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

Optimal. Leaf size=367 \[ \frac{\left (-a^4 b^2 (65 A+4 C)+68 a^2 A b^4+a^6 (6 A-11 C)-24 A b^6\right ) \sin (c+d x)}{6 a^4 d \left (a^2-b^2\right )^3}-\frac{\left (-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6-2 a^8 C+8 A b^8\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left (-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-2 a^6 C-4 A b^6\right ) \sin (c+d x)}{2 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac{\left (-a^2 b^2 (9 A+2 C)-3 a^4 C+4 A b^4\right ) \sin (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac{\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac{4 A b x}{a^5} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.82357, antiderivative size = 367, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {4101, 4100, 4104, 3919, 3831, 2659, 208} \[ \frac{\left (-a^4 b^2 (65 A+4 C)+68 a^2 A b^4+a^6 (6 A-11 C)-24 A b^6\right ) \sin (c+d x)}{6 a^4 d \left (a^2-b^2\right )^3}-\frac{\left (-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6-2 a^8 C+8 A b^8\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left (-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-2 a^6 C-4 A b^6\right ) \sin (c+d x)}{2 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac{\left (-a^2 b^2 (9 A+2 C)-3 a^4 C+4 A b^4\right ) \sin (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac{\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac{4 A b x}{a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4101

Int[((A_.) + 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^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)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e
 + f*x])^n*Simp[a^2*(A + C)*(m + 1) - (A*b^2 + a^2*C)*(m + n + 1) - a*b*(A + C)*(m + 1)*Csc[e + f*x] + (A*b^2
+ a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] && NeQ[a^2 - b^2, 0] &&
LtQ[m, -1] &&  !(ILtQ[m + 1/2, 0] && ILtQ[n, 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]

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 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 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 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]

Rubi steps

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

Mathematica [C]  time = 7.42231, size = 1089, normalized size = 2.97 \[ -\frac{8 A b x \sec ^2(c+d x) \left (C \sec ^2(c+d x)+A\right ) (b+a \cos (c+d x))^4}{a^5 (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}+\frac{\left (-2 C a^8-20 A b^2 a^6-3 b^2 C a^6+35 A b^4 a^4-28 A b^6 a^2+8 A b^8\right ) \sec ^2(c+d x) \left (C \sec ^2(c+d x)+A\right ) \left (-\frac{2 i \tan ^{-1}\left (\sec \left (\frac{d x}{2}\right ) \left (\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right ) \left (i a \sin \left (c+\frac{d x}{2}\right )-i b \sin \left (\frac{d x}{2}\right )\right )\right ) \cos (c)}{a^5 \sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{2 \tan ^{-1}\left (\sec \left (\frac{d x}{2}\right ) \left (\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right ) \left (i a \sin \left (c+\frac{d x}{2}\right )-i b \sin \left (\frac{d x}{2}\right )\right )\right ) \sin (c)}{a^5 \sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}\right ) (b+a \cos (c+d x))^4}{\left (b^2-a^2\right )^3 (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}+\frac{2 A \sec (c+d x) \left (C \sec ^2(c+d x)+A\right ) \tan (c+d x) (b+a \cos (c+d x))^4}{a^4 d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}+\frac{\sec (c) \sec ^2(c+d x) \left (C \sec ^2(c+d x)+A\right ) \left (36 A \sin (c) b^8-26 a A \sin (d x) b^7-96 a^2 A \sin (c) b^6+6 a^2 C \sin (c) b^6+71 a^3 A \sin (d x) b^5-2 a^3 C \sin (d x) b^5+75 a^4 A \sin (c) b^4-18 a^4 C \sin (c) b^4-60 a^5 A \sin (d x) b^3+5 a^5 C \sin (d x) b^3+27 a^6 C \sin (c) b^2-18 a^7 C \sin (d x) b\right ) (b+a \cos (c+d x))^3}{3 a^5 \left (a^2-b^2\right )^3 d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}+\frac{\sec (c) \sec ^2(c+d x) \left (C \sec ^2(c+d x)+A\right ) \left (12 A \sin (c) b^7-10 a A \sin (d x) b^6-17 a^2 A \sin (c) b^5+6 a^2 C \sin (c) b^5+15 a^3 A \sin (d x) b^4-4 a^3 C \sin (d x) b^4-11 a^4 C \sin (c) b^3+9 a^5 C \sin (d x) b^2\right ) (b+a \cos (c+d x))^2}{3 a^5 \left (a^2-b^2\right )^2 d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}+\frac{2 \sec (c) \sec ^2(c+d x) \left (C \sec ^2(c+d x)+A\right ) \left (A \sin (c) b^6-a A \sin (d x) b^5+a^2 C \sin (c) b^4-a^3 C \sin (d x) b^3\right ) (b+a \cos (c+d x))}{3 a^5 \left (a^2-b^2\right ) d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-8*A*b*x*(b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a^5*(A + 2*C + A*Cos[2*c + 2*d*x])*(a
 + b*Sec[c + d*x])^4) + ((-20*a^6*A*b^2 + 35*a^4*A*b^4 - 28*a^2*A*b^6 + 8*A*b^8 - 2*a^8*C - 3*a^6*b^2*C)*(b +
a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(((-2*I)*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*
Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2]
+ I*a*Sin[c + (d*x)/2])]*Cos[c])/(a^5*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (2*ArcTan[Sec[(d*x)/2]*
(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]
]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Sin[c])/(a^5*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]]))
)/((-a^2 + b^2)^3*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + (2*(b + a*Cos[c + d*x])*Sec[c]*Sec[
c + d*x]^2*(A + C*Sec[c + d*x]^2)*(A*b^6*Sin[c] + a^2*b^4*C*Sin[c] - a*A*b^5*Sin[d*x] - a^3*b^3*C*Sin[d*x]))/(
3*a^5*(a^2 - b^2)*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^2*Sec[c]*Se
c[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(-17*a^2*A*b^5*Sin[c] + 12*A*b^7*Sin[c] - 11*a^4*b^3*C*Sin[c] + 6*a^2*b^5*
C*Sin[c] + 15*a^3*A*b^4*Sin[d*x] - 10*a*A*b^6*Sin[d*x] + 9*a^5*b^2*C*Sin[d*x] - 4*a^3*b^4*C*Sin[d*x]))/(3*a^5*
(a^2 - b^2)^2*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^3*Sec[c]*Sec[c
+ d*x]^2*(A + C*Sec[c + d*x]^2)*(75*a^4*A*b^4*Sin[c] - 96*a^2*A*b^6*Sin[c] + 36*A*b^8*Sin[c] + 27*a^6*b^2*C*Si
n[c] - 18*a^4*b^4*C*Sin[c] + 6*a^2*b^6*C*Sin[c] - 60*a^5*A*b^3*Sin[d*x] + 71*a^3*A*b^5*Sin[d*x] - 26*a*A*b^7*S
in[d*x] - 18*a^7*b*C*Sin[d*x] + 5*a^5*b^3*C*Sin[d*x] - 2*a^3*b^5*C*Sin[d*x]))/(3*a^5*(a^2 - b^2)^3*d*(A + 2*C
+ A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + (2*A*(b + a*Cos[c + d*x])^4*Sec[c + d*x]*(A + C*Sec[c + d*x]^2
)*Tan[c + d*x])/(a^4*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4)

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Maple [B]  time = 0.152, size = 2283, normalized size = 6.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d/(tan(1/2*d*x+1/2*c)^2*a-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+6/d/a^4/(tan(1/2*d*x+1/2*c)^2*a-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/(tan(1/2*d*x+1/2*c)^2*a-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/(tan(1/2*d*x+1/2*c)^2*a-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)*tan(1/2*d*x+1/2*c)^3*A-12/d/a^4/(tan(1/2*d*x+1/2*c)^2*a-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+3/d*a/(tan(1/2*d*x+1/2*c)^2*a-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*b^2*C+2/d/a^3/(tan(1/2*d*x+1/2*c)^2*a-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/(tan(1/2*d*x+1/2*c)^2*a-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+6/d*b/(tan(1/2*d*x+1/2*c)^2*a
-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^2-12/d*b/(tan(1/2*d*x+1/
2*c)^2*a-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^2-18/d/a^2/(ta
n(1/2*d*x+1/2*c)^2*a-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-
2/d/a^3/(tan(1/2*d*x+1/2*c)^2*a-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^6-3/d*a/(tan(1/2*d*x+1/2*c)^2*a-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^2*C-18/d/a^2/(tan(1/2*d*x+1/2*c)^2*a-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^5-5/d/a/(tan(1/2*d*x+1/2*c)^2*a-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+2/d/(tan(1/2*d*x+1/2*c)^2*a-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+6/d*b/(tan(1/2*d*x+1/2*c)^2*a-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^2-40/d*b^3/(tan(1/2*d*x+1/2*c)^2*a-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+20/d*b^3/(tan(1/2*d*x+1/2*c)^2*a-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^2*a/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)
*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*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)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C+20/d*b^3/(tan(1/2*d*x+1/2*c)^2*a-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*A-35/d/a/(a^6-3*a^4*b^2+3*a^
2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A*b^4-8/d/a^5/(a^6-3*a^4*
b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A*b^8+28/d/a^3/(a
^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A*b^6-4/
3/d/(tan(1/2*d*x+1/2*c)^2*a-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^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b)
)^(1/2))*C+2/d*A/a^4*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)-8/d*A/a^5*b*arctan(tan(1/2*d*x+1/2*c))

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.06307, size = 4319, normalized size = 11.77 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/12*(48*(A*a^11*b - 4*A*a^9*b^3 + 6*A*a^7*b^5 - 4*A*a^5*b^7 + A*a^3*b^9)*d*x*cos(d*x + c)^3 + 144*(A*a^10*b
^2 - 4*A*a^8*b^4 + 6*A*a^6*b^6 - 4*A*a^4*b^8 + A*a^2*b^10)*d*x*cos(d*x + c)^2 + 144*(A*a^9*b^3 - 4*A*a^7*b^5 +
 6*A*a^5*b^7 - 4*A*a^3*b^9 + A*a*b^11)*d*x*cos(d*x + c) + 48*(A*a^8*b^4 - 4*A*a^6*b^6 + 6*A*a^4*b^8 - 4*A*a^2*
b^10 + A*b^12)*d*x - 3*(2*C*a^8*b^3 + (20*A + 3*C)*a^6*b^5 - 35*A*a^4*b^7 + 28*A*a^2*b^9 - 8*A*b^11 + (2*C*a^1
1 + (20*A + 3*C)*a^9*b^2 - 35*A*a^7*b^4 + 28*A*a^5*b^6 - 8*A*a^3*b^8)*cos(d*x + c)^3 + 3*(2*C*a^10*b + (20*A +
 3*C)*a^8*b^3 - 35*A*a^6*b^5 + 28*A*a^4*b^7 - 8*A*a^2*b^9)*cos(d*x + c)^2 + 3*(2*C*a^9*b^2 + (20*A + 3*C)*a^7*
b^4 - 35*A*a^5*b^6 + 28*A*a^3*b^8 - 8*A*a*b^10)*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*((6*A - 11*C)*a^9*b^3 - (71*A - 7*C)*a^7*b^5 + (133*A + 4*C)*a^5*b^7 - 92*A
*a^3*b^9 + 24*A*a*b^11 + 6*(A*a^12 - 4*A*a^10*b^2 + 6*A*a^8*b^4 - 4*A*a^6*b^6 + A*a^4*b^8)*cos(d*x + c)^3 + (1
8*(A - C)*a^11*b - (132*A - 23*C)*a^9*b^3 + (239*A - 7*C)*a^7*b^5 - (169*A - 2*C)*a^5*b^7 + 44*A*a^3*b^9)*cos(
d*x + c)^2 + 3*(3*(2*A - 3*C)*a^10*b^2 - (59*A - 8*C)*a^8*b^4 + (110*A + C)*a^6*b^6 - 77*A*a^4*b^8 + 20*A*a^2*
b^10)*cos(d*x + c))*sin(d*x + c))/((a^16 - 4*a^14*b^2 + 6*a^12*b^4 - 4*a^10*b^6 + a^8*b^8)*d*cos(d*x + c)^3 +
3*(a^15*b - 4*a^13*b^3 + 6*a^11*b^5 - 4*a^9*b^7 + a^7*b^9)*d*cos(d*x + c)^2 + 3*(a^14*b^2 - 4*a^12*b^4 + 6*a^1
0*b^6 - 4*a^8*b^8 + a^6*b^10)*d*cos(d*x + c) + (a^13*b^3 - 4*a^11*b^5 + 6*a^9*b^7 - 4*a^7*b^9 + a^5*b^11)*d),
-1/6*(24*(A*a^11*b - 4*A*a^9*b^3 + 6*A*a^7*b^5 - 4*A*a^5*b^7 + A*a^3*b^9)*d*x*cos(d*x + c)^3 + 72*(A*a^10*b^2
- 4*A*a^8*b^4 + 6*A*a^6*b^6 - 4*A*a^4*b^8 + A*a^2*b^10)*d*x*cos(d*x + c)^2 + 72*(A*a^9*b^3 - 4*A*a^7*b^5 + 6*A
*a^5*b^7 - 4*A*a^3*b^9 + A*a*b^11)*d*x*cos(d*x + c) + 24*(A*a^8*b^4 - 4*A*a^6*b^6 + 6*A*a^4*b^8 - 4*A*a^2*b^10
 + A*b^12)*d*x - 3*(2*C*a^8*b^3 + (20*A + 3*C)*a^6*b^5 - 35*A*a^4*b^7 + 28*A*a^2*b^9 - 8*A*b^11 + (2*C*a^11 +
(20*A + 3*C)*a^9*b^2 - 35*A*a^7*b^4 + 28*A*a^5*b^6 - 8*A*a^3*b^8)*cos(d*x + c)^3 + 3*(2*C*a^10*b + (20*A + 3*C
)*a^8*b^3 - 35*A*a^6*b^5 + 28*A*a^4*b^7 - 8*A*a^2*b^9)*cos(d*x + c)^2 + 3*(2*C*a^9*b^2 + (20*A + 3*C)*a^7*b^4
- 35*A*a^5*b^6 + 28*A*a^3*b^8 - 8*A*a*b^10)*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))) - ((6*A - 11*C)*a^9*b^3 - (71*A - 7*C)*a^7*b^5 + (133*A + 4*C)*a^5*b^7
- 92*A*a^3*b^9 + 24*A*a*b^11 + 6*(A*a^12 - 4*A*a^10*b^2 + 6*A*a^8*b^4 - 4*A*a^6*b^6 + A*a^4*b^8)*cos(d*x + c)^
3 + (18*(A - C)*a^11*b - (132*A - 23*C)*a^9*b^3 + (239*A - 7*C)*a^7*b^5 - (169*A - 2*C)*a^5*b^7 + 44*A*a^3*b^9
)*cos(d*x + c)^2 + 3*(3*(2*A - 3*C)*a^10*b^2 - (59*A - 8*C)*a^8*b^4 + (110*A + C)*a^6*b^6 - 77*A*a^4*b^8 + 20*
A*a^2*b^10)*cos(d*x + c))*sin(d*x + c))/((a^16 - 4*a^14*b^2 + 6*a^12*b^4 - 4*a^10*b^6 + a^8*b^8)*d*cos(d*x + c
)^3 + 3*(a^15*b - 4*a^13*b^3 + 6*a^11*b^5 - 4*a^9*b^7 + a^7*b^9)*d*cos(d*x + c)^2 + 3*(a^14*b^2 - 4*a^12*b^4 +
 6*a^10*b^6 - 4*a^8*b^8 + a^6*b^10)*d*cos(d*x + c) + (a^13*b^3 - 4*a^11*b^5 + 6*a^9*b^7 - 4*a^7*b^9 + a^5*b^11
)*d)]

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

[Out]

Timed out

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Giac [B]  time = 1.36366, size = 1143, normalized size = 3.11 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*(2*C*a^8 + 20*A*a^6*b^2 + 3*C*a^6*b^2 - 35*A*a^4*b^4 + 28*A*a^2*b^6 - 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^1
1 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*sqrt(-a^2 + b^2)) - 12*(d*x + c)*A*b/a^5 + (18*C*a^8*b*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 + 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 - 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 + 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 - 24*A*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 + 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 - 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 + 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 - 152*A*a^2*b^7*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) + 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) + 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) + 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) + 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) - 24*A*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) + 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) +
 6*A*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 + 1)*a^4))/d