3.1025 \(\int \frac{\sec ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=667 \[ -\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (-a^2 b^2 (3 A-61 C)+15 a^3 b B-35 a^4 C-33 a b^3 B+b^4 (21 A-8 C)\right )}{12 b^3 d \left (a^2-b^2\right )^2}-\frac{\sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (a^2 b^2 (A+13 C)+3 a^3 b B-7 a^4 C-9 a b^3 B+5 A b^4\right )}{4 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (-a^2 b^2 (3 A-61 C)+15 a^3 b B-35 a^4 C-33 a b^3 B+b^4 (21 A-8 C)\right )}{12 b^3 d \left (a^2-b^2\right )^2}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left (-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+15 a^4 b B-35 a^5 C+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{4 b^4 d \left (a^2-b^2\right )^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+15 a^4 b B-35 a^5 C+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{4 b^4 d \left (a^2-b^2\right )^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+38 a^3 b^3 B-15 a^5 b B+35 a^6 C-35 a b^5 B+15 A b^6\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^4 d (a-b)^2 (a+b)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 2.24793, antiderivative size = 667, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.209, Rules used = {4098, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ -\frac{\sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (a^2 b^2 (A+13 C)+3 a^3 b B-7 a^4 C-9 a b^3 B+5 A b^4\right )}{4 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (-a^2 b^2 (3 A-61 C)+15 a^3 b B-35 a^4 C-33 a b^3 B+b^4 (21 A-8 C)\right )}{12 b^3 d \left (a^2-b^2\right )^2}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left (-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+15 a^4 b B-35 a^5 C+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{4 b^4 d \left (a^2-b^2\right )^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-a^2 b^2 (3 A-61 C)+15 a^3 b B-35 a^4 C-33 a b^3 B+b^4 (21 A-8 C)\right )}{12 b^3 d \left (a^2-b^2\right )^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+15 a^4 b B-35 a^5 C+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{4 b^4 d \left (a^2-b^2\right )^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+38 a^3 b^3 B-15 a^5 b B+35 a^6 C-35 a b^5 B+15 A b^6\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^4 d (a-b)^2 (a+b)^3} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]

[Out]

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

Rule 4098

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

Rule 4102

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[(C*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m
+ 1)*(d*Csc[e + f*x])^(n - 1))/(b*f*(m + n + 1)), x] + Dist[d/(b*(m + n + 1)), Int[(a + b*Csc[e + f*x])^m*(d*
Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + (A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) - a*C
*n)*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] && GtQ[n, 0]

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{\sec ^{\frac{7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx &=-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac{\int \frac{\sec ^{\frac{5}{2}}(c+d x) \left (\frac{5}{2} \left (A b^2-a (b B-a C)\right )+2 b (b B-a (A+C)) \sec (c+d x)-\frac{1}{2} \left (3 A b^2-3 a b B+7 a^2 C-4 b^2 C\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left (\frac{3}{4} \left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right )+b \left (a^2 b B+2 b^3 B+a^3 C-a b^2 (3 A+4 C)\right ) \sec (c+d x)-\frac{1}{4} \left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{\int \frac{\sqrt{\sec (c+d x)} \left (-\frac{1}{8} a \left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right )+\frac{1}{2} b \left (3 a^3 b B-12 a b^3 B-7 a^4 C+2 b^4 (3 A+C)+a^2 b^2 (3 A+14 C)\right ) \sec (c+d x)+\frac{3}{8} \left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{3 b^3 \left (a^2-b^2\right )^2}\\ &=\frac{\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{2 \int \frac{-\frac{3}{16} a \left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right )-\frac{1}{4} b \left (15 a^4 b B-30 a^2 b^3 B+6 b^5 B-a^3 b^2 (3 A-64 C)+4 a b^4 (3 A-5 C)-35 a^5 C\right ) \sec (c+d x)-\frac{1}{16} \left (45 a^5 b B-99 a^3 b^3 B+72 a b^5 B-a^4 b^2 (9 A-223 C)+a^2 b^4 (15 A-128 C)-105 a^6 C-8 b^6 (3 A+C)\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{3 b^4 \left (a^2-b^2\right )^2}\\ &=\frac{\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{2 \int \frac{-\frac{3}{16} a^2 \left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right )-\left (-\frac{3}{16} a b \left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right )+\frac{1}{4} a b \left (15 a^4 b B-30 a^2 b^3 B+6 b^5 B-a^3 b^2 (3 A-64 C)+4 a b^4 (3 A-5 C)-35 a^5 C\right )\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{3 a^2 b^4 \left (a^2-b^2\right )^2}+\frac{\left (15 A b^6-15 a^5 b B+38 a^3 b^3 B-35 a b^5 B+a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 a^6 C\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{8 b^4 \left (a^2-b^2\right )^2}\\ &=\frac{\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \int \sqrt{\sec (c+d x)} \, dx}{24 b^3 \left (a^2-b^2\right )^2}-\frac{\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{8 b^4 \left (a^2-b^2\right )^2}+\frac{\left (\left (15 A b^6-15 a^5 b B+38 a^3 b^3 B-35 a b^5 B+a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 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 b^4 \left (a^2-b^2\right )^2}\\ &=\frac{\left (15 A b^6-15 a^5 b B+38 a^3 b^3 B-35 a b^5 B+a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 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-b)^2 b^4 (a+b)^3 d}+\frac{\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\left (\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{24 b^3 \left (a^2-b^2\right )^2}-\frac{\left (\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{8 b^4 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 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 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 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 b^3 \left (a^2-b^2\right )^2 d}+\frac{\left (15 A b^6-15 a^5 b B+38 a^3 b^3 B-35 a b^5 B+a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 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-b)^2 b^4 (a+b)^3 d}+\frac{\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}\\ \end{align*}

Mathematica [A]  time = 7.78109, size = 1161, normalized size = 1.74 \[ \frac{\sec (c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (-\frac{2 \left (-48 B b^6-96 a A b^5+160 a C b^5+240 a^2 B b^4+24 a^3 A b^3-512 a^3 C b^3-120 a^4 B b^2+280 a^5 C b\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 (315 C a^6-135 b B a^5+27 A b^2 a^4-641 b^2 C a^4+285 b^3 B a^3-57 A b^4 a^2+328 b^4 C a^2-168 b^5 B a+48 A b^6+16 b^6 C\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 (105 C a^6-45 b B a^5+9 A b^2 a^4-195 b^2 C a^4+87 b^3 B a^3-27 A b^4 a^2+72 b^4 C a^2-24 b^5 B a\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-b)^2 b^4 (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{\left (35 C a^5-15 b B a^4+3 A b^2 a^3-65 b^2 C a^3+29 b^3 B a^2-9 A b^4 a+24 b^4 C a-8 b^5 B\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^2}+\frac{-C \sin (c+d x) a^3+b B \sin (c+d x) a^2-A b^2 \sin (c+d x) a}{b^2 \left (b^2-a^2\right ) (b+a \cos (c+d x))^2}+\frac{9 C \sin (c+d x) a^5-5 b B \sin (c+d x) a^4+A b^2 \sin (c+d x) a^3-15 b^2 C \sin (c+d x) a^3+11 b^3 B \sin (c+d x) a^2-7 A b^4 \sin (c+d x) a}{2 b^3 \left (b^2-a^2\right )^2 (b+a \cos (c+d x))}+\frac{4 C \tan (c+d x)}{3 b^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[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^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*(24*a^3*A*b^3 - 96*a*A*b^5 -
120*a^4*b^2*B + 240*a^2*b^4*B - 48*b^6*B + 280*a^5*b*C - 512*a^3*b^3*C + 160*a*b^5*C)*Cos[c + d*x]^2*Elliptic
Pi[-(b/a), -ArcSin[Sqrt[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*(27*a^4*A*b^2 - 57*a^2*A*b^4 + 48*A*b^6 - 135*a^5*b*B + 285*a^3*b
^3*B - 168*a*b^5*B + 315*a^6*C - 641*a^4*b^2*C + 328*a^2*b^4*C + 16*b^6*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sq
rt[Sec[c + d*x]]], -1] + EllipticPi[-(b/a), -ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Se
c[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) - (2*(9*a^4*A*b^2 - 27*a^2*A*b^4 - 4
5*a^5*b*B + 87*a^3*b^3*B - 24*a*b^5*B + 105*a^6*C - 195*a^4*b^2*C + 72*a^2*b^4*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[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*S
qrt[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 - Se
c[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 - b)^2*b^4*(a + b)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[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)*(-((3*a^3*A*b^2 - 9*a*A*b^4 - 15*a
^4*b*B + 29*a^2*b^3*B - 8*b^5*B + 35*a^5*C - 65*a^3*b^2*C + 24*a*b^4*C)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^2) +
(-(a*A*b^2*Sin[c + d*x]) + a^2*b*B*Sin[c + d*x] - a^3*C*Sin[c + d*x])/(b^2*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2
) + (a^3*A*b^2*Sin[c + d*x] - 7*a*A*b^4*Sin[c + d*x] - 5*a^4*b*B*Sin[c + d*x] + 11*a^2*b^3*B*Sin[c + d*x] + 9*
a^5*C*Sin[c + d*x] - 15*a^3*b^2*C*Sin[c + d*x])/(2*b^3*(-a^2 + b^2)^2*(b + a*Cos[c + d*x])) + (4*C*Tan[c + d*x
])/(3*b^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 = 20.032, size = 2185, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^(7/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^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*a*(B*b-2*C*a)/b^3*(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)*E
llipticF(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^2*(B*b-3*C*a)/b^4/(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)*EllipticPi(cos(1/2*d*x+1/2*c),2*a/(a-b),2^(1/2))+2*(A*b^2-B*a*b+C*a^2)/b
^2*(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)*EllipticF(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)*Ellipt
icF(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*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*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)*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)*EllipticE(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)*Ell
ipticPi(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)))+2*C/b^3*(-1/6*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)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+1/3*(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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2*(B*b-3*C*a)/b^4*(-(
sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(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))+2*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*cos(1/2*d*x+1
/2*c)*sin(1/2*d*x+1/2*c)^2)/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1))/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x
+1/2*c)^2-1)^(1/2)/d

________________________________________________________________________________________

Maxima [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(sec(d*x+c)^(7/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

Timed out

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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(sec(d*x+c)^(7/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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(sec(d*x+c)**(7/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)**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{{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac{7}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(7/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^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)*sec(d*x + c)^(7/2)/(b*sec(d*x + c) + a)^3, x)