\(\int \frac {(a+b \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sqrt {\sec (c+d x)}} \, dx\) [1005]

Optimal result

Integrand size = 43, antiderivative size = 441 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=-\frac {2 \left (60 a^3 b B+36 a b^3 B-15 a^4 (A-C)+18 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (21 a^4 B+42 a^2 b^2 B+5 b^4 B+28 a^3 b (3 A+C)+4 a b^3 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (1098 a^3 b B+756 a b^3 B+192 a^4 C+21 b^4 (9 A+7 C)+7 a^2 b^2 (261 A+155 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 b \left (261 a^2 b B+75 b^3 B+64 a^3 C+2 a b^2 (147 A+101 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {2 \left (63 A b^2+117 a b B+48 a^2 C+49 b^2 C\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d}+\frac {2 (9 b B+8 a C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d} \]

[Out]

2/315*b*(261*B*a^2*b+75*B*b^3+64*a^3*C+2*a*b^2*(147*A+101*C))*sec(d*x+c)^(3/2)*sin(d*x+c)/d+2/315*(1098*B*a^3*
b+756*B*a*b^3+192*a^4*C+21*b^4*(9*A+7*C)+7*a^2*b^2*(261*A+155*C))*sin(d*x+c)*sec(d*x+c)^(1/2)/d+2/315*(63*A*b^
2+117*B*a*b+48*C*a^2+49*C*b^2)*(a+b*sec(d*x+c))^2*sin(d*x+c)*sec(d*x+c)^(1/2)/d+2/63*(9*B*b+8*C*a)*(a+b*sec(d*
x+c))^3*sin(d*x+c)*sec(d*x+c)^(1/2)/d+2/9*C*(a+b*sec(d*x+c))^4*sin(d*x+c)*sec(d*x+c)^(1/2)/d-2/15*(60*B*a^3*b+
36*B*a*b^3-15*a^4*(A-C)+18*a^2*b^2*(5*A+3*C)+b^4*(9*A+7*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*El
lipticE(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+2/21*(21*B*a^4+42*B*a^2*b^2+5*B*b^4+28
*a^3*b*(3*A+C)+4*a*b^3*(7*A+5*C))*(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

Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4181, 4161, 4132, 3856, 2720, 4131, 2719} \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^2}{315 d}+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{315 d}+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (192 a^4 C+1098 a^3 b B+7 a^2 b^2 (261 A+155 C)+756 a b^3 B+21 b^4 (9 A+7 C)\right )}{315 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right )}{21 d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right )}{15 d}+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{63 d}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d} \]

[In]

Int[((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]

[Out]

(-2*(60*a^3*b*B + 36*a*b^3*B - 15*a^4*(A - C) + 18*a^2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*E
llipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3*A +
C) + 4*a*b^3*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(1098*a
^3*b*B + 756*a*b^3*B + 192*a^4*C + 21*b^4*(9*A + 7*C) + 7*a^2*b^2*(261*A + 155*C))*Sqrt[Sec[c + d*x]]*Sin[c +
d*x])/(315*d) + (2*b*(261*a^2*b*B + 75*b^3*B + 64*a^3*C + 2*a*b^2*(147*A + 101*C))*Sec[c + d*x]^(3/2)*Sin[c +
d*x])/(315*d) + (2*(63*A*b^2 + 117*a*b*B + 48*a^2*C + 49*b^2*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*Sin[
c + d*x])/(315*d) + (2*(9*b*B + 8*a*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(63*d) + (2*C*S
qrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(9*d)

Rule 2719

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

Rule 2720

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

Rule 3856

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 4131

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

Rule 4132

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 4161

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

Rule 4181

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 C \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d}+\frac {2}{9} \int \frac {(a+b \sec (c+d x))^3 \left (\frac {1}{2} a (9 A-C)+\frac {1}{2} (9 A b+9 a B+7 b C) \sec (c+d x)+\frac {1}{2} (9 b B+8 a C) \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx \\ & = \frac {2 (9 b B+8 a C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d}+\frac {4}{63} \int \frac {(a+b \sec (c+d x))^2 \left (\frac {3}{4} a (21 a A-3 b B-5 a C)+\frac {1}{4} \left (126 a A b+63 a^2 B+45 b^2 B+82 a b C\right ) \sec (c+d x)+\frac {1}{4} \left (63 A b^2+117 a b B+48 a^2 C+49 b^2 C\right ) \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx \\ & = \frac {2 \left (63 A b^2+117 a b B+48 a^2 C+49 b^2 C\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d}+\frac {2 (9 b B+8 a C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d}+\frac {8}{315} \int \frac {(a+b \sec (c+d x)) \left (-\frac {1}{8} a \left (162 a b B-3 a^2 (105 A-41 C)+7 b^2 (9 A+7 C)\right )+\frac {1}{8} \left (315 a^3 B+531 a b^2 B+21 b^3 (9 A+7 C)+a^2 b (945 A+479 C)\right ) \sec (c+d x)+\frac {3}{8} \left (261 a^2 b B+75 b^3 B+64 a^3 C+2 a b^2 (147 A+101 C)\right ) \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx \\ & = \frac {2 b \left (261 a^2 b B+75 b^3 B+64 a^3 C+2 a b^2 (147 A+101 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {2 \left (63 A b^2+117 a b B+48 a^2 C+49 b^2 C\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d}+\frac {2 (9 b B+8 a C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d}+\frac {16}{945} \int \frac {-\frac {3}{16} a^2 \left (162 a b B-a^2 (315 A-123 C)+7 b^2 (9 A+7 C)\right )+\frac {45}{16} \left (21 a^4 B+42 a^2 b^2 B+5 b^4 B+28 a^3 b (3 A+C)+4 a b^3 (7 A+5 C)\right ) \sec (c+d x)+\frac {3}{16} \left (1098 a^3 b B+756 a b^3 B+192 a^4 C+21 b^4 (9 A+7 C)+7 a^2 b^2 (261 A+155 C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx \\ & = \frac {2 b \left (261 a^2 b B+75 b^3 B+64 a^3 C+2 a b^2 (147 A+101 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {2 \left (63 A b^2+117 a b B+48 a^2 C+49 b^2 C\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d}+\frac {2 (9 b B+8 a C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d}+\frac {16}{945} \int \frac {-\frac {3}{16} a^2 \left (162 a b B-a^2 (315 A-123 C)+7 b^2 (9 A+7 C)\right )+\frac {3}{16} \left (1098 a^3 b B+756 a b^3 B+192 a^4 C+21 b^4 (9 A+7 C)+7 a^2 b^2 (261 A+155 C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (21 a^4 B+42 a^2 b^2 B+5 b^4 B+28 a^3 b (3 A+C)+4 a b^3 (7 A+5 C)\right ) \int \sqrt {\sec (c+d x)} \, dx \\ & = \frac {2 \left (1098 a^3 b B+756 a b^3 B+192 a^4 C+21 b^4 (9 A+7 C)+7 a^2 b^2 (261 A+155 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 b \left (261 a^2 b B+75 b^3 B+64 a^3 C+2 a b^2 (147 A+101 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {2 \left (63 A b^2+117 a b B+48 a^2 C+49 b^2 C\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d}+\frac {2 (9 b B+8 a C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d}+\frac {1}{15} \left (-60 a^3 b B-36 a b^3 B+15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (\left (21 a^4 B+42 a^2 b^2 B+5 b^4 B+28 a^3 b (3 A+C)+4 a b^3 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 \left (21 a^4 B+42 a^2 b^2 B+5 b^4 B+28 a^3 b (3 A+C)+4 a b^3 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (1098 a^3 b B+756 a b^3 B+192 a^4 C+21 b^4 (9 A+7 C)+7 a^2 b^2 (261 A+155 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 b \left (261 a^2 b B+75 b^3 B+64 a^3 C+2 a b^2 (147 A+101 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {2 \left (63 A b^2+117 a b B+48 a^2 C+49 b^2 C\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d}+\frac {2 (9 b B+8 a C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d}+\frac {1}{15} \left (\left (-60 a^3 b B-36 a b^3 B+15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = -\frac {2 \left (60 a^3 b B+36 a b^3 B-15 a^4 (A-C)+18 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \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 {2 \left (21 a^4 B+42 a^2 b^2 B+5 b^4 B+28 a^3 b (3 A+C)+4 a b^3 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (1098 a^3 b B+756 a b^3 B+192 a^4 C+21 b^4 (9 A+7 C)+7 a^2 b^2 (261 A+155 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 b \left (261 a^2 b B+75 b^3 B+64 a^3 C+2 a b^2 (147 A+101 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {2 \left (63 A b^2+117 a b B+48 a^2 C+49 b^2 C\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d}+\frac {2 (9 b B+8 a C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 12.30 (sec) , antiderivative size = 609, normalized size of antiderivative = 1.38 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {2 \cos ^6(c+d x) \left (\frac {2 \left (105 a^4 A-630 a^2 A b^2-63 A b^4-420 a^3 b B-252 a b^3 B-105 a^4 C-378 a^2 b^2 C-49 b^4 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}+2 \left (420 a^3 A b+140 a A b^3+105 a^4 B+210 a^2 b^2 B+25 b^4 B+140 a^3 b C+100 a b^3 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}\right ) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{105 d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {4}{15} \left (90 a^2 A b^2+9 A b^4+60 a^3 b B+36 a b^3 B+15 a^4 C+54 a^2 b^2 C+7 b^4 C\right ) \sin (c+d x)+\frac {4}{7} \sec ^3(c+d x) \left (b^4 B \sin (c+d x)+4 a b^3 C \sin (c+d x)\right )+\frac {4}{21} \sec (c+d x) \left (28 a A b^3 \sin (c+d x)+42 a^2 b^2 B \sin (c+d x)+5 b^4 B \sin (c+d x)+28 a^3 b C \sin (c+d x)+20 a b^3 C \sin (c+d x)\right )+\frac {4}{45} \sec ^2(c+d x) \left (9 A b^4 \sin (c+d x)+36 a b^3 B \sin (c+d x)+54 a^2 b^2 C \sin (c+d x)+7 b^4 C \sin (c+d x)\right )+\frac {4}{9} b^4 C \sec ^3(c+d x) \tan (c+d x)\right )}{d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {11}{2}}(c+d x)} \]

[In]

Integrate[((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]

[Out]

(2*Cos[c + d*x]^6*((2*(105*a^4*A - 630*a^2*A*b^2 - 63*A*b^4 - 420*a^3*b*B - 252*a*b^3*B - 105*a^4*C - 378*a^2*
b^2*C - 49*b^4*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(420*a^3*A*b + 140*a*
A*b^3 + 105*a^4*B + 210*a^2*b^2*B + 25*b^4*B + 140*a^3*b*C + 100*a*b^3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*
x)/2, 2]*Sqrt[Sec[c + d*x]])*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(105*d*(b + a*Cos
[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]
+ C*Sec[c + d*x]^2)*((4*(90*a^2*A*b^2 + 9*A*b^4 + 60*a^3*b*B + 36*a*b^3*B + 15*a^4*C + 54*a^2*b^2*C + 7*b^4*C)
*Sin[c + d*x])/15 + (4*Sec[c + d*x]^3*(b^4*B*Sin[c + d*x] + 4*a*b^3*C*Sin[c + d*x]))/7 + (4*Sec[c + d*x]*(28*a
*A*b^3*Sin[c + d*x] + 42*a^2*b^2*B*Sin[c + d*x] + 5*b^4*B*Sin[c + d*x] + 28*a^3*b*C*Sin[c + d*x] + 20*a*b^3*C*
Sin[c + d*x]))/21 + (4*Sec[c + d*x]^2*(9*A*b^4*Sin[c + d*x] + 36*a*b^3*B*Sin[c + d*x] + 54*a^2*b^2*C*Sin[c + d
*x] + 7*b^4*C*Sin[c + d*x]))/45 + (4*b^4*C*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C
+ 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(11/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1522\) vs. \(2(461)=922\).

Time = 9.78 (sec) , antiderivative size = 1523, normalized size of antiderivative = 3.45

method	result	size
default	\(\text {Expression too large to display}\)	\(1523\)
parts	\(\text {Expression too large to display}\)	\(1742\)

[In]

int((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*B*a^4*(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*A*a^4*(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))+8*A*a^3*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))+2*
A*a^4*(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))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))+2*C*b^4*(-1/144*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)^5-7/180*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)^3-14/15*sin(1/2*d*
x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)+7/15*(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(c
os(1/2*d*x+1/2*c),2^(1/2))-7/15*(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))-EllipticE(cos(1/2*d*x+1/2*c),2^(1
/2))))+2*b^3*(B*b+4*C*a)*(-1/56*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)^4-5/42*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+5/21*(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)))+2/5*b^2*(A*b^2+4*B*a*b+6*C*a^2)/(8*sin(1/
2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+1/2*c)^2*(24*sin(1/2*d*x+1/2*c)^6
*cos(1/2*d*x+1/2*c)-12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2
*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4-24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+12*EllipticE(cos(1/2*d*x+1/2*c),2
^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^2+8*sin(1/2*d*x+1/2*c
)^2*cos(1/2*d*x+1/2*c)-3*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(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)+2*a^2*(6*A*b^2+4*B*a*b+C*a^2)/sin(1/2*d*x
+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c
)^2*cos(1/2*d*x+1/2*c)-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2
*c)^2)^(1/2))+4*a*b*(2*A*b^2+3*B*a*b+2*C*a^2)*(-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))))/sin(1/2*d*x+1/2*c
)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.17 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=-\frac {15 \, \sqrt {2} {\left (21 i \, B a^{4} + 28 i \, {\left (3 \, A + C\right )} a^{3} b + 42 i \, B a^{2} b^{2} + 4 i \, {\left (7 \, A + 5 \, C\right )} a b^{3} + 5 i \, B b^{4}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-21 i \, B a^{4} - 28 i \, {\left (3 \, A + C\right )} a^{3} b - 42 i \, B a^{2} b^{2} - 4 i \, {\left (7 \, A + 5 \, C\right )} a b^{3} - 5 i \, B b^{4}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-15 i \, {\left (A - C\right )} a^{4} + 60 i \, B a^{3} b + 18 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b^{2} + 36 i \, B a b^{3} + i \, {\left (9 \, A + 7 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (15 i \, {\left (A - C\right )} a^{4} - 60 i \, B a^{3} b - 18 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b^{2} - 36 i \, B a b^{3} - i \, {\left (9 \, A + 7 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (35 \, C b^{4} + 21 \, {\left (15 \, C a^{4} + 60 \, B a^{3} b + 18 \, {\left (5 \, A + 3 \, C\right )} a^{2} b^{2} + 36 \, B a b^{3} + {\left (9 \, A + 7 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left (28 \, C a^{3} b + 42 \, B a^{2} b^{2} + 4 \, {\left (7 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left (54 \, C a^{2} b^{2} + 36 \, B a b^{3} + {\left (9 \, A + 7 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 45 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d \cos \left (d x + c\right )^{4}} \]

[In]

integrate((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

-1/315*(15*sqrt(2)*(21*I*B*a^4 + 28*I*(3*A + C)*a^3*b + 42*I*B*a^2*b^2 + 4*I*(7*A + 5*C)*a*b^3 + 5*I*B*b^4)*co
s(d*x + c)^4*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-21*I*B*a^4 - 28*I*(3*A +
C)*a^3*b - 42*I*B*a^2*b^2 - 4*I*(7*A + 5*C)*a*b^3 - 5*I*B*b^4)*cos(d*x + c)^4*weierstrassPInverse(-4, 0, cos(
d*x + c) - I*sin(d*x + c)) + 21*sqrt(2)*(-15*I*(A - C)*a^4 + 60*I*B*a^3*b + 18*I*(5*A + 3*C)*a^2*b^2 + 36*I*B*
a*b^3 + I*(9*A + 7*C)*b^4)*cos(d*x + c)^4*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*s
in(d*x + c))) + 21*sqrt(2)*(15*I*(A - C)*a^4 - 60*I*B*a^3*b - 18*I*(5*A + 3*C)*a^2*b^2 - 36*I*B*a*b^3 - I*(9*A
+ 7*C)*b^4)*cos(d*x + c)^4*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)))
- 2*(35*C*b^4 + 21*(15*C*a^4 + 60*B*a^3*b + 18*(5*A + 3*C)*a^2*b^2 + 36*B*a*b^3 + (9*A + 7*C)*b^4)*cos(d*x + c
)^4 + 15*(28*C*a^3*b + 42*B*a^2*b^2 + 4*(7*A + 5*C)*a*b^3 + 5*B*b^4)*cos(d*x + c)^3 + 7*(54*C*a^2*b^2 + 36*B*a
*b^3 + (9*A + 7*C)*b^4)*cos(d*x + c)^2 + 45*(4*C*a*b^3 + B*b^4)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))
/(d*cos(d*x + c)^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*sec(d*x+c))**4*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/sec(d*x+c)**(1/2),x)

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1/2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^4/sqrt(sec(d*x + c)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]

[In]

int(((a + b/cos(c + d*x))^4*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/cos(c + d*x))^(1/2),x)

[Out]

int(((a + b/cos(c + d*x))^4*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/cos(c + d*x))^(1/2), x)