3.1009 \(\int \frac {(a+b \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=426 \[ \frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{315 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 b^2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (7 a^2 (7 A+9 C)+162 a b B+3 b^2 (41 A-105 C)\right )}{315 d}+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{315 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right )}{21 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^4 (7 A+9 C)+36 a^3 b B+18 a^2 b^2 (3 A+5 C)+60 a b^3 B+15 b^4 (A-C)\right )}{15 d}+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]

[Out]

2/315*(48*A*b^2+117*a*b*B+7*a^2*(7*A+9*C))*(a+b*sec(d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/63*(8*A*b+9*B*a)
*(a+b*sec(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/9*A*(a+b*sec(d*x+c))^4*sin(d*x+c)/d/sec(d*x+c)^(7/2)+2/315
*a*(64*A*b^3+75*a^3*B+261*a*b^2*B+a^2*(202*A*b+294*C*b))*sin(d*x+c)/d/sec(d*x+c)^(1/2)-2/315*b^2*(162*a*b*B+3*
b^2*(41*A-105*C)+7*a^2*(7*A+9*C))*sin(d*x+c)*sec(d*x+c)^(1/2)/d+2/15*(36*a^3*b*B+60*a*b^3*B+15*b^4*(A-C)+18*a^
2*b^2*(3*A+5*C)+a^4*(7*A+9*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(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*(5*a^4*B+42*a^2*b^2*B+21*b^4*B+28*a*b^3*(A+3*C)+4*a^3*b*(5*A+7
*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)*se
c(d*x+c)^(1/2)/d

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Rubi [A]  time = 1.31, antiderivative size = 426, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4094, 4074, 4047, 3771, 2641, 4046, 2639} \[ \frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{315 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 b^2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (7 a^2 (7 A+9 C)+162 a b B+3 b^2 (41 A-105 C)\right )}{315 d}+\frac {2 a \sin (c+d x) \left (a^2 (202 A b+294 b C)+75 a^3 B+261 a b^2 B+64 A b^3\right )}{315 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (4 a^3 b (5 A+7 C)+42 a^2 b^2 B+5 a^4 B+28 a b^3 (A+3 C)+21 b^4 B\right )}{21 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+36 a^3 b B+60 a b^3 B+15 b^4 (A-C)\right )}{15 d}+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]

[Out]

(2*(36*a^3*b*B + 60*a*b^3*B + 15*b^4*(A - C) + 18*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*El
lipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(5*a^4*B + 42*a^2*b^2*B + 21*b^4*B + 28*a*b^3*(A + 3*C
) + 4*a^3*b*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(64*A*
b^3 + 75*a^3*B + 261*a*b^2*B + a^2*(202*A*b + 294*b*C))*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]) - (2*b^2*(162
*a*b*B + 3*b^2*(41*A - 105*C) + 7*a^2*(7*A + 9*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (2*(48*A*b^2 + 1
17*a*b*B + 7*a^2*(7*A + 9*C))*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*(8*A*b + 9*
a*B)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*A*(a + b*Sec[c + d*x])^4*Sin[c + d*x]
)/(9*d*Sec[c + d*x]^(7/2))

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 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 4046

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 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]

Rule 4094

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

Rubi steps

\begin {align*} \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx &=\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2}{9} \int \frac {(a+b \sec (c+d x))^3 \left (\frac {1}{2} (8 A b+9 a B)+\frac {1}{2} (7 a A+9 b B+9 a C) \sec (c+d x)-\frac {1}{2} b (A-9 C) \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {4}{63} \int \frac {(a+b \sec (c+d x))^2 \left (\frac {1}{4} \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right )+\frac {1}{4} \left (82 a A b+45 a^2 B+63 b^2 B+126 a b C\right ) \sec (c+d x)-\frac {3}{4} b (5 A b+3 a B-21 b C) \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {8}{315} \int \frac {(a+b \sec (c+d x)) \left (\frac {3}{8} \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right )+\frac {1}{8} \left (531 a^2 b B+315 b^3 B+21 a^3 (7 A+9 C)+a b^2 (479 A+945 C)\right ) \sec (c+d x)-\frac {1}{8} b \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right ) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)}}+\frac {2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {16}{945} \int \frac {-\frac {3}{16} \left (192 A b^4+756 a^3 b B+1098 a b^3 B+21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)\right )-\frac {45}{16} \left (5 a^4 B+42 a^2 b^2 B+21 b^4 B+28 a b^3 (A+3 C)+4 a^3 b (5 A+7 C)\right ) \sec (c+d x)+\frac {3}{16} b^2 \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {2 a \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right ) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)}}+\frac {2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {16}{945} \int \frac {-\frac {3}{16} \left (192 A b^4+756 a^3 b B+1098 a b^3 B+21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)\right )+\frac {3}{16} b^2 \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx-\frac {1}{21} \left (-5 a^4 B-42 a^2 b^2 B-21 b^4 B-28 a b^3 (A+3 C)-4 a^3 b (5 A+7 C)\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=\frac {2 a \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right ) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)}}-\frac {2 b^2 \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {1}{15} \left (-36 a^3 b B-60 a b^3 B-15 b^4 (A-C)-18 a^2 b^2 (3 A+5 C)-a^4 (7 A+9 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx-\frac {1}{21} \left (\left (-5 a^4 B-42 a^2 b^2 B-21 b^4 B-28 a b^3 (A+3 C)-4 a^3 b (5 A+7 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 (5 a^4 B+42 a^2 b^2 B+21 b^4 B+28 a b^3 (A+3 C)+4 a^3 b (5 A+7 C)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 a \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right ) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)}}-\frac {2 b^2 \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {1}{15} \left (\left (-36 a^3 b B-60 a b^3 B-15 b^4 (A-C)-18 a^2 b^2 (3 A+5 C)-a^4 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 \left (36 a^3 b B+60 a b^3 B+15 b^4 (A-C)+18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 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 (5 a^4 B+42 a^2 b^2 B+21 b^4 B+28 a b^3 (A+3 C)+4 a^3 b (5 A+7 C)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 a \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right ) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)}}-\frac {2 b^2 \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\\ \end {align*}

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Mathematica [A]  time = 7.34, size = 517, normalized size = 1.21 \[ \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {1}{36} a^4 A \sin (5 (c+d x))+\frac {1}{14} a^3 (a B+4 A b) \sin (4 (c+d x))+\frac {1}{180} a^2 \sin (3 (c+d x)) \left (43 a^2 A+36 a^2 C+144 a b B+216 A b^2\right )+\frac {1}{21} a \sin (2 (c+d x)) \left (13 a^3 B+52 a^2 A b+56 a^2 b C+84 a b^2 B+56 A b^3\right )+\frac {1}{90} \sin (c+d x) \left (19 a^4 A+18 a^4 C+72 a^3 b B+108 a^2 A b^2+360 b^4 C\right )\right )}{d \sec ^{\frac {11}{2}}(c+d x) (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac {2 \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (25 a^4 B+100 a^3 A b+140 a^3 b C+210 a^2 b^2 B+140 a A b^3+420 a b^3 C+105 b^4 B\right )+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (49 a^4 A+63 a^4 C+252 a^3 b B+378 a^2 A b^2+630 a^2 b^2 C+420 a b^3 B+105 A b^4-105 b^4 C\right )}{\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}\right )}{105 d (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]

[Out]

(2*Cos[c + d*x]^6*((2*(49*a^4*A + 378*a^2*A*b^2 + 105*A*b^4 + 252*a^3*b*B + 420*a*b^3*B + 63*a^4*C + 630*a^2*b
^2*C - 105*b^4*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(100*a^3*A*b + 140*a*
A*b^3 + 25*a^4*B + 210*a^2*b^2*B + 105*b^4*B + 140*a^3*b*C + 420*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)*(((19*a^4*A + 108*a^2*A*b^2 + 72*a^3*b*B + 18*a^4*C + 360*b^4*C)*Sin[c + d*x])/90 + (a*(52
*a^2*A*b + 56*A*b^3 + 13*a^3*B + 84*a*b^2*B + 56*a^2*b*C)*Sin[2*(c + d*x)])/21 + (a^2*(43*a^2*A + 216*A*b^2 +
144*a*b*B + 36*a^2*C)*Sin[3*(c + d*x)])/180 + (a^3*(4*A*b + a*B)*Sin[4*(c + d*x)])/14 + (a^4*A*Sin[5*(c + d*x)
])/36))/(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))

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fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b^{4} \sec \left (d x + c\right )^{6} + {\left (4 \, C a b^{3} + B b^{4}\right )} \sec \left (d x + c\right )^{5} + A a^{4} + {\left (6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \sec \left (d x + c\right )^{4} + 2 \, {\left (2 \, C a^{3} b + 3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \sec \left (d x + c\right )^{3} + {\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} \sec \left (d x + c\right )^{2} + {\left (B a^{4} + 4 \, A a^{3} b\right )} \sec \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac {9}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)^(9/2),x, algorithm="fricas")

[Out]

integral((C*b^4*sec(d*x + c)^6 + (4*C*a*b^3 + B*b^4)*sec(d*x + c)^5 + A*a^4 + (6*C*a^2*b^2 + 4*B*a*b^3 + A*b^4
)*sec(d*x + c)^4 + 2*(2*C*a^3*b + 3*B*a^2*b^2 + 2*A*a*b^3)*sec(d*x + c)^3 + (C*a^4 + 4*B*a^3*b + 6*A*a^2*b^2)*
sec(d*x + c)^2 + (B*a^4 + 4*A*a^3*b)*sec(d*x + c))/sec(d*x + c)^(9/2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{4}}{\sec \left (d x + c\right )^{\frac {9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)^(9/2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^4/sec(d*x + c)^(9/2), x)

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maple [B]  time = 6.80, size = 1652, normalized size = 3.88 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)^(9/2),x)

[Out]

-2/315*(-1120*A*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*a^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)
^10+80*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(28*A*a+36*A*b+9*B*a)*sin(1/2*d*x+1/2*c)^8*cos
(1/2*d*x+1/2*c)-8*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*a^2*(259*A*a^2+540*A*a*b+378*A*b^2+135*
B*a^2+252*B*a*b+63*C*a^2)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+56*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*
c)^2)^(1/2)*a*(17*A*a^3+60*A*a^2*b+54*A*a*b^2+30*A*b^3+15*B*a^3+36*B*a^2*b+45*B*a*b^2+9*C*a^3+30*C*a^2*b)*sin(
1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-6*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(28*A*a^4+160*A*a^3
*b+126*A*a^2*b^2+140*A*a*b^3+40*B*a^4+84*B*a^3*b+210*B*a^2*b^2+21*C*a^4+140*C*a^3*b+105*C*b^4)*sin(1/2*d*x+1/2
*c)^2*cos(1/2*d*x+1/2*c)+300*A*a^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)*EllipticF(c
os(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)+420*a*A*b^3*(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)-147*A*(-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)^(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^4-1134*A*(-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)^(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*b^2-315*A*(-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)^(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^4+75*a^4*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))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2
*d*x+1/2*c)^2)^(1/2)+630*a^2*b^2*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))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+315*B*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)*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)-756*B*(-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)^(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^3*b-1260*B*(-2*sin(1/2*d*x+1/2*c)^4+si
n(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+
1/2*c),2^(1/2))*a*b^3+420*a^3*b*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)+1260*C*a*b^3*(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)-189*C*(-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)^(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^4-1890*C*(-2*sin(1/2*d*x+1/2*c)^4+si
n(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+
1/2*c),2^(1/2))*a^2*b^2+315*C*(-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)^(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^4)/(-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))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x, algorithm="maxima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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))^(9/2),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))^(9/2), 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))**4*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/sec(d*x+c)**(9/2),x)

[Out]

Timed out

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