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

Optimal. Leaf size=369 \[ -\frac{b \left (-a^2 b^2 (21 A-2 C)+a^4 (6 A-5 C)+12 A b^4\right ) \sin (c+d x)}{2 a^4 d \left (a^2-b^2\right )^2}+\frac{\left (-a^2 b^2 (10 A-C)+a^4 (A-4 C)+6 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a^3 d \left (a^2-b^2\right )^2}-\frac{b \left (5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+6 a^6 C+12 A b^6\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{\left (7 a^2 A b^2+3 a^4 C-4 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac{\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac{x \left (a^2 (A+2 C)+12 A b^2\right )}{2 a^5} \]

[Out]

((12*A*b^2 + a^2*(A + 2*C))*x)/(2*a^5) - (b*(12*A*b^6 - 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) - (b*(12*A*b^4 + a^4*
(6*A - 5*C) - a^2*b^2*(21*A - 2*C))*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((6*A*b^4 + 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^2*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 + 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]))

________________________________________________________________________________________

Rubi [A]  time = 1.61451, antiderivative size = 369, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {4101, 4100, 4104, 3919, 3831, 2659, 208} \[ -\frac{b \left (-a^2 b^2 (21 A-2 C)+a^4 (6 A-5 C)+12 A b^4\right ) \sin (c+d x)}{2 a^4 d \left (a^2-b^2\right )^2}+\frac{\left (-a^2 b^2 (10 A-C)+a^4 (A-4 C)+6 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a^3 d \left (a^2-b^2\right )^2}-\frac{b \left (5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+6 a^6 C+12 A b^6\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{\left (7 a^2 A b^2+3 a^4 C-4 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac{\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac{x \left (a^2 (A+2 C)+12 A b^2\right )}{2 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((12*A*b^2 + a^2*(A + 2*C))*x)/(2*a^5) - (b*(12*A*b^6 - 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) - (b*(12*A*b^4 + a^4*
(6*A - 5*C) - a^2*b^2*(21*A - 2*C))*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((6*A*b^4 + 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^2*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 + 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 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 ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx &=\frac{\left (A b^2+a^2 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^2 (A-C)\right )+2 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))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac{\left (A b^2+a^2 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+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+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right )+a b \left (A b^2-a^2 (4 A+3 C)\right ) \sec (c+d x)+2 \left (7 a^2 A b^2-4 A b^4+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+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^2 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+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 b \left (12 A b^4+a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)\right )+2 a \left (2 A b^4-a^2 b^2 (4 A+C)-a^4 (A+2 C)\right ) \sec (c+d x)-2 b \left (6 A b^4+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{b \left (12 A b^4+a^4 (6 A-5 C)-a^2 b^2 (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+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^2 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+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+a^2 (A+2 C)\right )+2 a b \left (6 A b^4+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+a^2 (A+2 C)\right ) x}{2 a^5}-\frac{b \left (12 A b^4+a^4 (6 A-5 C)-a^2 b^2 (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+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^2 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+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-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+a^2 (A+2 C)\right ) x}{2 a^5}-\frac{b \left (12 A b^4+a^4 (6 A-5 C)-a^2 b^2 (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+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^2 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+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-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+a^2 (A+2 C)\right ) x}{2 a^5}-\frac{b \left (12 A b^4+a^4 (6 A-5 C)-a^2 b^2 (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+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^2 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+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-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+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+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{b \left (12 A b^4+a^4 (6 A-5 C)-a^2 b^2 (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+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^2 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+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*}

Mathematica [A]  time = 2.43933, size = 256, normalized size = 0.69 \[ \frac{2 (c+d x) \left (a^2 (A+2 C)+12 A b^2\right )-\frac{2 a b^3 \left (a^2 C+A b^2\right ) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)^2}+\frac{2 a b^2 \left (a^2 b^2 (10 A-3 C)+6 a^4 C-7 A b^4\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a \cos (c+d x)+b)}+\frac{4 b \left (5 a^4 b^2 (4 A-C)+a^2 b^4 (2 C-29 A)+6 a^6 C+12 A b^6\right ) \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+a^2 A \sin (2 (c+d x))-12 a A b \sin (c+d x)}{4 a^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(12*A*b^2 + a^2*(A + 2*C))*(c + d*x) + (4*b*(12*A*b^6 + 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) - 12*a*A*b*Sin[c + d*x] - (2*a*b^3
*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])^2) + (2*a*b^2*(-7*A*b^4 + a^2*b^2*(10*A -
3*C) + 6*a^4*C)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])) + a^2*A*Sin[2*(c + d*x)])/(4*a^5*d)

________________________________________________________________________________________

Maple [B]  time = 0.149, size = 1478, normalized size = 4. \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)^2*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x)

[Out]

1/d*A/a^3*arctan(tan(1/2*d*x+1/2*c))+5/d*b^3/a/(a^4-2*a^2*b^2+b^4)/((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*b^5/a^3/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x
+1/2*c)/((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)*arctanh((a-b)*tan(1/2*d*x+
1/2*c)/((a+b)*(a-b))^(1/2))*A+6/d*b^6/a^4/(tan(1/2*d*x+1/2*c)^2*a-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-1/d*b^3/a/(tan(1/2*d*x+1/2*c)^2*a-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/(tan(1/2*d*x+1/2*c)^2*a-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-1/d*b^5/a^3/(tan(1/2*d*x+1/2*c)^2*a-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-6/d*b^6/a^4/(tan(1/2*d*x+1/2*c)^2*a-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-1/d*b^3/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*
C-2/d*b^4/a^2/(tan(1/2*d*x+1/2*c)^2*a-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*a
/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C+10/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)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A*b^4-10/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)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A*b^4-6/d/(tan(1/2*d*x+1/
2*c)^2*a-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/(tan(1/2*d*x+1/2*c
)^2*a-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^2-20/d/a*b^3/(a^4-2*a^2*b^2+b^4)/((a+
b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A+29/d/a^3*b^5/(a^4-2*a^2*b^2+b^4)/((a+b
)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A-1/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)^2*b^5/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+2/d/a^3*arctan(tan(1/2*d*x+1/2*c))*C+1
/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)*A+12/d/a^5*arctan(tan(1/2*d*x+1/2*c))*A*b^2-1/d/a^3/(1+ta
n(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3*A-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., 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)^2*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 0.943741, size = 3407, normalized size = 9.23 \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)^2*(A+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 + 3*(3*A - 2*C)*a^8*b^2 - 3*(11*A - 2*C)*a^6*b^4 + (35*A - 2*C)*a^4*b^6 - 12*A*a^2*b^8
)*d*x*cos(d*x + c)^2 + 4*((A + 2*C)*a^9*b + 3*(3*A - 2*C)*a^7*b^3 - 3*(11*A - 2*C)*a^5*b^5 + (35*A - 2*C)*a^3*
b^7 - 12*A*a*b^9)*d*x*cos(d*x + c) + 2*((A + 2*C)*a^8*b^2 + 3*(3*A - 2*C)*a^6*b^4 - 3*(11*A - 2*C)*a^4*b^6 + (
35*A - 2*C)*a^2*b^8 - 12*A*b^10)*d*x + (6*C*a^6*b^3 + 5*(4*A - C)*a^4*b^5 - (29*A - 2*C)*a^2*b^7 + 12*A*b^9 +
(6*C*a^8*b + 5*(4*A - C)*a^6*b^3 - (29*A - 2*C)*a^4*b^5 + 12*A*a^2*b^7)*cos(d*x + c)^2 + 2*(6*C*a^7*b^2 + 5*(4
*A - C)*a^5*b^4 - (29*A - 2*C)*a^3*b^6 + 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*((6*A - 5*C)*a^7*b^3 - (27*A - 7*C)*a^5*b^5 + (33*A - 2*C)*a^3*b^7 - 1
2*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 + 4*(A*a^9*b - 3*A*a^7*b^3 + 3*A*a
^5*b^5 - A*a^3*b^7)*cos(d*x + c)^2 + ((11*A - 6*C)*a^8*b^2 - (43*A - 9*C)*a^6*b^4 + (50*A - 3*C)*a^4*b^6 - 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 + 3*(3*A - 2*C)*a^8*b^2 - 3*(11*A - 2*C)*a^6*b^4 + (35*A - 2*C)*a^4*b^6 - 12*A*a^2*b^8)*d*x*cos(
d*x + c)^2 + 2*((A + 2*C)*a^9*b + 3*(3*A - 2*C)*a^7*b^3 - 3*(11*A - 2*C)*a^5*b^5 + (35*A - 2*C)*a^3*b^7 - 12*A
*a*b^9)*d*x*cos(d*x + c) + ((A + 2*C)*a^8*b^2 + 3*(3*A - 2*C)*a^6*b^4 - 3*(11*A - 2*C)*a^4*b^6 + (35*A - 2*C)*
a^2*b^8 - 12*A*b^10)*d*x - (6*C*a^6*b^3 + 5*(4*A - C)*a^4*b^5 - (29*A - 2*C)*a^2*b^7 + 12*A*b^9 + (6*C*a^8*b +
5*(4*A - C)*a^6*b^3 - (29*A - 2*C)*a^4*b^5 + 12*A*a^2*b^7)*cos(d*x + c)^2 + 2*(6*C*a^7*b^2 + 5*(4*A - C)*a^5*
b^4 - (29*A - 2*C)*a^3*b^6 + 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))) - ((6*A - 5*C)*a^7*b^3 - (27*A - 7*C)*a^5*b^5 + (33*A - 2*C)*a^3*b^7 - 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 + 4*(A*a^9*b - 3*A*a^7*b^3 + 3*A*a^5
*b^5 - A*a^3*b^7)*cos(d*x + c)^2 + ((11*A - 6*C)*a^8*b^2 - (43*A - 9*C)*a^6*b^4 + (50*A - 3*C)*a^4*b^6 - 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)]

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

[Out]

Timed out

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Giac [B]  time = 1.30796, size = 1551, normalized size = 4.2 \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)^2*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x, algorithm="giac")

[Out]

-1/2*(2*(6*C*a^6*b + 20*A*a^4*b^3 - 5*C*a^4*b^3 - 29*A*a^2*b^5 + 2*C*a^2*b^5 + 12*A*b^7)*(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^9 - 2*a^7*b^2 + a^5*b^4)*sqrt(-a^2 + b^2)) + 2*(A*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 - 13*A*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 - 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 - 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 + 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 + 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 - 4*A*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 + 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 - 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 - 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 + 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 + 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 - 4*A*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 - 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
- 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 + 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 + 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 + 36*A*b^7*tan(1/2*d*x + 1/2*c)^3 - A*a^7*tan(1/2*d*x + 1/2*c) + 4*A*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) - 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) - 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) + 3*C*a^3*b^4*tan(1/2*d*x + 1/2*c) - 17*A*a^2*b^5*ta
n(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) + 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) - (A*a^2 + 2*C*a^2 + 12*A*b^2)*(d*x + c)/a^5)/d