\(\int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [1317]

Optimal result

Integrand size = 43, antiderivative size = 401 \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \left (60 a^3 b B+36 a b^3 B-15 a^4 (A-C)+18 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (21 a^4 B+42 a^2 b^2 B+5 b^4 B+28 a^3 b (3 A+C)+4 a b^3 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 b \left (261 a^2 b B+75 b^3 B+64 a^3 C+2 a b^2 (147 A+101 C)\right ) \sin (c+d x)}{315 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (1098 a^3 b B+756 a b^3 B+192 a^4 C+21 b^4 (9 A+7 C)+7 a^2 b^2 (261 A+155 C)\right ) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)}}+\frac {2 \left (63 A b^2+117 a b B+48 a^2 C+49 b^2 C\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (9 b B+8 a C) (b+a \cos (c+d x))^3 \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 C (b+a \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)} \]

[Out]

-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)*EllipticE(sin(1/2*d*x+1/2*c),2^(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))/d+2/31
5*b*(261*B*a^2*b+75*B*b^3+64*a^3*C+2*a*b^2*(147*A+101*C))*sin(d*x+c)/d/cos(d*x+c)^(3/2)+2/315*(63*A*b^2+117*B*
a*b+48*C*a^2+49*C*b^2)*(b+a*cos(d*x+c))^2*sin(d*x+c)/d/cos(d*x+c)^(5/2)+2/63*(9*B*b+8*C*a)*(b+a*cos(d*x+c))^3*
sin(d*x+c)/d/cos(d*x+c)^(7/2)+2/9*C*(b+a*cos(d*x+c))^4*sin(d*x+c)/d/cos(d*x+c)^(9/2)+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)/d/cos(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 1.44 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4197, 3126, 3110, 3100, 2827, 2720, 2719} \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \sin (c+d x) \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a \cos (c+d x)+b)^2}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 b \sin (c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{315 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right )}{21 d}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right )}{15 d}+\frac {2 \sin (c+d x) \left (192 a^4 C+1098 a^3 b B+7 a^2 b^2 (261 A+155 C)+756 a b^3 B+21 b^4 (9 A+7 C)\right )}{315 d \sqrt {\cos (c+d x)}}+\frac {2 (8 a C+9 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)} \]

[In]

Int[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),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))*EllipticE[(c + d*x)/
2, 2])/(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))*EllipticF[(c
+ d*x)/2, 2])/(21*d) + (2*b*(261*a^2*b*B + 75*b^3*B + 64*a^3*C + 2*a*b^2*(147*A + 101*C))*Sin[c + d*x])/(315*
d*Cos[c + d*x]^(3/2)) + (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 + 1
55*C))*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]) + (2*(63*A*b^2 + 117*a*b*B + 48*a^2*C + 49*b^2*C)*(b + a*Cos[c
+ d*x])^2*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*(9*b*B + 8*a*C)*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/
(63*d*Cos[c + d*x]^(7/2)) + (2*C*(b + a*Cos[c + d*x])^4*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))

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 2827

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 3100

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m
+ 1)*(a^2 - b^2))), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B +
a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b,
e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

Rule 3110

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)
*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)
), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d +
b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f*x] - b*C*d*(m
+ 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] &&
NeQ[a^2 - b^2, 0] && LtQ[m, -1]

Rule 3126

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(d*(n + 1)
*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) +
(c*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*
c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n +
1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2
, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 4197

Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sec[(e_.)
+ (f_.)*(x_)] + (C_.)*sec[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[d^(m + 2), Int[(b + a*Cos[e + f*x])^m*(d*
Cos[e + f*x])^(n - m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}
, x] &&  !IntegerQ[n] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(b+a \cos (c+d x))^4 \left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx \\ & = \frac {2 C (b+a \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2}{9} \int \frac {(b+a \cos (c+d x))^3 \left (\frac {1}{2} (9 b B+8 a C)+\frac {1}{2} (9 A b+9 a B+7 b C) \cos (c+d x)+\frac {1}{2} a (9 A-C) \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx \\ & = \frac {2 (9 b B+8 a C) (b+a \cos (c+d x))^3 \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 C (b+a \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {4}{63} \int \frac {(b+a \cos (c+d x))^2 \left (\frac {1}{4} \left (63 A b^2+117 a b B+48 a^2 C+49 b^2 C\right )+\frac {1}{4} \left (126 a A b+63 a^2 B+45 b^2 B+82 a b C\right ) \cos (c+d x)+\frac {3}{4} a (21 a A-3 b B-5 a C) \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 \left (63 A b^2+117 a b B+48 a^2 C+49 b^2 C\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (9 b B+8 a C) (b+a \cos (c+d x))^3 \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 C (b+a \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {8}{315} \int \frac {(b+a \cos (c+d x)) \left (\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 )+\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 ) \cos (c+d x)-\frac {1}{8} a \left (162 a b B-3 a^2 (105 A-41 C)+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(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 ) \sin (c+d x)}{315 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (63 A b^2+117 a b B+48 a^2 C+49 b^2 C\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (9 b B+8 a C) (b+a \cos (c+d x))^3 \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 C (b+a \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}-\frac {16}{945} \int \frac {-\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 )-\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 ) \cos (c+d x)+\frac {3}{16} a^2 \left (162 a b B-a^2 (315 A-123 C)+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(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 ) \sin (c+d x)}{315 d \cos ^{\frac {3}{2}}(c+d x)}+\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 ) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)}}+\frac {2 \left (63 A b^2+117 a b B+48 a^2 C+49 b^2 C\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (9 b B+8 a C) (b+a \cos (c+d x))^3 \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 C (b+a \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}-\frac {32}{945} \int \frac {-\frac {45}{32} \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 )+\frac {63}{32} \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 ) \cos (c+d x)}{\sqrt {\cos (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 ) \sin (c+d x)}{315 d \cos ^{\frac {3}{2}}(c+d x)}+\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 ) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)}}+\frac {2 \left (63 A b^2+117 a b B+48 a^2 C+49 b^2 C\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (9 b B+8 a C) (b+a \cos (c+d x))^3 \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 C (b+a \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}-\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 \frac {1}{\sqrt {\cos (c+d x)}} \, dx-\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 \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 ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{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 ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 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 ) \sin (c+d x)}{315 d \cos ^{\frac {3}{2}}(c+d x)}+\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 ) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)}}+\frac {2 \left (63 A b^2+117 a b B+48 a^2 C+49 b^2 C\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (9 b B+8 a C) (b+a \cos (c+d x))^3 \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 C (b+a \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 16.46 (sec) , antiderivative size = 4150, normalized size of antiderivative = 10.35 \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(Cos[c + d*x]^(13/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(15*a^4*A - 180*a^2*A
*b^2 - 18*A*b^4 - 120*a^3*b*B - 72*a*b^3*B - 30*a^4*C - 108*a^2*b^2*C - 14*b^4*C + 15*a^4*A*Cos[2*c])*Csc[c]*S
ec[c])/(15*d) + (4*b^4*C*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(9*d) + (4*Sec[c]*Sec[c + d*x]^4*(7*b^4*C*Sin[c] + 9*
b^4*B*Sin[d*x] + 36*a*b^3*C*Sin[d*x]))/(63*d) + (4*Sec[c]*Sec[c + d*x]^2*(63*A*b^4*Sin[c] + 252*a*b^3*B*Sin[c]
+ 378*a^2*b^2*C*Sin[c] + 49*b^4*C*Sin[c] + 420*a*A*b^3*Sin[d*x] + 630*a^2*b^2*B*Sin[d*x] + 75*b^4*B*Sin[d*x]
+ 420*a^3*b*C*Sin[d*x] + 300*a*b^3*C*Sin[d*x]))/(315*d) + (4*Sec[c]*Sec[c + d*x]^3*(45*b^4*B*Sin[c] + 180*a*b^
3*C*Sin[c] + 63*A*b^4*Sin[d*x] + 252*a*b^3*B*Sin[d*x] + 378*a^2*b^2*C*Sin[d*x] + 49*b^4*C*Sin[d*x]))/(315*d) +
(4*Sec[c]*Sec[c + d*x]*(140*a*A*b^3*Sin[c] + 210*a^2*b^2*B*Sin[c] + 25*b^4*B*Sin[c] + 140*a^3*b*C*Sin[c] + 10
0*a*b^3*C*Sin[c] + 630*a^2*A*b^2*Sin[d*x] + 63*A*b^4*Sin[d*x] + 420*a^3*b*B*Sin[d*x] + 252*a*b^3*B*Sin[d*x] +
105*a^4*C*Sin[d*x] + 378*a^2*b^2*C*Sin[d*x] + 49*b^4*C*Sin[d*x]))/(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])) - (16*a^3*A*b*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/
4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcT
an[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*S
qrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]
)*Sqrt[1 + Cot[c]^2]) - (16*a*A*b^3*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTa
n[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1
- Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - A
rcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]
^2]) - (4*a^4*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b
*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[C
ot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*
(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (8*a^2*b^2*B*Co
s[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(
A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sq
rt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x
])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (20*b^4*B*Cos[c + d*x]^6*Csc[c]*H
ypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] +
C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[
c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*
B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (16*a^3*b*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ
[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2
)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - Arc
Tan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] +
A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (80*a*b^3*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4
}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTa
n[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sq
rt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*
x])*Sqrt[1 + Cot[c]^2]) - (2*a^4*A*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c
+ d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[
c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[
c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos
[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]
]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (12*a^
2*A*b^2*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricP
FQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + Arc
Tan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sq
rt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[
c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d
*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (6*A*b^4*Cos[c + d*x]^6*Csc[c]*(a
+ b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x
+ ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x +
ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x +
ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]
^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^4*(A + 2
*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (8*a^3*b*B*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*
Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x
+ ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c
]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt
[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*
Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[
2*c + 2*d*x])) + (24*a*b^3*B*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]
^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(S
qrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*S
qrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*
Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqr
t[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (2*a^4*C*C
os[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2,
-1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c
]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Ta
n[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt
[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*C
os[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (36*a^2*b^2*C*Cos[c + d*x]^6*Csc[c]*(a + b
*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + Ar
cTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + Arc
Tan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcT
an[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 +
Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^4*(A + 2*C +
2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (14*b^4*C*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c
+ d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + Ar
cTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos
[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 +
Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d
*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*
c + 2*d*x]))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1522\) vs. \(2(429)=858\).

Time = 6.32 (sec) , antiderivative size = 1523, normalized size of antiderivative = 3.80

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

[In]

int((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*cos(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 = 0.18 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.31 \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, 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 )^{5} {\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 )^{5} {\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 )^{5} {\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 )^{5} {\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 ) - 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 )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{315 \, d \cos \left (d x + c\right )^{5}} \]

[In]

integrate((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*cos(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)^5*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)^5*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)^5*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)^5*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))*sqrt(cos(d*x + c))*sin(d*x + c))
/(d*cos(d*x + c)^5)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F]

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

[In]

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

Mupad [B] (verification not implemented)

Time = 30.44 (sec) , antiderivative size = 866, normalized size of antiderivative = 2.16 \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

(8*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2)*((4*C*a*b^3*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^
(1/2)) + (7*C*a^3*b*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (3*C*a*b^3*sin(c + d*x))/(cos(
c + d*x)^(7/2)*(sin(c + d*x)^2)^(1/2))))/(21*d) - (8*hypergeom([-1/4, 1/2], 7/4, cos(c + d*x)^2)*((7*C*b^4*sin
(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (5*C*b^4*sin(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d
*x)^2)^(1/2)) + (54*C*a^2*b^2*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2))))/(135*d) + (2*hyperge
om([-1/4, 1/2], 3/4, cos(c + d*x)^2)*((45*C*a^4*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (2
8*C*b^4*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (12*C*b^4*sin(c + d*x))/(cos(c + d*x)^(5/2
)*(sin(c + d*x)^2)^(1/2)) + (5*C*b^4*sin(c + d*x))/(cos(c + d*x)^(9/2)*(sin(c + d*x)^2)^(1/2)) + (216*C*a^2*b^
2*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (54*C*a^2*b^2*sin(c + d*x))/(cos(c + d*x)^(5/2)*
(sin(c + d*x)^2)^(1/2))))/(45*d) + (2*A*a^4*ellipticE(c/2 + (d*x)/2, 2))/d + (2*B*a^4*ellipticF(c/2 + (d*x)/2,
2))/d + (8*A*a^3*b*ellipticF(c/2 + (d*x)/2, 2))/d + (2*A*b^4*sin(c + d*x)*hypergeom([-5/4, 1/2], -1/4, cos(c
+ d*x)^2))/(5*d*cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)) + (2*B*b^4*sin(c + d*x)*hypergeom([-7/4, 1/2], -3/4
, cos(c + d*x)^2))/(7*d*cos(c + d*x)^(7/2)*(sin(c + d*x)^2)^(1/2)) + (8*A*a*b^3*sin(c + d*x)*hypergeom([-3/4,
1/2], 1/4, cos(c + d*x)^2))/(3*d*cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (8*B*a^3*b*sin(c + d*x)*hypergeo
m([-1/4, 1/2], 3/4, cos(c + d*x)^2))/(d*cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (8*B*a*b^3*sin(c + d*x)*h
ypergeom([-5/4, 1/2], -1/4, cos(c + d*x)^2))/(5*d*cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)) + (32*C*a*b^3*sin
(c + d*x)*hypergeom([-3/4, 1/2], 5/4, cos(c + d*x)^2))/(21*d*cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (12*
A*a^2*b^2*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2))/(d*cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/
2)) + (4*B*a^2*b^2*sin(c + d*x)*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2))/(d*cos(c + d*x)^(3/2)*(sin(c + d*
x)^2)^(1/2))