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

[Out]

2/11*a*A*sin(d*x+c)/d/sec(d*x+c)^(9/2)+2/9*(A*b+B*a)*sin(d*x+c)/d/sec(d*x+c)^(7/2)+2/77*(9*A*a+11*B*b+11*C*a)*
sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/45*(7*A*b+7*B*a+9*C*b)*sin(d*x+c)/d/sec(d*x+c)^(3/2)+10/231*(9*A*a+11*B*b+11*C
*a)*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2/15*(7*A*b+7*B*a+9*C*b)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ell
ipticE(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+10/231*(9*A*a+11*B*b+11*C*a)*(cos(1/2*d
*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/
d

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Rubi [A]  time = 0.31, antiderivative size = 266, normalized size of antiderivative = 1.00, 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 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]

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 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 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 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]

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*}

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Mathematica [C]  time = 6.96, size = 1371, normalized size = 5.15 \[ \frac {60 a A F\left (\left .\frac {1}{2} (c+d x)\right |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 F\left (\left .\frac {1}{2} (c+d x)\right |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 F\left (\left .\frac {1}{2} (c+d x)\right |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 ) \, _2F_1\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 ) \, _2F_1\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 ) \, _2F_1\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)*(-1/180*((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])*Co
s[d*x]*Csc[c])/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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fricas [F]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b \sec \left (d x + c\right )^{3} + {\left (C a + B b\right )} \sec \left (d x + c\right )^{2} + A a + {\left (B a + A b\right )} \sec \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac {11}{2}}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}}{\sec \left (d x + c\right )^{\frac {11}{2}}}\,{d x} \]

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)

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maple [B]  time = 5.31, size = 611, normalized size = 2.30 \[ -\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (20160 A a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-50400 a A -12320 A b -12320 a B \right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (56880 a A +24640 A b +24640 a B +7920 B b +7920 a C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-34920 a A -22792 A b -22792 a B -11880 B b -11880 a C -5544 C b \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (13860 a A +10472 A b +10472 a B +9240 B b +9240 a C +5544 C b \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-2790 a A -1848 A b -1848 a B -2640 B b -2640 a C -1386 C b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-1617 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b +675 a A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-1617 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a +825 B b \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-2079 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b +825 a C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{3465 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b/cos(c + d*x))*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/cos(c + d*x))^(11/2),x)

[Out]

int(((a + b/cos(c + d*x))*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/cos(c + d*x))^(11/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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