3.1030 \(\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=598 \[ \frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (a^4 b^2 (128 A-15 C)-a^2 b^4 (223 A-9 C)+8 a^6 (A+3 C)+99 a^3 b^3 B-72 a^5 b B-45 a b^5 B+105 A b^6\right )}{12 a^5 d \left (a^2-b^2\right )^2}+\frac{\sin (c+d x) \left (-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+33 a^3 b B-15 a b^3 B+35 A b^4\right )}{12 a^3 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x) \left (-a^2 b^2 (13 A+C)+9 a^3 b B-5 a^4 C-3 a b^3 B+7 A b^4\right )}{4 a^2 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}+\frac{\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-a^2 b^3 (65 A-3 C)+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right )}{4 a^4 d \left (a^2-b^2\right )^2}-\frac{b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (3 a^4 b^2 (21 A-2 C)-a^2 b^4 (86 A-3 C)+38 a^3 b^3 B-35 a^5 b B+15 a^6 C-15 a b^5 B+35 A b^6\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^5 d (a-b)^2 (a+b)^3} \]

[Out]

-((35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B + 3*a^4*b*(8*A - 3*C) - a^2*b^3*(65*A - 3*C))*Sqrt[Cos[c + d
*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a^2 - b^2)^2*d) + ((105*A*b^6 - 72*a^5*b*B + 99*a^3
*b^3*B - 45*a*b^5*B + a^4*b^2*(128*A - 15*C) - a^2*b^4*(223*A - 9*C) + 8*a^6*(A + 3*C))*Sqrt[Cos[c + d*x]]*Ell
ipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*a^5*(a^2 - b^2)^2*d) - (b*(35*A*b^6 - 35*a^5*b*B + 38*a^3*b^3*B
- 15*a*b^5*B - a^2*b^4*(86*A - 3*C) + 3*a^4*b^2*(21*A - 2*C) + 15*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/
(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^5*(a - b)^2*(a + b)^3*d) + ((35*A*b^4 + 33*a^3*b*B - 15*a*b^
3*B + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]) + ((A
*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2) - ((7*A*b^4
+ 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*(13*A + C))*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]
]*(a + b*Sec[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.8052, antiderivative size = 598, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.209, Rules used = {4100, 4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac{\sin (c+d x) \left (-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+33 a^3 b B-15 a b^3 B+35 A b^4\right )}{12 a^3 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x) \left (-a^2 b^2 (13 A+C)+9 a^3 b B-5 a^4 C-3 a b^3 B+7 A b^4\right )}{4 a^2 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}+\frac{\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^4 b^2 (128 A-15 C)-a^2 b^4 (223 A-9 C)+8 a^6 (A+3 C)+99 a^3 b^3 B-72 a^5 b B-45 a b^5 B+105 A b^6\right )}{12 a^5 d \left (a^2-b^2\right )^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-a^2 b^3 (65 A-3 C)+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right )}{4 a^4 d \left (a^2-b^2\right )^2}-\frac{b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (3 a^4 b^2 (21 A-2 C)-a^2 b^4 (86 A-3 C)+38 a^3 b^3 B-35 a^5 b B+15 a^6 C-15 a b^5 B+35 A b^6\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^5 d (a-b)^2 (a+b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B + 3*a^4*b*(8*A - 3*C) - a^2*b^3*(65*A - 3*C))*Sqrt[Cos[c + d
*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a^2 - b^2)^2*d) + ((105*A*b^6 - 72*a^5*b*B + 99*a^3
*b^3*B - 45*a*b^5*B + a^4*b^2*(128*A - 15*C) - a^2*b^4*(223*A - 9*C) + 8*a^6*(A + 3*C))*Sqrt[Cos[c + d*x]]*Ell
ipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*a^5*(a^2 - b^2)^2*d) - (b*(35*A*b^6 - 35*a^5*b*B + 38*a^3*b^3*B
- 15*a*b^5*B - a^2*b^4*(86*A - 3*C) + 3*a^4*b^2*(21*A - 2*C) + 15*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/
(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^5*(a - b)^2*(a + b)^3*d) + ((35*A*b^4 + 33*a^3*b*B - 15*a*b^
3*B + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]) + ((A
*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2) - ((7*A*b^4
+ 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*(13*A + C))*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]
]*(a + b*Sec[c + d*x]))

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 4106

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d
_.)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))), x_Symbol] :> Dist[(A*b^2 - a*b*B + a^2*C)/(a^2*d^2), Int[(d*Csc[
e + f*x])^(3/2)/(a + b*Csc[e + f*x]), x], x] + Dist[1/a^2, Int[(a*A - (A*b - a*B)*Csc[e + f*x])/Sqrt[d*Csc[e +
f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]

Rule 3849

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[d*Sqrt[d*S
in[e + f*x]]*Sqrt[d*Csc[e + f*x]], Int[1/(Sqrt[d*Sin[e + f*x]]*(b + a*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d
, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 2805

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[c + d]), x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx &=\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac{\int \frac{\frac{1}{2} \left (7 A b^2-3 a b B-a^2 (4 A-3 C)\right )+2 a (A b-a B+b C) \sec (c+d x)-\frac{5}{2} \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (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 ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac{\left (7 A b^4+9 a^3 b B-3 a b^3 B-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}+\frac{\int \frac{\frac{1}{4} \left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 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)-\frac{3}{4} \left (7 A b^4+9 a^3 b B-3 a b^3 B-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{\left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}+\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac{\left (7 A b^4+9 a^3 b B-3 a b^3 B-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}-\frac{\int \frac{\frac{3}{8} \left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B+3 a^4 b (8 A-3 C)-a^2 b^3 (65 A-3 C)\right )+\frac{1}{2} a \left (7 A b^4+12 a^3 b B-3 a b^3 B-2 a^4 (A+3 C)-a^2 b^2 (14 A+3 C)\right ) \sec (c+d x)-\frac{1}{8} b \left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{3 a^3 \left (a^2-b^2\right )^2}\\ &=\frac{\left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}+\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac{\left (7 A b^4+9 a^3 b B-3 a b^3 B-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}-\frac{\int \frac{\frac{3}{8} a \left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B+3 a^4 b (8 A-3 C)-a^2 b^3 (65 A-3 C)\right )-\left (\frac{3}{8} b \left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B+3 a^4 b (8 A-3 C)-a^2 b^3 (65 A-3 C)\right )-\frac{1}{2} a^2 \left (7 A b^4+12 a^3 b B-3 a b^3 B-2 a^4 (A+3 C)-a^2 b^2 (14 A+3 C)\right )\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{3 a^5 \left (a^2-b^2\right )^2}-\frac{\left (b \left (35 A b^6-35 a^5 b B+38 a^3 b^3 B-15 a b^5 B-a^2 b^4 (86 A-3 C)+3 a^4 b^2 (21 A-2 C)+15 a^6 C\right )\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{8 a^5 \left (a^2-b^2\right )^2}\\ &=\frac{\left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}+\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac{\left (7 A b^4+9 a^3 b B-3 a b^3 B-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}-\frac{\left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B+3 a^4 b (8 A-3 C)-a^2 b^3 (65 A-3 C)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{8 a^4 \left (a^2-b^2\right )^2}+\frac{\left (105 A b^6-72 a^5 b B+99 a^3 b^3 B-45 a b^5 B+a^4 b^2 (128 A-15 C)-a^2 b^4 (223 A-9 C)+8 a^6 (A+3 C)\right ) \int \sqrt{\sec (c+d x)} \, dx}{24 a^5 \left (a^2-b^2\right )^2}-\frac{\left (b \left (35 A b^6-35 a^5 b B+38 a^3 b^3 B-15 a b^5 B-a^2 b^4 (86 A-3 C)+3 a^4 b^2 (21 A-2 C)+15 a^6 C\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{8 a^5 \left (a^2-b^2\right )^2}\\ &=-\frac{b \left (35 A b^6-35 a^5 b B+38 a^3 b^3 B-15 a b^5 B-a^2 b^4 (86 A-3 C)+3 a^4 b^2 (21 A-2 C)+15 a^6 C\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{4 a^5 (a-b)^2 (a+b)^3 d}+\frac{\left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}+\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac{\left (7 A b^4+9 a^3 b B-3 a b^3 B-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}-\frac{\left (\left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B+3 a^4 b (8 A-3 C)-a^2 b^3 (65 A-3 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{8 a^4 \left (a^2-b^2\right )^2}+\frac{\left (\left (105 A b^6-72 a^5 b B+99 a^3 b^3 B-45 a b^5 B+a^4 b^2 (128 A-15 C)-a^2 b^4 (223 A-9 C)+8 a^6 (A+3 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{24 a^5 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B+3 a^4 b (8 A-3 C)-a^2 b^3 (65 A-3 C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac{\left (105 A b^6-72 a^5 b B+99 a^3 b^3 B-45 a b^5 B+a^4 b^2 (128 A-15 C)-a^2 b^4 (223 A-9 C)+8 a^6 (A+3 C)\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{12 a^5 \left (a^2-b^2\right )^2 d}-\frac{b \left (35 A b^6-35 a^5 b B+38 a^3 b^3 B-15 a b^5 B-a^2 b^4 (86 A-3 C)+3 a^4 b^2 (21 A-2 C)+15 a^6 C\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{4 a^5 (a-b)^2 (a+b)^3 d}+\frac{\left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}+\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac{\left (7 A b^4+9 a^3 b B-3 a b^3 B-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}\\ \end{align*}

Mathematica [A]  time = 7.59956, size = 1121, normalized size = 1.87 \[ \frac{\sec (c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (-\frac{2 \left (16 A a^5+48 C a^5-96 b B a^4+112 A b^2 a^3+24 b^2 C a^3+24 b^3 B a^2-56 A b^4 a\right ) \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac{2 \left (24 B a^5-56 A b a^4-15 b C a^4-21 b^2 B a^3+73 A b^3 a^2-3 b^3 C a^2+15 b^4 B a-35 A b^5\right ) \left (\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right )+\Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right ) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}-\frac{2 \left (24 B a^5-72 A b a^4+27 b C a^4-87 b^2 B a^3+195 A b^3 a^2-9 b^3 C a^2+45 b^4 B a-105 A b^5\right ) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left (\Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2-2 b \sec ^2(c+d x) a+2 b a+2 b E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+(a-2 b) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 b^2 \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt{\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}\right ) (b+a \cos (c+d x))^3}{24 a^3 (a-b)^2 (a+b)^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (\frac{b \left (-9 C a^4+13 b B a^3-17 A b^2 a^2+3 b^2 C a^2-7 b^3 B a+11 A b^4\right ) \sin (c+d x)}{2 a^4 \left (b^2-a^2\right )^2}-\frac{A \sin (c+d x) b^5-a B \sin (c+d x) b^4+a^2 C \sin (c+d x) b^3}{a^4 \left (a^2-b^2\right ) (b+a \cos (c+d x))^2}+\frac{-13 A \sin (c+d x) b^6+9 a B \sin (c+d x) b^5+19 a^2 A \sin (c+d x) b^4-5 a^2 C \sin (c+d x) b^4-15 a^3 B \sin (c+d x) b^3+11 a^4 C \sin (c+d x) b^2}{2 a^4 \left (a^2-b^2\right )^2 (b+a \cos (c+d x))}+\frac{2 A \sin (2 (c+d x))}{3 a^3}\right ) (b+a \cos (c+d x))^3}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(16*a^5*A + 112*a^3*A*b^2 -
56*a*A*b^4 - 96*a^4*b*B + 24*a^2*b^3*B + 48*a^5*C + 24*a^3*b^2*C)*Cos[c + d*x]^2*EllipticPi[-(b/a), -ArcSin[Sq
rt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1
- Cos[c + d*x]^2)) + (2*(-56*a^4*A*b + 73*a^2*A*b^3 - 35*A*b^5 + 24*a^5*B - 21*a^3*b^2*B + 15*a*b^4*B - 15*a^4
*b*C - 3*a^2*b^3*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] + EllipticPi[-(b/a), -ArcSin[Sqr
t[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1
- Cos[c + d*x]^2)) - (2*(-72*a^4*A*b + 195*a^2*A*b^3 - 105*A*b^5 + 24*a^5*B - 87*a^3*b^2*B + 45*a*b^4*B + 27*a
^4*b*C - 9*a^2*b^3*C)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(2*a*b - 2*a*b*Sec[c + d*x]^2 + 2*a*b*EllipticE[Ar
cSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + a*(a - 2*b)*EllipticF[ArcSin[Sqrt[
Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + a^2*EllipticPi[-(b/a), -ArcSin[Sqrt[Sec[c +
d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*b^2*EllipticPi[-(b/a), -ArcSin[Sqrt[Sec[c + d*x]]]
, -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]
^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(24*a^3*(a - b)^2*(a + b)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*C
os[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*
Sec[c + d*x]^2)*((b*(-17*a^2*A*b^2 + 11*A*b^4 + 13*a^3*b*B - 7*a*b^3*B - 9*a^4*C + 3*a^2*b^2*C)*Sin[c + d*x])/
(2*a^4*(-a^2 + b^2)^2) - (A*b^5*Sin[c + d*x] - a*b^4*B*Sin[c + d*x] + a^2*b^3*C*Sin[c + d*x])/(a^4*(a^2 - b^2)
*(b + a*Cos[c + d*x])^2) + (19*a^2*A*b^4*Sin[c + d*x] - 13*A*b^6*Sin[c + d*x] - 15*a^3*b^3*B*Sin[c + d*x] + 9*
a*b^5*B*Sin[c + d*x] + 11*a^4*b^2*C*Sin[c + d*x] - 5*a^2*b^4*C*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^2*(b + a*Cos[c
+ d*x])) + (2*A*Sin[2*(c + d*x)])/(3*a^3)))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c
+ d*x])^3)

________________________________________________________________________________________

Maple [B]  time = 13.84, size = 2289, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2/3/a^5*(4*A*a^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/
2*c)^4+a^2*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2
))+18*A*b^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)
)+9*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b-
2*A*a^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2-9*B*a*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1
)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-3*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2
)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2+3*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*
EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+2/a^5*b^2*(5*A
*b^2-4*B*a*b+3*C*a^2)*(a^2/b/(a^2-b^2)*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)
/(2*cos(1/2*d*x+1/2*c)^2*a-a+b)-1/2/(a+b)/b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2
*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+1/2*a/b/(a^2-b^2)*(sin
(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2
)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-1/2*a/b/(a^2-b^2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^
2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-1/2/b/(a
^2-b^2)/(a^2-a*b)*a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+
sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2*a/(a-b),2^(1/2))+3/2*b/(a^2-b^2)/(a^2-a*b)*a*(sin(
1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)
*EllipticPi(cos(1/2*d*x+1/2*c),2*a/(a-b),2^(1/2)))+2/a^4*b*(10*A*b^2-6*B*a*b+3*C*a^2)/(a^2-a*b)*(sin(1/2*d*x+1
/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*Elliptic
Pi(cos(1/2*d*x+1/2*c),2*a/(a-b),2^(1/2))-2*b^3*(A*b^2-B*a*b+C*a^2)/a^5*(1/2*a^2/b/(a^2-b^2)*cos(1/2*d*x+1/2*c)
*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*cos(1/2*d*x+1/2*c)^2*a-a+b)^2+3/4*a^2*(a^2-3*b^2)/b^2
/(a^2-b^2)^2*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*cos(1/2*d*x+1/2*c)^2*a
-a+b)-3/8/(a+b)/(a^2-b^2)/b^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1
/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^2-1/4/(a+b)/(a^2-b^2)/b*(sin(1/2*d
*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*Elli
pticF(cos(1/2*d*x+1/2*c),2^(1/2))*a+7/8/(a+b)/(a^2-b^2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+
1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+3/8*a^3/b^
2/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*
d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-9/8*a/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*c
os(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c)
,2^(1/2))-3/8*a^3/b^2/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d
*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+9/8*a/(a^2-b^2)^2*(sin(1/2*d*x+1
/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*Elliptic
E(cos(1/2*d*x+1/2*c),2^(1/2))-3/8/(a-b)/(a+b)/(a^2-b^2)/b^2/(a^2-a*b)*a^5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos
(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),
2*a/(a-b),2^(1/2))+3/4/(a-b)/(a+b)/(a^2-b^2)/(a^2-a*b)*a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)
^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2*a/(a-b),2^(1/
2))-15/8/(a-b)/(a+b)/(a^2-b^2)*b^2/(a^2-a*b)*a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/
(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2*a/(a-b),2^(1/2))))/sin(1/
2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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