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

Optimal. Leaf size=411 \[ \frac{\left (-65 a^4 A b^2+68 a^2 A b^4+6 a^6 A-17 a^3 b^3 B+26 a^5 b B+6 a b^5 B-24 A b^6\right ) \sin (c+d x)}{6 a^4 d \left (a^2-b^2\right )^3}+\frac{b \left (-35 a^4 A b^3+28 a^2 A b^5+20 a^6 A b+8 a^5 b^2 B-7 a^3 b^4 B-8 a^7 B+2 a b^6 B-8 A b^7\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b \left (-11 a^2 A b^3+12 a^4 A b+2 a^3 b^2 B-6 a^5 B-a b^4 B+4 A b^5\right ) \sin (c+d x)}{2 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}+\frac{b \left (9 a^2 A b-6 a^3 B+a b^2 B-4 A b^3\right ) \sin (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac{b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac{x (4 A b-a B)}{a^5} \]

[Out]

-(((4*A*b - a*B)*x)/a^5) + (b*(20*a^6*A*b - 35*a^4*A*b^3 + 28*a^2*A*b^5 - 8*A*b^7 - 8*a^7*B + 8*a^5*b^2*B - 7*
a^3*b^4*B + 2*a*b^6*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d
) + ((6*a^6*A - 65*a^4*A*b^2 + 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)*Sin[c + d*x])/
(6*a^4*(a^2 - b^2)^3*d) + (b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (b*(9*a^2*
A*b - 4*A*b^3 - 6*a^3*B + a*b^2*B)*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b*(12*a^4*A
*b - 11*a^2*A*b^3 + 4*A*b^5 - 6*a^5*B + 2*a^3*b^2*B - a*b^4*B)*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Sec
[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 5.59506, antiderivative size = 411, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {4030, 4100, 4104, 3919, 3831, 2659, 208} \[ \frac{\left (-65 a^4 A b^2+68 a^2 A b^4+6 a^6 A-17 a^3 b^3 B+26 a^5 b B+6 a b^5 B-24 A b^6\right ) \sin (c+d x)}{6 a^4 d \left (a^2-b^2\right )^3}+\frac{b \left (-35 a^4 A b^3+28 a^2 A b^5+20 a^6 A b+8 a^5 b^2 B-7 a^3 b^4 B-8 a^7 B+2 a b^6 B-8 A b^7\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b \left (-11 a^2 A b^3+12 a^4 A b+2 a^3 b^2 B-6 a^5 B-a b^4 B+4 A b^5\right ) \sin (c+d x)}{2 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}+\frac{b \left (9 a^2 A b-6 a^3 B+a b^2 B-4 A b^3\right ) \sin (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac{b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac{x (4 A b-a B)}{a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((4*A*b - a*B)*x)/a^5) + (b*(20*a^6*A*b - 35*a^4*A*b^3 + 28*a^2*A*b^5 - 8*A*b^7 - 8*a^7*B + 8*a^5*b^2*B - 7*
a^3*b^4*B + 2*a*b^6*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d
) + ((6*a^6*A - 65*a^4*A*b^2 + 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)*Sin[c + d*x])/
(6*a^4*(a^2 - b^2)^3*d) + (b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (b*(9*a^2*
A*b - 4*A*b^3 - 6*a^3*B + a*b^2*B)*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b*(12*a^4*A
*b - 11*a^2*A*b^3 + 4*A*b^5 - 6*a^5*B + 2*a^3*b^2*B - a*b^4*B)*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Sec
[c + d*x]))

Rule 4030

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[(b*(A*b - a*B)*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*(a^2*(m + 1) - b^2*(m + n + 1)) + a*b*B*n - a*(A*b - a*B)*(m + 1)*Csc[e + f*x] + b*(A*b - a*B)*(m
+ n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && 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) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx &=\frac{b (A b-a B) \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 (-3 a^2 A+4 A b^2-a b B+3 a (A b-a B) \sec (c+d x)-3 b (A b-a B) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac{b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\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 (6 a^4 A-23 a^2 A b^2+12 A b^4+8 a^3 b B-3 a b^3 B-2 a \left (6 a^2 A b-A b^3-3 a^3 B-2 a b^2 B\right ) \sec (c+d x)+2 b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\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{b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\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 (-6 a^6 A+65 a^4 A b^2-68 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 \left (18 a^4 A b-7 a^2 A b^3+4 A b^5-6 a^5 B-8 a^3 b^2 B-a b^4 B\right ) \sec (c+d x)-3 b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\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 (6 a^6 A-65 a^4 A b^2+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\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac{b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\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{-6 \left (a^2-b^2\right )^3 (4 A b-a B)+3 a b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\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-a B) x}{a^5}+\frac{\left (6 a^6 A-65 a^4 A b^2+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\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac{b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac{\left (b \left (20 a^6 A b-35 a^4 A b^3+28 a^2 A b^5-8 A b^7-8 a^7 B+8 a^5 b^2 B-7 a^3 b^4 B+2 a b^6 B\right )\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-a B) x}{a^5}+\frac{\left (6 a^6 A-65 a^4 A b^2+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\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac{b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac{\left (20 a^6 A b-35 a^4 A b^3+28 a^2 A b^5-8 A b^7-8 a^7 B+8 a^5 b^2 B-7 a^3 b^4 B+2 a b^6 B\right ) \int \frac{1}{1+\frac{a \cos (c+d x)}{b}} \, dx}{2 a^5 \left (a^2-b^2\right )^3}\\ &=-\frac{(4 A b-a B) x}{a^5}+\frac{\left (6 a^6 A-65 a^4 A b^2+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\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac{b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac{\left (20 a^6 A b-35 a^4 A b^3+28 a^2 A b^5-8 A b^7-8 a^7 B+8 a^5 b^2 B-7 a^3 b^4 B+2 a b^6 B\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 )^3 d}\\ &=-\frac{(4 A b-a B) x}{a^5}+\frac{b \left (20 a^6 A b-35 a^4 A b^3+28 a^2 A b^5-8 A b^7-8 a^7 B+8 a^5 b^2 B-7 a^3 b^4 B+2 a b^6 B\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 (6 a^6 A-65 a^4 A b^2+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\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac{b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\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 [B]  time = 6.08404, size = 1205, normalized size = 2.93 \[ \frac{(b+a \cos (c+d x)) \sec ^3(c+d x) (A+B \sec (c+d x)) \left (\frac{24 b \left (8 B a^7-20 A b a^6-8 b^2 B a^5+35 A b^3 a^4+7 b^4 B a^3-28 A b^5 a^2-2 b^6 B a+8 A b^7\right ) \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right ) (b+a \cos (c+d x))^3}{\left (a^2-b^2\right )^{7/2}}+\frac{6 B c \cos (3 (c+d x)) a^{10}+6 B d x \cos (3 (c+d x)) a^{10}+6 A \sin (2 (c+d x)) a^{10}+3 A \sin (4 (c+d x)) a^{10}+36 b B c a^9+36 b B d x a^9-24 A b c \cos (3 (c+d x)) a^9-24 A b d x \cos (3 (c+d x)) a^9+18 A b \sin (c+d x) a^9+18 A b \sin (3 (c+d x)) a^9-144 A b^2 c a^8-144 A b^2 d x a^8-18 b^2 B c \cos (3 (c+d x)) a^8-18 b^2 B d x \cos (3 (c+d x)) a^8+36 b^2 B \sin (c+d x) a^8+18 A b^2 \sin (2 (c+d x)) a^8+36 b^2 B \sin (3 (c+d x)) a^8-9 A b^2 \sin (4 (c+d x)) a^8-84 b^3 B c a^7-84 b^3 B d x a^7+72 A b^3 c \cos (3 (c+d x)) a^7+72 A b^3 d x \cos (3 (c+d x)) a^7-90 A b^3 \sin (c+d x) a^7+120 b^3 B \sin (2 (c+d x)) a^7-114 A b^3 \sin (3 (c+d x)) a^7+336 A b^4 c a^6+336 A b^4 d x a^6+18 b^4 B c \cos (3 (c+d x)) a^6+18 b^4 B d x \cos (3 (c+d x)) a^6+72 b^4 B \sin (c+d x) a^6-300 A b^4 \sin (2 (c+d x)) a^6-32 b^4 B \sin (3 (c+d x)) a^6+9 A b^4 \sin (4 (c+d x)) a^6+36 b^5 B c a^5+36 b^5 B d x a^5-72 A b^5 c \cos (3 (c+d x)) a^5-72 A b^5 d x \cos (3 (c+d x)) a^5-135 A b^5 \sin (c+d x) a^5-90 b^5 B \sin (2 (c+d x)) a^5+125 A b^5 \sin (3 (c+d x)) a^5-144 A b^6 c a^4-144 A b^6 d x a^4-6 b^6 B c \cos (3 (c+d x)) a^4-6 b^6 B d x \cos (3 (c+d x)) a^4-57 b^6 B \sin (c+d x) a^4+336 A b^6 \sin (2 (c+d x)) a^4+11 b^6 B \sin (3 (c+d x)) a^4-3 A b^6 \sin (4 (c+d x)) a^4+36 b^7 B c a^3+36 b^7 B d x a^3+24 A b^7 c \cos (3 (c+d x)) a^3+24 A b^7 d x \cos (3 (c+d x)) a^3+228 A b^7 \sin (c+d x) a^3+30 b^7 B \sin (2 (c+d x)) a^3-44 A b^7 \sin (3 (c+d x)) a^3-144 A b^8 c a^2-144 A b^8 d x a^2+36 b \left (a^2-b^2\right )^3 (a B-4 A b) (c+d x) \cos (2 (c+d x)) a^2+24 b^8 B \sin (c+d x) a^2-120 A b^8 \sin (2 (c+d x)) a^2-24 b^9 B c a-24 b^9 B d x a+18 \left (a^2-b^2\right )^3 \left (a^2+4 b^2\right ) (a B-4 A b) (c+d x) \cos (c+d x) a-96 A b^9 \sin (c+d x) a+96 A b^{10} c+96 A b^{10} d x}{\left (a^2-b^2\right )^3}\right )}{24 a^5 d (B+A \cos (c+d x)) (a+b \sec (c+d x))^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(A + B*Sec[c + d*x])*((24*b*(-20*a^6*A*b + 35*a^4*A*b^3 - 28*a^2*A*b^5 +
8*A*b^7 + 8*a^7*B - 8*a^5*b^2*B + 7*a^3*b^4*B - 2*a*b^6*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]
]*(b + a*Cos[c + d*x])^3)/(a^2 - b^2)^(7/2) + (-144*a^8*A*b^2*c + 336*a^6*A*b^4*c - 144*a^4*A*b^6*c - 144*a^2*
A*b^8*c + 96*A*b^10*c + 36*a^9*b*B*c - 84*a^7*b^3*B*c + 36*a^5*b^5*B*c + 36*a^3*b^7*B*c - 24*a*b^9*B*c - 144*a
^8*A*b^2*d*x + 336*a^6*A*b^4*d*x - 144*a^4*A*b^6*d*x - 144*a^2*A*b^8*d*x + 96*A*b^10*d*x + 36*a^9*b*B*d*x - 84
*a^7*b^3*B*d*x + 36*a^5*b^5*B*d*x + 36*a^3*b^7*B*d*x - 24*a*b^9*B*d*x + 18*a*(a^2 - b^2)^3*(a^2 + 4*b^2)*(-4*A
*b + a*B)*(c + d*x)*Cos[c + d*x] + 36*a^2*b*(a^2 - b^2)^3*(-4*A*b + a*B)*(c + d*x)*Cos[2*(c + d*x)] - 24*a^9*A
*b*c*Cos[3*(c + d*x)] + 72*a^7*A*b^3*c*Cos[3*(c + d*x)] - 72*a^5*A*b^5*c*Cos[3*(c + d*x)] + 24*a^3*A*b^7*c*Cos
[3*(c + d*x)] + 6*a^10*B*c*Cos[3*(c + d*x)] - 18*a^8*b^2*B*c*Cos[3*(c + d*x)] + 18*a^6*b^4*B*c*Cos[3*(c + d*x)
] - 6*a^4*b^6*B*c*Cos[3*(c + d*x)] - 24*a^9*A*b*d*x*Cos[3*(c + d*x)] + 72*a^7*A*b^3*d*x*Cos[3*(c + d*x)] - 72*
a^5*A*b^5*d*x*Cos[3*(c + d*x)] + 24*a^3*A*b^7*d*x*Cos[3*(c + d*x)] + 6*a^10*B*d*x*Cos[3*(c + d*x)] - 18*a^8*b^
2*B*d*x*Cos[3*(c + d*x)] + 18*a^6*b^4*B*d*x*Cos[3*(c + d*x)] - 6*a^4*b^6*B*d*x*Cos[3*(c + d*x)] + 18*a^9*A*b*S
in[c + d*x] - 90*a^7*A*b^3*Sin[c + d*x] - 135*a^5*A*b^5*Sin[c + d*x] + 228*a^3*A*b^7*Sin[c + d*x] - 96*a*A*b^9
*Sin[c + d*x] + 36*a^8*b^2*B*Sin[c + d*x] + 72*a^6*b^4*B*Sin[c + d*x] - 57*a^4*b^6*B*Sin[c + d*x] + 24*a^2*b^8
*B*Sin[c + d*x] + 6*a^10*A*Sin[2*(c + d*x)] + 18*a^8*A*b^2*Sin[2*(c + d*x)] - 300*a^6*A*b^4*Sin[2*(c + d*x)] +
336*a^4*A*b^6*Sin[2*(c + d*x)] - 120*a^2*A*b^8*Sin[2*(c + d*x)] + 120*a^7*b^3*B*Sin[2*(c + d*x)] - 90*a^5*b^5
*B*Sin[2*(c + d*x)] + 30*a^3*b^7*B*Sin[2*(c + d*x)] + 18*a^9*A*b*Sin[3*(c + d*x)] - 114*a^7*A*b^3*Sin[3*(c + d
*x)] + 125*a^5*A*b^5*Sin[3*(c + d*x)] - 44*a^3*A*b^7*Sin[3*(c + d*x)] + 36*a^8*b^2*B*Sin[3*(c + d*x)] - 32*a^6
*b^4*B*Sin[3*(c + d*x)] + 11*a^4*b^6*B*Sin[3*(c + d*x)] + 3*a^10*A*Sin[4*(c + d*x)] - 9*a^8*A*b^2*Sin[4*(c + d
*x)] + 9*a^6*A*b^4*Sin[4*(c + d*x)] - 3*a^4*A*b^6*Sin[4*(c + d*x)])/(a^2 - b^2)^3))/(24*a^5*d*(B + A*Cos[c + d
*x])*(a + b*Sec[c + d*x])^4)

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Maple [B]  time = 0.144, size = 2891, normalized size = 7. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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-35/d*b^4/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+28/d*b^6/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-8/d*b^8/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+2/d/a^4*B*arctan(tan(1/2*d*x+1/2*c))-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)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*B+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*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*A-12/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*b^2/(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^3/(tan(1/
2*d*x+1/2*c)^2*a-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)*tan(1/2*d*x+1/2*c)*A-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*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*A-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*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*A-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*b^4/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^
3)*tan(1/2*d*x+1/2*c)*A-12/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*b^2/(a+b)/(a^3-3*a^2*b+3*
a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B-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*b^5/(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*a*b^2/(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+24/d*b^2*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^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-44/3/d*b^4/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^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+6/d*b^7/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/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A+4/d*b^6/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^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*b^6/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)*B-2/d*b^6/
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
*B+116/3/d*b^5/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^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^7/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/(a^2-2*a*b+b^2)/(a^2+2*a*
b+b^2)*tan(1/2*d*x+1/2*c)^3*A+6/d*b^4/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+1/d*b^5/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)^5*B+6/d*b^4/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-1/d*b^5/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)*B+6/d*b^7/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/(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/(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-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A-4/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-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B+20/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+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A+4/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+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B+2/d*b^7/a
^4/(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))*B+8
/d*b^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))
*B-7/d*b^5/a^2/(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))*B+2/d/a^4*A*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)-8/d/a^5*A*arctan(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(cos(d*x+c)*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.36104, size = 5736, normalized size = 13.96 \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+B*sec(d*x+c))/(a+b*sec(d*x+c))^4,x, algorithm="fricas")

[Out]

[1/12*(12*(B*a^12 - 4*A*a^11*b - 4*B*a^10*b^2 + 16*A*a^9*b^3 + 6*B*a^8*b^4 - 24*A*a^7*b^5 - 4*B*a^6*b^6 + 16*A
*a^5*b^7 + B*a^4*b^8 - 4*A*a^3*b^9)*d*x*cos(d*x + c)^3 + 36*(B*a^11*b - 4*A*a^10*b^2 - 4*B*a^9*b^3 + 16*A*a^8*
b^4 + 6*B*a^7*b^5 - 24*A*a^6*b^6 - 4*B*a^5*b^7 + 16*A*a^4*b^8 + B*a^3*b^9 - 4*A*a^2*b^10)*d*x*cos(d*x + c)^2 +
36*(B*a^10*b^2 - 4*A*a^9*b^3 - 4*B*a^8*b^4 + 16*A*a^7*b^5 + 6*B*a^6*b^6 - 24*A*a^5*b^7 - 4*B*a^4*b^8 + 16*A*a
^3*b^9 + B*a^2*b^10 - 4*A*a*b^11)*d*x*cos(d*x + c) + 12*(B*a^9*b^3 - 4*A*a^8*b^4 - 4*B*a^7*b^5 + 16*A*a^6*b^6
+ 6*B*a^5*b^7 - 24*A*a^4*b^8 - 4*B*a^3*b^9 + 16*A*a^2*b^10 + B*a*b^11 - 4*A*b^12)*d*x - 3*(8*B*a^7*b^4 - 20*A*
a^6*b^5 - 8*B*a^5*b^6 + 35*A*a^4*b^7 + 7*B*a^3*b^8 - 28*A*a^2*b^9 - 2*B*a*b^10 + 8*A*b^11 + (8*B*a^10*b - 20*A
*a^9*b^2 - 8*B*a^8*b^3 + 35*A*a^7*b^4 + 7*B*a^6*b^5 - 28*A*a^5*b^6 - 2*B*a^4*b^7 + 8*A*a^3*b^8)*cos(d*x + c)^3
+ 3*(8*B*a^9*b^2 - 20*A*a^8*b^3 - 8*B*a^7*b^4 + 35*A*a^6*b^5 + 7*B*a^5*b^6 - 28*A*a^4*b^7 - 2*B*a^3*b^8 + 8*A
*a^2*b^9)*cos(d*x + c)^2 + 3*(8*B*a^8*b^3 - 20*A*a^7*b^4 - 8*B*a^6*b^5 + 35*A*a^5*b^6 + 7*B*a^4*b^7 - 28*A*a^3
*b^8 - 2*B*a^2*b^9 + 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*a^9*b^3 + 26*B*a^8*b^4 - 71*A*a^7*b^5 - 43*B*a^6*b^6 + 133*A*a^5*b^7 + 23*B*a^4*b^8 -
92*A*a^3*b^9 - 6*B*a^2*b^10 + 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*a^11*b + 36*B*a^10*b^2 - 132*A*a^9*b^3 - 68*B*a^8*b^4 + 239*A*a^7*b^5 + 43*B*a^6*b^6 -
169*A*a^5*b^7 - 11*B*a^4*b^8 + 44*A*a^3*b^9)*cos(d*x + c)^2 + 3*(6*A*a^10*b^2 + 20*B*a^9*b^3 - 59*A*a^8*b^4 -
35*B*a^7*b^5 + 110*A*a^6*b^6 + 20*B*a^5*b^7 - 77*A*a^4*b^8 - 5*B*a^3*b^9 + 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), 1/6*(6*(B*a^12 - 4*A*a^11
*b - 4*B*a^10*b^2 + 16*A*a^9*b^3 + 6*B*a^8*b^4 - 24*A*a^7*b^5 - 4*B*a^6*b^6 + 16*A*a^5*b^7 + B*a^4*b^8 - 4*A*a
^3*b^9)*d*x*cos(d*x + c)^3 + 18*(B*a^11*b - 4*A*a^10*b^2 - 4*B*a^9*b^3 + 16*A*a^8*b^4 + 6*B*a^7*b^5 - 24*A*a^6
*b^6 - 4*B*a^5*b^7 + 16*A*a^4*b^8 + B*a^3*b^9 - 4*A*a^2*b^10)*d*x*cos(d*x + c)^2 + 18*(B*a^10*b^2 - 4*A*a^9*b^
3 - 4*B*a^8*b^4 + 16*A*a^7*b^5 + 6*B*a^6*b^6 - 24*A*a^5*b^7 - 4*B*a^4*b^8 + 16*A*a^3*b^9 + B*a^2*b^10 - 4*A*a*
b^11)*d*x*cos(d*x + c) + 6*(B*a^9*b^3 - 4*A*a^8*b^4 - 4*B*a^7*b^5 + 16*A*a^6*b^6 + 6*B*a^5*b^7 - 24*A*a^4*b^8
- 4*B*a^3*b^9 + 16*A*a^2*b^10 + B*a*b^11 - 4*A*b^12)*d*x - 3*(8*B*a^7*b^4 - 20*A*a^6*b^5 - 8*B*a^5*b^6 + 35*A*
a^4*b^7 + 7*B*a^3*b^8 - 28*A*a^2*b^9 - 2*B*a*b^10 + 8*A*b^11 + (8*B*a^10*b - 20*A*a^9*b^2 - 8*B*a^8*b^3 + 35*A
*a^7*b^4 + 7*B*a^6*b^5 - 28*A*a^5*b^6 - 2*B*a^4*b^7 + 8*A*a^3*b^8)*cos(d*x + c)^3 + 3*(8*B*a^9*b^2 - 20*A*a^8*
b^3 - 8*B*a^7*b^4 + 35*A*a^6*b^5 + 7*B*a^5*b^6 - 28*A*a^4*b^7 - 2*B*a^3*b^8 + 8*A*a^2*b^9)*cos(d*x + c)^2 + 3*
(8*B*a^8*b^3 - 20*A*a^7*b^4 - 8*B*a^6*b^5 + 35*A*a^5*b^6 + 7*B*a^4*b^7 - 28*A*a^3*b^8 - 2*B*a^2*b^9 + 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*a^9*b^3 + 26*B*a^8*b^4 - 71*A*a^7*b^5 - 43*B*a^6*b^6 + 133*A*a^5*b^7 + 23*B*a^4*b^8 - 92*A*a^3*b^9 - 6*B
*a^2*b^10 + 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*a^11*b + 36*B*a^10*b^2 - 132*A*a^9*b^3 - 68*B*a^8*b^4 + 239*A*a^7*b^5 + 43*B*a^6*b^6 - 169*A*a^5*b^7 - 11
*B*a^4*b^8 + 44*A*a^3*b^9)*cos(d*x + c)^2 + 3*(6*A*a^10*b^2 + 20*B*a^9*b^3 - 59*A*a^8*b^4 - 35*B*a^7*b^5 + 110
*A*a^6*b^6 + 20*B*a^5*b^7 - 77*A*a^4*b^8 - 5*B*a^3*b^9 + 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+B*sec(d*x+c))/(a+b*sec(d*x+c))**4,x)

[Out]

Timed out

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Giac [B]  time = 1.5629, size = 1304, normalized size = 3.17 \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+B*sec(d*x+c))/(a+b*sec(d*x+c))^4,x, algorithm="giac")

[Out]

-1/3*(3*(8*B*a^7*b - 20*A*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)) + (36*B*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 + 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 + 24*A*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 + 45*B*a^4*b^5*ta
n(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 - 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 + 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 - 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 + 36*B*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) - 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) + 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) + 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*t
an(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)*(d*x + c)
/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