos ^2(c+d x) \left (2 \left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right )+a \left (A b^3+2 a^3 B+a b^2 B-a^2 b (4 A+3 C)\right ) \sec (c+d x)+2 \left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (2 \left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right )+2 a \left (2 A b^4+4 a^3 b B-a b^3 B-a^2 b^2 (4 A+C)-a^4 (A+2 C)\right ) \sec (c+d x)-2 b \left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {2 \left (a^2-b^2\right )^2 \left (12 A b^2-6 a b B+a^2 (A+2 C)\right )+2 a b \left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{4 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {\left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) x}{2 a^5}-\frac {\left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (b \left (12 A b^6-12 a^5 b B+15 a^3 b^3 B-6 a b^5 B-a^2 b^4 (29 A-2 C)+5 a^4 b^2 (4 A-C)+6 a^6 C\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )^2}\\ &=\frac {\left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) x}{2 a^5}-\frac {\left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (12 A b^6-12 a^5 b B+15 a^3 b^3 B-6 a b^5 B-a^2 b^4 (29 A-2 C)+5 a^4 b^2 (4 A-C)+6 a^6 C\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 a^5 \left (a^2-b^2\right )^2}\\ &=\frac {\left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) x}{2 a^5}-\frac {\left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (12 A b^6-12 a^5 b B+15 a^3 b^3 B-6 a b^5 B-a^2 b^4 (29 A-2 C)+5 a^4 b^2 (4 A-C)+6 a^6 C\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 \left (a^2-b^2\right )^2 d}\\ &=\frac {\left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) x}{2 a^5}-\frac {b \left (20 a^4 A b^2-29 a^2 A b^4+12 A b^6-12 a^5 b B+15 a^3 b^3 B-6 a b^5 B+6 a^6 C-5 a^4 b^2 C+2 a^2 b^4 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{5/2} (a+b)^{5/2} d}-\frac {\left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}\\ \end {align*}

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Mathematica [A] time = 4.92, size = 881, normalized size = 1.94 \[ \frac {\frac {16 b \left (6 C a^6-12 b B a^5+5 b^2 (4 A-C) a^4+15 b^3 B a^3+b^4 (2 C-29 A) a^2-6 b^5 B a+12 A b^6\right ) \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {4 A c a^8+8 c C a^8+4 A d x a^8+8 C d x a^8+4 B \sin (c+d x) a^8+2 A \sin (2 (c+d x)) a^8+4 B \sin (3 (c+d x)) a^8+A \sin (4 (c+d x)) a^8-24 b B c a^7-24 b B d x a^7-8 A b \sin (c+d x) a^7+16 b B \sin (2 (c+d x)) a^7-8 A b \sin (3 (c+d x)) a^7+48 A b^2 c a^6+48 A b^2 d x a^6+8 b^2 B \sin (c+d x) a^6-48 A b^2 \sin (2 (c+d x)) a^6+24 b^2 C \sin (2 (c+d x)) a^6-8 b^2 B \sin (3 (c+d x)) a^6-2 A b^2 \sin (4 (c+d x)) a^6-32 A b^3 \sin (c+d x) a^5+40 b^3 C \sin (c+d x) a^5-64 b^3 B \sin (2 (c+d x)) a^5+16 A b^3 \sin (3 (c+d x)) a^5-12 A b^4 c a^4-24 b^4 c C a^4-12 A b^4 d x a^4-24 b^4 C d x a^4-84 b^4 B \sin (c+d x) a^4+130 A b^4 \sin (2 (c+d x)) a^4-12 b^4 C \sin (2 (c+d x)) a^4+4 b^4 B \sin (3 (c+d x)) a^4+A b^4 \sin (4 (c+d x)) a^4+72 b^5 B c a^3+72 b^5 B d x a^3+160 A b^5 \sin (c+d x) a^3-16 b^5 C \sin (c+d x) a^3+36 b^5 B \sin (2 (c+d x)) a^3-8 A b^5 \sin (3 (c+d x)) a^3-136 A b^6 c a^2+16 b^6 c C a^2-136 A b^6 d x a^2+16 b^6 C d x a^2+48 b^6 B \sin (c+d x) a^2-72 A b^6 \sin (2 (c+d x)) a^2-48 b^7 B c a-48 b^7 B d x a+16 b \left (a^2-b^2\right )^2 \left ((A+2 C) a^2-6 b B a+12 A b^2\right ) (c+d x) \cos (c+d x) a-96 A b^7 \sin (c+d x) a+96 A b^8 c+96 A b^8 d x+4 \left (a^3-a b^2\right )^2 \left ((A+2 C) a^2-6 b B a+12 A b^2\right ) (c+d x) \cos (2 (c+d x))}{\left (a^2-b^2\right )^2 (b+a \cos (c+d x))^2}}{16 a^5 d} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((16*b*(12*A*b^6 - 12*a^5*b*B + 15*a^3*b^3*B - 6*a*b^5*B + 5*a^4*b^2*(4*A - C) + 6*a^6*C + a^2*b^4*(-29*A + 2*

C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (4*a^8*A*c + 48*a^6*A*b^2*c - 12

*a^4*A*b^4*c - 136*a^2*A*b^6*c + 96*A*b^8*c - 24*a^7*b*B*c + 72*a^3*b^5*B*c - 48*a*b^7*B*c + 8*a^8*c*C - 24*a^

4*b^4*c*C + 16*a^2*b^6*c*C + 4*a^8*A*d*x + 48*a^6*A*b^2*d*x - 12*a^4*A*b^4*d*x - 136*a^2*A*b^6*d*x + 96*A*b^8*

d*x - 24*a^7*b*B*d*x + 72*a^3*b^5*B*d*x - 48*a*b^7*B*d*x + 8*a^8*C*d*x - 24*a^4*b^4*C*d*x + 16*a^2*b^6*C*d*x +

16*a*b*(a^2 - b^2)^2*(12*A*b^2 - 6*a*b*B + a^2*(A + 2*C))*(c + d*x)*Cos[c + d*x] + 4*(a^3 - a*b^2)^2*(12*A*b^

2 - 6*a*b*B + a^2*(A + 2*C))*(c + d*x)*Cos[2*(c + d*x)] - 8*a^7*A*b*Sin[c + d*x] - 32*a^5*A*b^3*Sin[c + d*x] +

160*a^3*A*b^5*Sin[c + d*x] - 96*a*A*b^7*Sin[c + d*x] + 4*a^8*B*Sin[c + d*x] + 8*a^6*b^2*B*Sin[c + d*x] - 84*a

^4*b^4*B*Sin[c + d*x] + 48*a^2*b^6*B*Sin[c + d*x] + 40*a^5*b^3*C*Sin[c + d*x] - 16*a^3*b^5*C*Sin[c + d*x] + 2*

a^8*A*Sin[2*(c + d*x)] - 48*a^6*A*b^2*Sin[2*(c + d*x)] + 130*a^4*A*b^4*Sin[2*(c + d*x)] - 72*a^2*A*b^6*Sin[2*(

c + d*x)] + 16*a^7*b*B*Sin[2*(c + d*x)] - 64*a^5*b^3*B*Sin[2*(c + d*x)] + 36*a^3*b^5*B*Sin[2*(c + d*x)] + 24*a

^6*b^2*C*Sin[2*(c + d*x)] - 12*a^4*b^4*C*Sin[2*(c + d*x)] - 8*a^7*A*b*Sin[3*(c + d*x)] + 16*a^5*A*b^3*Sin[3*(c

+ d*x)] - 8*a^3*A*b^5*Sin[3*(c + d*x)] + 4*a^8*B*Sin[3*(c + d*x)] - 8*a^6*b^2*B*Sin[3*(c + d*x)] + 4*a^4*b^4*

B*Sin[3*(c + d*x)] + a^8*A*Sin[4*(c + d*x)] - 2*a^6*A*b^2*Sin[4*(c + d*x)] + a^4*A*b^4*Sin[4*(c + d*x)])/((a^2

- b^2)^2*(b + a*Cos[c + d*x])^2))/(16*a^5*d)

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fricas [B] time = 0.76, size = 2127, normalized size = 4.70 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*((A + 2*C)*a^10 - 6*B*a^9*b + 3*(3*A - 2*C)*a^8*b^2 + 18*B*a^7*b^3 - 3*(11*A - 2*C)*a^6*b^4 - 18*B*a^5

*b^5 + (35*A - 2*C)*a^4*b^6 + 6*B*a^3*b^7 - 12*A*a^2*b^8)*d*x*cos(d*x + c)^2 + 4*((A + 2*C)*a^9*b - 6*B*a^8*b^

2 + 3*(3*A - 2*C)*a^7*b^3 + 18*B*a^6*b^4 - 3*(11*A - 2*C)*a^5*b^5 - 18*B*a^4*b^6 + (35*A - 2*C)*a^3*b^7 + 6*B*

a^2*b^8 - 12*A*a*b^9)*d*x*cos(d*x + c) + 2*((A + 2*C)*a^8*b^2 - 6*B*a^7*b^3 + 3*(3*A - 2*C)*a^6*b^4 + 18*B*a^5

*b^5 - 3*(11*A - 2*C)*a^4*b^6 - 18*B*a^3*b^7 + (35*A - 2*C)*a^2*b^8 + 6*B*a*b^9 - 12*A*b^10)*d*x + (6*C*a^6*b^

3 - 12*B*a^5*b^4 + 5*(4*A - C)*a^4*b^5 + 15*B*a^3*b^6 - (29*A - 2*C)*a^2*b^7 - 6*B*a*b^8 + 12*A*b^9 + (6*C*a^8

*b - 12*B*a^7*b^2 + 5*(4*A - C)*a^6*b^3 + 15*B*a^5*b^4 - (29*A - 2*C)*a^4*b^5 - 6*B*a^3*b^6 + 12*A*a^2*b^7)*co

s(d*x + c)^2 + 2*(6*C*a^7*b^2 - 12*B*a^6*b^3 + 5*(4*A - C)*a^5*b^4 + 15*B*a^4*b^5 - (29*A - 2*C)*a^3*b^6 - 6*B

*a^2*b^7 + 12*A*a*b^8)*cos(d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 -

2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) +

b^2)) + 2*(2*B*a^8*b^2 - (6*A - 5*C)*a^7*b^3 - 13*B*a^6*b^4 + (27*A - 7*C)*a^5*b^5 + 17*B*a^4*b^6 - (33*A - 2*

C)*a^3*b^7 - 6*B*a^2*b^8 + 12*A*a*b^9 + (A*a^10 - 3*A*a^8*b^2 + 3*A*a^6*b^4 - A*a^4*b^6)*cos(d*x + c)^3 + 2*(B

*a^10 - 2*A*a^9*b - 3*B*a^8*b^2 + 6*A*a^7*b^3 + 3*B*a^6*b^4 - 6*A*a^5*b^5 - B*a^4*b^6 + 2*A*a^3*b^7)*cos(d*x +

c)^2 + (4*B*a^9*b - (11*A - 6*C)*a^8*b^2 - 20*B*a^7*b^3 + (43*A - 9*C)*a^6*b^4 + 25*B*a^5*b^5 - (50*A - 3*C)*

a^4*b^6 - 9*B*a^3*b^7 + 18*A*a^2*b^8)*cos(d*x + c))*sin(d*x + c))/((a^13 - 3*a^11*b^2 + 3*a^9*b^4 - a^7*b^6)*d

*cos(d*x + c)^2 + 2*(a^12*b - 3*a^10*b^3 + 3*a^8*b^5 - a^6*b^7)*d*cos(d*x + c) + (a^11*b^2 - 3*a^9*b^4 + 3*a^7

*b^6 - a^5*b^8)*d), 1/2*(((A + 2*C)*a^10 - 6*B*a^9*b + 3*(3*A - 2*C)*a^8*b^2 + 18*B*a^7*b^3 - 3*(11*A - 2*C)*a

^6*b^4 - 18*B*a^5*b^5 + (35*A - 2*C)*a^4*b^6 + 6*B*a^3*b^7 - 12*A*a^2*b^8)*d*x*cos(d*x + c)^2 + 2*((A + 2*C)*a

^9*b - 6*B*a^8*b^2 + 3*(3*A - 2*C)*a^7*b^3 + 18*B*a^6*b^4 - 3*(11*A - 2*C)*a^5*b^5 - 18*B*a^4*b^6 + (35*A - 2*

C)*a^3*b^7 + 6*B*a^2*b^8 - 12*A*a*b^9)*d*x*cos(d*x + c) + ((A + 2*C)*a^8*b^2 - 6*B*a^7*b^3 + 3*(3*A - 2*C)*a^6

*b^4 + 18*B*a^5*b^5 - 3*(11*A - 2*C)*a^4*b^6 - 18*B*a^3*b^7 + (35*A - 2*C)*a^2*b^8 + 6*B*a*b^9 - 12*A*b^10)*d*

x - (6*C*a^6*b^3 - 12*B*a^5*b^4 + 5*(4*A - C)*a^4*b^5 + 15*B*a^3*b^6 - (29*A - 2*C)*a^2*b^7 - 6*B*a*b^8 + 12*A

*b^9 + (6*C*a^8*b - 12*B*a^7*b^2 + 5*(4*A - C)*a^6*b^3 + 15*B*a^5*b^4 - (29*A - 2*C)*a^4*b^5 - 6*B*a^3*b^6 + 1

2*A*a^2*b^7)*cos(d*x + c)^2 + 2*(6*C*a^7*b^2 - 12*B*a^6*b^3 + 5*(4*A - C)*a^5*b^4 + 15*B*a^4*b^5 - (29*A - 2*C

)*a^3*b^6 - 6*B*a^2*b^7 + 12*A*a*b^8)*cos(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c)

+ a)/((a^2 - b^2)*sin(d*x + c))) + (2*B*a^8*b^2 - (6*A - 5*C)*a^7*b^3 - 13*B*a^6*b^4 + (27*A - 7*C)*a^5*b^5 +

17*B*a^4*b^6 - (33*A - 2*C)*a^3*b^7 - 6*B*a^2*b^8 + 12*A*a*b^9 + (A*a^10 - 3*A*a^8*b^2 + 3*A*a^6*b^4 - A*a^4*b

^6)*cos(d*x + c)^3 + 2*(B*a^10 - 2*A*a^9*b - 3*B*a^8*b^2 + 6*A*a^7*b^3 + 3*B*a^6*b^4 - 6*A*a^5*b^5 - B*a^4*b^6

+ 2*A*a^3*b^7)*cos(d*x + c)^2 + (4*B*a^9*b - (11*A - 6*C)*a^8*b^2 - 20*B*a^7*b^3 + (43*A - 9*C)*a^6*b^4 + 25*

B*a^5*b^5 - (50*A - 3*C)*a^4*b^6 - 9*B*a^3*b^7 + 18*A*a^2*b^8)*cos(d*x + c))*sin(d*x + c))/((a^13 - 3*a^11*b^2

+ 3*a^9*b^4 - a^7*b^6)*d*cos(d*x + c)^2 + 2*(a^12*b - 3*a^10*b^3 + 3*a^8*b^5 - a^6*b^7)*d*cos(d*x + c) + (a^1

1*b^2 - 3*a^9*b^4 + 3*a^7*b^6 - a^5*b^8)*d)]

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giac [B] time = 10.01, size = 3408, normalized size = 7.52 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(((a^6 - a^5*b + 10*a^4*b^2 + 10*a^3*b^3 - 23*a^2*b^4 - 6*a*b^5 + 12*b^6)*sqrt(-a^2 + b^2)*A*abs(a^9 - 2*

a^7*b^2 + a^5*b^4)*abs(-a + b) - 3*(2*a^5*b + 2*a^4*b^2 - 4*a^3*b^3 - a^2*b^4 + 2*a*b^5)*sqrt(-a^2 + b^2)*B*ab

s(a^9 - 2*a^7*b^2 + a^5*b^4)*abs(-a + b) + (2*a^6 + 4*a^5*b - 4*a^4*b^2 - a^3*b^3 + 2*a^2*b^4)*sqrt(-a^2 + b^2

)*C*abs(a^9 - 2*a^7*b^2 + a^5*b^4)*abs(-a + b) - (a^15 - a^14*b + 8*a^13*b^2 - 28*a^12*b^3 - 42*a^11*b^4 + 111

*a^10*b^5 + 68*a^9*b^6 - 158*a^8*b^7 - 47*a^7*b^8 + 100*a^6*b^9 + 12*a^5*b^10 - 24*a^4*b^11)*sqrt(-a^2 + b^2)*

A*abs(-a + b) + 3*(2*a^14*b - 6*a^13*b^2 - 8*a^12*b^3 + 21*a^11*b^4 + 12*a^10*b^5 - 28*a^9*b^6 - 8*a^8*b^7 + 1

7*a^7*b^8 + 2*a^6*b^9 - 4*a^5*b^10)*sqrt(-a^2 + b^2)*B*abs(-a + b) - (2*a^15 - 8*a^14*b - 8*a^13*b^2 + 25*a^12

*b^3 + 12*a^11*b^4 - 30*a^10*b^5 - 8*a^9*b^6 + 17*a^8*b^7 + 2*a^7*b^8 - 4*a^6*b^9)*sqrt(-a^2 + b^2)*C*abs(-a +

b))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(tan(1/2*d*x + 1/2*c)/sqrt(-(a^8*b - 2*a^6*b^3 + a^4*b^5 + sqrt

((a^9 + a^8*b - 2*a^7*b^2 - 2*a^6*b^3 + a^5*b^4 + a^4*b^5)*(a^9 - a^8*b - 2*a^7*b^2 + 2*a^6*b^3 + a^5*b^4 - a^

4*b^5) + (a^8*b - 2*a^6*b^3 + a^4*b^5)^2))/(a^9 - a^8*b - 2*a^7*b^2 + 2*a^6*b^3 + a^5*b^4 - a^4*b^5))))/((a^9

- 2*a^7*b^2 + a^5*b^4)^2*(a^2 - 2*a*b + b^2) + (a^10*b - 2*a^9*b^2 - a^8*b^3 + 4*a^7*b^4 - a^6*b^5 - 2*a^5*b^6

+ a^4*b^7)*abs(a^9 - 2*a^7*b^2 + a^5*b^4)) + (A*a^15 + 2*C*a^15 - A*a^14*b - 6*B*a^14*b - 8*C*a^14*b + 8*A*a^

13*b^2 + 18*B*a^13*b^2 - 8*C*a^13*b^2 - 28*A*a^12*b^3 + 24*B*a^12*b^3 + 25*C*a^12*b^3 - 42*A*a^11*b^4 - 63*B*a

^11*b^4 + 12*C*a^11*b^4 + 111*A*a^10*b^5 - 36*B*a^10*b^5 - 30*C*a^10*b^5 + 68*A*a^9*b^6 + 84*B*a^9*b^6 - 8*C*a

^9*b^6 - 158*A*a^8*b^7 + 24*B*a^8*b^7 + 17*C*a^8*b^7 - 47*A*a^7*b^8 - 51*B*a^7*b^8 + 2*C*a^7*b^8 + 100*A*a^6*b

^9 - 6*B*a^6*b^9 - 4*C*a^6*b^9 + 12*A*a^5*b^10 + 12*B*a^5*b^10 - 24*A*a^4*b^11 + A*a^6*abs(a^9 - 2*a^7*b^2 + a

^5*b^4) + 2*C*a^6*abs(a^9 - 2*a^7*b^2 + a^5*b^4) - A*a^5*b*abs(a^9 - 2*a^7*b^2 + a^5*b^4) - 6*B*a^5*b*abs(a^9

- 2*a^7*b^2 + a^5*b^4) + 4*C*a^5*b*abs(a^9 - 2*a^7*b^2 + a^5*b^4) + 10*A*a^4*b^2*abs(a^9 - 2*a^7*b^2 + a^5*b^4

) - 6*B*a^4*b^2*abs(a^9 - 2*a^7*b^2 + a^5*b^4) - 4*C*a^4*b^2*abs(a^9 - 2*a^7*b^2 + a^5*b^4) + 10*A*a^3*b^3*abs

(a^9 - 2*a^7*b^2 + a^5*b^4) + 12*B*a^3*b^3*abs(a^9 - 2*a^7*b^2 + a^5*b^4) - C*a^3*b^3*abs(a^9 - 2*a^7*b^2 + a^

5*b^4) - 23*A*a^2*b^4*abs(a^9 - 2*a^7*b^2 + a^5*b^4) + 3*B*a^2*b^4*abs(a^9 - 2*a^7*b^2 + a^5*b^4) + 2*C*a^2*b^

4*abs(a^9 - 2*a^7*b^2 + a^5*b^4) - 6*A*a*b^5*abs(a^9 - 2*a^7*b^2 + a^5*b^4) - 6*B*a*b^5*abs(a^9 - 2*a^7*b^2 +

a^5*b^4) + 12*A*b^6*abs(a^9 - 2*a^7*b^2 + a^5*b^4))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(tan(1/2*d*x + 1

/2*c)/sqrt(-(a^8*b - 2*a^6*b^3 + a^4*b^5 - sqrt((a^9 + a^8*b - 2*a^7*b^2 - 2*a^6*b^3 + a^5*b^4 + a^4*b^5)*(a^9

- a^8*b - 2*a^7*b^2 + 2*a^6*b^3 + a^5*b^4 - a^4*b^5) + (a^8*b - 2*a^6*b^3 + a^4*b^5)^2))/(a^9 - a^8*b - 2*a^7

*b^2 + 2*a^6*b^3 + a^5*b^4 - a^4*b^5))))/(a^8*b*abs(a^9 - 2*a^7*b^2 + a^5*b^4) - 2*a^6*b^3*abs(a^9 - 2*a^7*b^2

+ a^5*b^4) + a^4*b^5*abs(a^9 - 2*a^7*b^2 + a^5*b^4) - (a^9 - 2*a^7*b^2 + a^5*b^4)^2) + 2*(A*a^7*tan(1/2*d*x +

1/2*c)^7 - 2*B*a^7*tan(1/2*d*x + 1/2*c)^7 + 4*A*a^6*b*tan(1/2*d*x + 1/2*c)^7 + 4*B*a^6*b*tan(1/2*d*x + 1/2*c)

^7 - 13*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 + 2*B*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 + 6*C*a^5*b^2*tan(1/2*d*x + 1/2*

c)^7 - 2*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 - 16*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 - 5*C*a^4*b^3*tan(1/2*d*x + 1/

2*c)^7 + 33*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 + 9*B*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 - 3*C*a^3*b^4*tan(1/2*d*x +

1/2*c)^7 - 17*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 + 9*B*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 + 2*C*a^2*b^5*tan(1/2*d*x

+ 1/2*c)^7 - 18*A*a*b^6*tan(1/2*d*x + 1/2*c)^7 - 6*B*a*b^6*tan(1/2*d*x + 1/2*c)^7 + 12*A*b^7*tan(1/2*d*x + 1/2

*c)^7 - 3*A*a^7*tan(1/2*d*x + 1/2*c)^5 + 2*B*a^7*tan(1/2*d*x + 1/2*c)^5 - 4*A*a^6*b*tan(1/2*d*x + 1/2*c)^5 + 4

*B*a^6*b*tan(1/2*d*x + 1/2*c)^5 - 5*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 - 10*B*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 + 6

*C*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 + 26*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 - 16*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^5

- 15*C*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 + 29*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 + 35*B*a^3*b^4*tan(1/2*d*x + 1/2*c

)^5 - 3*C*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 - 67*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 + 9*B*a^2*b^5*tan(1/2*d*x + 1/2

*c)^5 + 6*C*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 - 18*A*a*b^6*tan(1/2*d*x + 1/2*c)^5 - 18*B*a*b^6*tan(1/2*d*x + 1/2*

c)^5 + 36*A*b^7*tan(1/2*d*x + 1/2*c)^5 + 3*A*a^7*tan(1/2*d*x + 1/2*c)^3 + 2*B*a^7*tan(1/2*d*x + 1/2*c)^3 - 4*A

*a^6*b*tan(1/2*d*x + 1/2*c)^3 - 4*B*a^6*b*tan(1/2*d*x + 1/2*c)^3 + 5*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 - 10*B*a

^5*b^2*tan(1/2*d*x + 1/2*c)^3 - 6*C*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 + 26*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 + 16*

B*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 - 15*C*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 - 29*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 +

35*B*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 + 3*C*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 - 67*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^

3 - 9*B*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 + 6*C*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 + 18*A*a*b^6*tan(1/2*d*x + 1/2*c)^

3 - 18*B*a*b^6*tan(1/2*d*x + 1/2*c)^3 + 36*A*b^7*tan(1/2*d*x + 1/2*c)^3 - A*a^7*tan(1/2*d*x + 1/2*c) - 2*B*a^7

*tan(1/2*d*x + 1/2*c) + 4*A*a^6*b*tan(1/2*d*x + 1/2*c) - 4*B*a^6*b*tan(1/2*d*x + 1/2*c) + 13*A*a^5*b^2*tan(1/2

*d*x + 1/2*c) + 2*B*a^5*b^2*tan(1/2*d*x + 1/2*c) - 6*C*a^5*b^2*tan(1/2*d*x + 1/2*c) - 2*A*a^4*b^3*tan(1/2*d*x

+ 1/2*c) + 16*B*a^4*b^3*tan(1/2*d*x + 1/2*c) - 5*C*a^4*b^3*tan(1/2*d*x + 1/2*c) - 33*A*a^3*b^4*tan(1/2*d*x + 1

/2*c) + 9*B*a^3*b^4*tan(1/2*d*x + 1/2*c) + 3*C*a^3*b^4*tan(1/2*d*x + 1/2*c) - 17*A*a^2*b^5*tan(1/2*d*x + 1/2*c

) - 9*B*a^2*b^5*tan(1/2*d*x + 1/2*c) + 2*C*a^2*b^5*tan(1/2*d*x + 1/2*c) + 18*A*a*b^6*tan(1/2*d*x + 1/2*c) - 6*

B*a*b^6*tan(1/2*d*x + 1/2*c) + 12*A*b^7*tan(1/2*d*x + 1/2*c))/((a^8 - 2*a^6*b^2 + a^4*b^4)*(a*tan(1/2*d*x + 1/

2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*b*tan(1/2*d*x + 1/2*c)^2 - a - b)^2))/d

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maple [B] time = 1.14, size = 2206, normalized size = 4.87 \[ \text {Expression too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/d*b/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*C*a+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-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*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-b)^2*tan(1/2*d*x+1/2*c)*B+10/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/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A*b^4-1/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^

2*tan(1/2*d*x+1/2*c)^3*A+1/d/a^3*arctan(tan(1/2*d*x+1/2*c))*A-10/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/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A*b^4-6/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*C*b^2+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-b)^2*tan(1/2*d*x+1/2*c)*C*b^2+2/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^2*B*tan(1/2*d*x+1/2*c)^3+1/d/

a^3/(1+tan(1/2*d*x+1/2*c)^2)^2*A*tan(1/2*d*x+1/2*c)+2/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^2*B*tan(1/2*d*x+1/2*c)-1/

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)^2*b^5/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A-1/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)^2*b^5/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+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*b^6/a^4/(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*A+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-2/d*b^4/a^2/(a*ta

n(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-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-6/d*b^6/a^4/(a*tan(1/2*d*x+1/2*c)^2-t

an(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)*A-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-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+29/d/a^3/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/

2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A*b^5-20/d/a/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*

arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A*b^3-15/d*b^4/a^2/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/

2)*arctanh(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)

*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B+5/d*b^3/a/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arc

tanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*C-12/d*b^7/a^5/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arct

anh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A-2/d*b^5/a^3/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan

h(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*C+12/d*b^2/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctanh(tan

(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B-6/d/a^4*arctan(tan(1/2*d*x+1/2*c))*B*b+12/d/a^5*arctan(tan(1/2*d*

x+1/2*c))*A*b^2+2/d/a^3*arctan(tan(1/2*d*x+1/2*c))*C-6/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3*A

*b-6/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)*A*b

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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(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(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*a^2-4*b^2>0)', see `assume?`

for more details)Is 4*a^2-4*b^2 positive or negative?

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mupad [B] time = 21.93, size = 16016, normalized size = 35.36 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((tan(c/2 + (d*x)/2)^5*(3*A*a^7 - 36*A*b^7 - 2*B*a^7 + 67*A*a^2*b^5 - 29*A*a^3*b^4 - 26*A*a^4*b^3 + 5*A*a^5*b^

2 - 9*B*a^2*b^5 - 35*B*a^3*b^4 + 16*B*a^4*b^3 + 10*B*a^5*b^2 - 6*C*a^2*b^5 + 3*C*a^3*b^4 + 15*C*a^4*b^3 - 6*C*

a^5*b^2 + 18*A*a*b^6 + 4*A*a^6*b + 18*B*a*b^6 - 4*B*a^6*b))/((a + b)^2*(a^6 - 2*a^5*b + a^4*b^2)) - (tan(c/2 +

(d*x)/2)^3*(3*A*a^7 + 36*A*b^7 + 2*B*a^7 - 67*A*a^2*b^5 - 29*A*a^3*b^4 + 26*A*a^4*b^3 + 5*A*a^5*b^2 - 9*B*a^2

*b^5 + 35*B*a^3*b^4 + 16*B*a^4*b^3 - 10*B*a^5*b^2 + 6*C*a^2*b^5 + 3*C*a^3*b^4 - 15*C*a^4*b^3 - 6*C*a^5*b^2 + 1

8*A*a*b^6 - 4*A*a^6*b - 18*B*a*b^6 - 4*B*a^6*b))/((a + b)^2*(a^6 - 2*a^5*b + a^4*b^2)) + (tan(c/2 + (d*x)/2)^7

*(A*a^6 - 12*A*b^6 - 2*B*a^6 + 23*A*a^2*b^4 - 10*A*a^3*b^3 - 8*A*a^4*b^2 - 3*B*a^2*b^4 - 12*B*a^3*b^3 + 4*B*a^

4*b^2 - 2*C*a^2*b^4 + C*a^3*b^3 + 6*C*a^4*b^2 + 6*A*a*b^5 + 5*A*a^5*b + 6*B*a*b^5 + 2*B*a^5*b))/((a^4*b - a^5)

*(a + b)^2) + (tan(c/2 + (d*x)/2)*(A*a^6 - 12*A*b^6 + 2*B*a^6 + 23*A*a^2*b^4 + 10*A*a^3*b^3 - 8*A*a^4*b^2 + 3*

B*a^2*b^4 - 12*B*a^3*b^3 - 4*B*a^4*b^2 - 2*C*a^2*b^4 - C*a^3*b^3 + 6*C*a^4*b^2 - 6*A*a*b^5 - 5*A*a^5*b + 6*B*a

*b^5 + 2*B*a^5*b))/((a + b)*(a^6 - 2*a^5*b + a^4*b^2)))/(d*(2*a*b - tan(c/2 + (d*x)/2)^4*(2*a^2 - 6*b^2) + tan

(c/2 + (d*x)/2)^2*(4*a*b + 4*b^2) - tan(c/2 + (d*x)/2)^6*(4*a*b - 4*b^2) + tan(c/2 + (d*x)/2)^8*(a^2 - 2*a*b +

b^2) + a^2 + b^2)) - (atan(((((((4*(4*A*a^21 + 8*C*a^21 - 48*A*a^10*b^11 + 24*A*a^11*b^10 + 212*A*a^12*b^9 -

100*A*a^13*b^8 - 360*A*a^14*b^7 + 164*A*a^15*b^6 + 276*A*a^16*b^5 - 120*A*a^17*b^4 - 80*A*a^18*b^3 + 28*A*a^19

*b^2 + 24*B*a^11*b^10 - 12*B*a^12*b^9 - 108*B*a^13*b^8 + 48*B*a^14*b^7 + 192*B*a^15*b^6 - 84*B*a^16*b^5 - 156*

B*a^17*b^4 + 72*B*a^18*b^3 + 48*B*a^19*b^2 - 8*C*a^12*b^9 + 4*C*a^13*b^8 + 36*C*a^14*b^7 - 8*C*a^15*b^6 - 72*C

*a^16*b^5 + 12*C*a^17*b^4 + 68*C*a^18*b^3 - 16*C*a^19*b^2 - 24*B*a^20*b - 24*C*a^20*b))/(a^18*b + a^19 - a^12*

b^7 - a^13*b^6 + 3*a^14*b^5 + 3*a^15*b^4 - 3*a^16*b^3 - 3*a^17*b^2) - (8*tan(c/2 + (d*x)/2)*(A*b^2*6i + a^2*((

A*1i)/2 + C*1i) - B*a*b*3i)*(8*a^19*b - 8*a^10*b^10 + 8*a^11*b^9 + 32*a^12*b^8 - 32*a^13*b^7 - 48*a^14*b^6 + 4

8*a^15*b^5 + 32*a^16*b^4 - 32*a^17*b^3 - 8*a^18*b^2))/(a^5*(a^14*b + a^15 - a^8*b^7 - a^9*b^6 + 3*a^10*b^5 + 3

*a^11*b^4 - 3*a^12*b^3 - 3*a^13*b^2)))*(A*b^2*6i + a^2*((A*1i)/2 + C*1i) - B*a*b*3i))/a^5 + (8*tan(c/2 + (d*x)

/2)*(A^2*a^14 + 288*A^2*b^14 + 4*C^2*a^14 - 288*A^2*a*b^13 - 2*A^2*a^13*b - 8*C^2*a^13*b - 1104*A^2*a^2*b^12 +

1104*A^2*a^3*b^11 + 1538*A^2*a^4*b^10 - 1538*A^2*a^5*b^9 - 827*A^2*a^6*b^8 + 872*A^2*a^7*b^7 + 18*A^2*a^8*b^6

- 108*A^2*a^9*b^5 + 74*A^2*a^10*b^4 - 40*A^2*a^11*b^3 + 21*A^2*a^12*b^2 + 72*B^2*a^2*b^12 - 72*B^2*a^3*b^11 -

288*B^2*a^4*b^10 + 288*B^2*a^5*b^9 + 441*B^2*a^6*b^8 - 432*B^2*a^7*b^7 - 288*B^2*a^8*b^6 + 288*B^2*a^9*b^5 +

36*B^2*a^10*b^4 - 72*B^2*a^11*b^3 + 36*B^2*a^12*b^2 + 8*C^2*a^4*b^10 - 8*C^2*a^5*b^9 - 32*C^2*a^6*b^8 + 32*C^2

*a^7*b^7 + 57*C^2*a^8*b^6 - 48*C^2*a^9*b^5 - 52*C^2*a^10*b^4 + 32*C^2*a^11*b^3 + 24*C^2*a^12*b^2 + 4*A*C*a^14

- 288*A*B*a*b^13 - 12*A*B*a^13*b - 8*A*C*a^13*b - 24*B*C*a^13*b + 288*A*B*a^2*b^12 + 1128*A*B*a^3*b^11 - 1128*

A*B*a^4*b^10 - 1650*A*B*a^5*b^9 + 1632*A*B*a^6*b^8 + 984*A*B*a^7*b^7 - 1008*A*B*a^8*b^6 - 72*A*B*a^9*b^5 + 192

*A*B*a^10*b^4 - 108*A*B*a^11*b^3 + 24*A*B*a^12*b^2 + 96*A*C*a^2*b^12 - 96*A*C*a^3*b^11 - 376*A*C*a^4*b^10 + 37

6*A*C*a^5*b^9 + 598*A*C*a^6*b^8 - 544*A*C*a^7*b^7 - 444*A*C*a^8*b^6 + 336*A*C*a^9*b^5 + 104*A*C*a^10*b^4 - 64*

A*C*a^11*b^3 + 36*A*C*a^12*b^2 - 48*B*C*a^3*b^11 + 48*B*C*a^4*b^10 + 192*B*C*a^5*b^9 - 192*B*C*a^6*b^8 - 318*B

*C*a^7*b^7 + 288*B*C*a^8*b^6 + 252*B*C*a^9*b^5 - 192*B*C*a^10*b^4 - 72*B*C*a^11*b^3 + 48*B*C*a^12*b^2))/(a^14*

b + a^15 - a^8*b^7 - a^9*b^6 + 3*a^10*b^5 + 3*a^11*b^4 - 3*a^12*b^3 - 3*a^13*b^2))*(A*b^2*6i + a^2*((A*1i)/2 +

C*1i) - B*a*b*3i)*1i)/a^5 - (((((4*(4*A*a^21 + 8*C*a^21 - 48*A*a^10*b^11 + 24*A*a^11*b^10 + 212*A*a^12*b^9 -

100*A*a^13*b^8 - 360*A*a^14*b^7 + 164*A*a^15*b^6 + 276*A*a^16*b^5 - 120*A*a^17*b^4 - 80*A*a^18*b^3 + 28*A*a^19

*b^2 + 24*B*a^11*b^10 - 12*B*a^12*b^9 - 108*B*a^13*b^8 + 48*B*a^14*b^7 + 192*B*a^15*b^6 - 84*B*a^16*b^5 - 156*

B*a^17*b^4 + 72*B*a^18*b^3 + 48*B*a^19*b^2 - 8*C*a^12*b^9 + 4*C*a^13*b^8 + 36*C*a^14*b^7 - 8*C*a^15*b^6 - 72*C

*a^16*b^5 + 12*C*a^17*b^4 + 68*C*a^18*b^3 - 16*C*a^19*b^2 - 24*B*a^20*b - 24*C*a^20*b))/(a^18*b + a^19 - a^12*

b^7 - a^13*b^6 + 3*a^14*b^5 + 3*a^15*b^4 - 3*a^16*b^3 - 3*a^17*b^2) + (8*tan(c/2 + (d*x)/2)*(A*b^2*6i + a^2*((

A*1i)/2 + C*1i) - B*a*b*3i)*(8*a^19*b - 8*a^10*b^10 + 8*a^11*b^9 + 32*a^12*b^8 - 32*a^13*b^7 - 48*a^14*b^6 + 4

8*a^15*b^5 + 32*a^16*b^4 - 32*a^17*b^3 - 8*a^18*b^2))/(a^5*(a^14*b + a^15 - a^8*b^7 - a^9*b^6 + 3*a^10*b^5 + 3

*a^11*b^4 - 3*a^12*b^3 - 3*a^13*b^2)))*(A*b^2*6i + a^2*((A*1i)/2 + C*1i) - B*a*b*3i))/a^5 - (8*tan(c/2 + (d*x)