3.989 \(\int \frac{(a+b \sec (c+d x)) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sec ^{\frac{11}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=266 \[ \frac{10 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) (9 a A+11 a C+11 b B)}{231 d}+\frac{2 \sin (c+d x) (7 a B+7 A b+9 b C)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) (9 a A+11 a C+11 b B)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{10 \sin (c+d x) (9 a A+11 a C+11 b B)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (7 a B+7 A b+9 b C)}{15 d}+\frac{2 (a B+A b) \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)} \]

[Out]

(2*(7*A*b + 7*a*B + 9*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (10*(9*a*
A + 11*b*B + 11*a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a*A*Sin[c +
d*x])/(11*d*Sec[c + d*x]^(9/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(9*a*A + 11*b*B
+ 11*a*C)*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (2*(7*A*b + 7*a*B + 9*b*C)*Sin[c + d*x])/(45*d*Sec[c + d*x
]^(3/2)) + (10*(9*a*A + 11*b*B + 11*a*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.308012, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4074, 4047, 3769, 3771, 2641, 4045, 2639} \[ \frac{2 \sin (c+d x) (7 a B+7 A b+9 b C)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) (9 a A+11 a C+11 b B)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{10 \sin (c+d x) (9 a A+11 a C+11 b B)}{231 d \sqrt{\sec (c+d x)}}+\frac{10 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (9 a A+11 a C+11 b B)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (7 a B+7 A b+9 b C)}{15 d}+\frac{2 (a B+A b) \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]

[Out]

(2*(7*A*b + 7*a*B + 9*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (10*(9*a*
A + 11*b*B + 11*a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a*A*Sin[c +
d*x])/(11*d*Sec[c + d*x]^(9/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(9*a*A + 11*b*B
+ 11*a*C)*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (2*(7*A*b + 7*a*B + 9*b*C)*Sin[c + d*x])/(45*d*Sec[c + d*x
]^(3/2)) + (10*(9*a*A + 11*b*B + 11*a*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])

Rule 4074

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_)), x_Symbol] :> Simp[(A*a*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*n), x]
+ Dist[1/(d*n), Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (n*(a*C + B*b) + A*a*(n + 1))*Csc[e + f*x]
+ b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]

Rule 4047

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(
C_.)), x_Symbol] :> Dist[B/b, Int[(b*Csc[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x
]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, x]

Rule 3769

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Csc[c + d*x])^(n + 1))/(b*d*n), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[2*n]

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 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]

Rule 4045

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(A*Cot[e
+ f*x]*(b*Csc[e + f*x])^m)/(f*m), x] + Dist[(C*m + A*(m + 1))/(b^2*m), Int[(b*Csc[e + f*x])^(m + 2), x], x] /
; FreeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]

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]

Rubi steps

\begin{align*} \int \frac{(a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac{11}{2}}(c+d x)} \, dx &=\frac{2 a A \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}-\frac{2}{11} \int \frac{-\frac{11}{2} (A b+a B)-\frac{1}{2} (9 a A+11 b B+11 a C) \sec (c+d x)-\frac{11}{2} b C \sec ^2(c+d x)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 a A \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}-\frac{2}{11} \int \frac{-\frac{11}{2} (A b+a B)-\frac{11}{2} b C \sec ^2(c+d x)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx-\frac{1}{11} (-9 a A-11 b B-11 a C) \int \frac{1}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a A \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 (A b+a B) \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (9 a A+11 b B+11 a C) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{1}{77} (5 (9 a A+11 b B+11 a C)) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx-\frac{1}{9} (-7 A b-7 a B-9 b C) \int \frac{1}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a A \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 (A b+a B) \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (9 a A+11 b B+11 a C) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (7 A b+7 a B+9 b C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{10 (9 a A+11 b B+11 a C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{1}{231} (5 (9 a A+11 b B+11 a C)) \int \sqrt{\sec (c+d x)} \, dx-\frac{1}{15} (-7 A b-7 a B-9 b C) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a A \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 (A b+a B) \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (9 a A+11 b B+11 a C) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (7 A b+7 a B+9 b C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{10 (9 a A+11 b B+11 a C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{1}{231} \left (5 (9 a A+11 b B+11 a C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx-\frac{1}{15} \left ((-7 A b-7 a B-9 b C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 (7 A b+7 a B+9 b C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{15 d}+\frac{10 (9 a A+11 b B+11 a C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{231 d}+\frac{2 a A \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 (A b+a B) \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (9 a A+11 b B+11 a C) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (7 A b+7 a B+9 b C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{10 (9 a A+11 b B+11 a C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 6.91667, size = 1371, normalized size = 5.15 \[ \frac{60 a A \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^{\frac{7}{2}}(c+d x)}{77 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{20 b B \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^{\frac{7}{2}}(c+d x)}{21 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{20 a C \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^{\frac{7}{2}}(c+d x)}{21 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{14 \sqrt{2} A b e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc (c) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right ) (a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^3(c+d x)}{45 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{14 \sqrt{2} a B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc (c) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right ) (a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^3(c+d x)}{45 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{2 \sqrt{2} b C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc (c) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right ) (a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^3(c+d x)}{5 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(a+b \sec (c+d x)) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (-\frac{(149 A b+198 C b+187 A \cos (2 c) b+234 C \cos (2 c) b+149 a B+187 a B \cos (2 c)) \cos (d x) \csc (c)}{180 d}+\frac{(1041 a A+1144 b B+1144 a C) \cos (2 d x) \sin (2 c)}{1848 d}+\frac{(43 A b+36 C b+43 a B) \cos (3 d x) \sin (3 c)}{180 d}+\frac{(16 a A+11 b B+11 a C) \cos (4 d x) \sin (4 c)}{154 d}+\frac{(A b+a B) \cos (5 d x) \sin (5 c)}{36 d}+\frac{a A \cos (6 d x) \sin (6 c)}{88 d}+\frac{(187 A b+234 C b+187 a B) \cos (c) \sin (d x)}{90 d}+\frac{(1041 a A+1144 b B+1144 a C) \cos (2 c) \sin (2 d x)}{1848 d}+\frac{(43 A b+36 C b+43 a B) \cos (3 c) \sin (3 d x)}{180 d}+\frac{(16 a A+11 b B+11 a C) \cos (4 c) \sin (4 d x)}{154 d}+\frac{(A b+a B) \cos (5 c) \sin (5 d x)}{36 d}+\frac{a A \cos (6 c) \sin (6 d x)}{88 d}\right )}{(b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]

[Out]

(-14*Sqrt[2]*A*b*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^3*
Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -
E^((2*I)*(c + d*x))])*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(45*d*E^(I*d*x)*(b + a*Cos
[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (14*Sqrt[2]*a*B*Sqrt[E^(I*(c + d*x))/(1 + E^((
2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((
2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(a + b*Sec[c + d*x])*(A +
B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(45*d*E^(I*d*x)*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[
2*c + 2*d*x])) - (2*Sqrt[2]*b*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*
Cos[c + d*x]^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1
/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d
*x)*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (60*a*A*Cos[c + d*x]^(7/2)*Ellip
ticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(77*d*(b
+ a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (20*b*B*Cos[c + d*x]^(7/2)*EllipticF[(
c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(21*d*(b + a*C
os[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (20*a*C*Cos[c + d*x]^(7/2)*EllipticF[(c + d*
x)/2, 2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(21*d*(b + a*Cos[c +
d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[
c + d*x]^2)*(-((149*A*b + 149*a*B + 198*b*C + 187*A*b*Cos[2*c] + 187*a*B*Cos[2*c] + 234*b*C*Cos[2*c])*Cos[d*x]
*Csc[c])/(180*d) + ((1041*a*A + 1144*b*B + 1144*a*C)*Cos[2*d*x]*Sin[2*c])/(1848*d) + ((43*A*b + 43*a*B + 36*b*
C)*Cos[3*d*x]*Sin[3*c])/(180*d) + ((16*a*A + 11*b*B + 11*a*C)*Cos[4*d*x]*Sin[4*c])/(154*d) + ((A*b + a*B)*Cos[
5*d*x]*Sin[5*c])/(36*d) + (a*A*Cos[6*d*x]*Sin[6*c])/(88*d) + ((187*A*b + 187*a*B + 234*b*C)*Cos[c]*Sin[d*x])/(
90*d) + ((1041*a*A + 1144*b*B + 1144*a*C)*Cos[2*c]*Sin[2*d*x])/(1848*d) + ((43*A*b + 43*a*B + 36*b*C)*Cos[3*c]
*Sin[3*d*x])/(180*d) + ((16*a*A + 11*b*B + 11*a*C)*Cos[4*c]*Sin[4*d*x])/(154*d) + ((A*b + a*B)*Cos[5*c]*Sin[5*
d*x])/(36*d) + (a*A*Cos[6*c]*Sin[6*d*x])/(88*d)))/((b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*
c + 2*d*x])*Sec[c + d*x]^(5/2))

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Maple [B]  time = 2.638, size = 611, normalized size = 2.3 \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))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(11/2),x)

[Out]

-2/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(20160*A*a*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*
c)^12+(-50400*A*a-12320*A*b-12320*B*a)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(56880*A*a+24640*A*b+24640*B*a
+7920*B*b+7920*C*a)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-34920*A*a-22792*A*b-22792*B*a-11880*B*b-11880*C*
a-5544*C*b)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(13860*A*a+10472*A*b+10472*B*a+9240*B*b+9240*C*a+5544*C*b)
*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-2790*A*a-1848*A*b-1848*B*a-2640*B*b-2640*C*a-1386*C*b)*sin(1/2*d*x+
1/2*c)^2*cos(1/2*d*x+1/2*c)-1617*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))*b+675*A*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))-1617*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+825*B*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))-2079*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)*EllipticE(cos(1/2*d
*x+1/2*c),2^(1/2))*b+825*a*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)))/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^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(-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))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(11/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{C b \sec \left (d x + c\right )^{3} +{\left (C a + B b\right )} \sec \left (d x + c\right )^{2} + A a +{\left (B a + A b\right )} \sec \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac{11}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(11/2),x, algorithm="fricas")

[Out]

integral((C*b*sec(d*x + c)^3 + (C*a + B*b)*sec(d*x + c)^2 + A*a + (B*a + A*b)*sec(d*x + c))/sec(d*x + c)^(11/2
), x)

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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))*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/sec(d*x+c)**(11/2),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 )}{\left (b \sec \left (d x + c\right ) + a\right )}}{\sec \left (d x + c\right )^{\frac{11}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(11/2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)/sec(d*x + c)^(11/2), x)