3.1510 \(\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^{\frac{11}{2}}(c+d x) \, dx\)

Optimal. Leaf size=590 \[ \frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt{a+b \cos (c+d x)}}{315 a d}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (-2 a^2 b (44 A+63 C)-75 a^3 B-9 a b^2 B+4 A b^3\right ) \sqrt{a+b \cos (c+d x)}}{315 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (3 a^2 b (13 A-57 B+21 C)-3 a^3 (49 A-25 B+63 C)+6 a b^2 (A-3 B)+8 A b^3\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{315 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (3 a^2 b^2 (11 A+21 C)+21 a^4 (7 A+9 C)+246 a^3 b B-18 a b^3 B+8 A b^4\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{315 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 (3 a B+A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{9 d} \]

[Out]

(2*(a - b)*Sqrt[a + b]*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqr
t[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a
+ b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^4*d*Sqrt[Sec
[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(8*A*b^3 + 6*a*b^2*(A - 3*B) + 3*a^2*b*(13*A - 57*B + 21*C) - 3*a^3*(49*A
- 25*B + 63*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[C
os[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])
/(315*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(4*A*b^3 - 75*a^3*B - 9*a*b^2*B - 2*a^2*b*(44*A + 63*C))*Sqrt[a + b*Cos[c
+ d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a^2*d) + (2*(3*A*b^2 + 72*a*b*B + 7*a^2*(7*A + 9*C))*Sqrt[a + b
*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*a*d) + (2*(A*b + 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c +
d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)

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Rubi [A]  time = 2.21878, antiderivative size = 590, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4221, 3047, 3055, 2998, 2816, 2994} \[ \frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt{a+b \cos (c+d x)}}{315 a d}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (-2 a^2 b (44 A+63 C)-75 a^3 B-9 a b^2 B+4 A b^3\right ) \sqrt{a+b \cos (c+d x)}}{315 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (3 a^2 b (13 A-57 B+21 C)-3 a^3 (49 A-25 B+63 C)+6 a b^2 (A-3 B)+8 A b^3\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{315 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (3 a^2 b^2 (11 A+21 C)+21 a^4 (7 A+9 C)+246 a^3 b B-18 a b^3 B+8 A b^4\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{315 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 (3 a B+A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{9 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]

[Out]

(2*(a - b)*Sqrt[a + b]*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqr
t[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a
+ b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^4*d*Sqrt[Sec
[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(8*A*b^3 + 6*a*b^2*(A - 3*B) + 3*a^2*b*(13*A - 57*B + 21*C) - 3*a^3*(49*A
- 25*B + 63*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[C
os[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])
/(315*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(4*A*b^3 - 75*a^3*B - 9*a*b^2*B - 2*a^2*b*(44*A + 63*C))*Sqrt[a + b*Cos[c
+ d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a^2*d) + (2*(3*A*b^2 + 72*a*b*B + 7*a^2*(7*A + 9*C))*Sqrt[a + b
*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*a*d) + (2*(A*b + 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c +
d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)

Rule 4221

Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Dist[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u,
x]

Rule 3047

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 3055

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[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

Rule 2998

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*s
in[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[(A - B)/(a - b), Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e
+ f*x]]), x], x] - Dist[(A*b - a*B)/(a - b), Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin
[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2
- d^2, 0] && NeQ[A, B]

Rule 2816

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-2*
Tan[e + f*x]*Rt[(a + b)/d, 2]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*Ellipt
icF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[d*Sin[e + f*x]]*Rt[(a + b)/d, 2])], -((a + b)/(a - b))])/(a*f), x] /
; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]

Rule 2994

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]]), x_Symbol] :> Simp[(-2*A*(c - d)*Tan[e + f*x]*Rt[(c + d)/b, 2]*Sqrt[(c*(1 + Csc[e + f*x]))/(c
- d)]*Sqrt[(c*(1 - Csc[e + f*x]))/(c + d)]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/(Sqrt[b*Sin[e + f*x]]*Rt[
(c + d)/b, 2])], -((c + d)/(c - d))])/(f*b*c^2), x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] &&
EqQ[A, B] && PosQ[(c + d)/b]

Rubi steps

\begin{align*} \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac{11}{2}}(c+d x) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx\\ &=\frac{2 A (a+b \cos (c+d x))^{3/2} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{9} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+b \cos (c+d x)} \left (\frac{3}{2} (A b+3 a B)+\frac{1}{2} (7 a A+9 b B+9 a C) \cos (c+d x)+\frac{1}{2} b (4 A+9 C) \cos ^2(c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 (A b+3 a B) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 A (a+b \cos (c+d x))^{3/2} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{63} \left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{4} \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right )+\frac{1}{4} \left (92 a A b+45 a^2 B+63 b^2 B+126 a b C\right ) \cos (c+d x)+\frac{1}{4} b (40 A b+36 a B+63 b C) \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx\\ &=\frac{2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{315 a d}+\frac{2 (A b+3 a B) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 A (a+b \cos (c+d x))^{3/2} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{3}{8} \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right )+\frac{1}{8} a \left (396 a b B+21 a^2 (7 A+9 C)+b^2 (209 A+315 C)\right ) \cos (c+d x)+\frac{1}{4} b \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{315 a}\\ &=-\frac{2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 a^2 d}+\frac{2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{315 a d}+\frac{2 (A b+3 a B) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 A (a+b \cos (c+d x))^{3/2} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{\left (16 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{3}{16} \left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\right )+\frac{3}{16} a \left (2 A b^3+75 a^3 B+153 a b^2 B+6 a^2 b (31 A+42 C)\right ) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{945 a^2}\\ &=-\frac{2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 a^2 d}+\frac{2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{315 a d}+\frac{2 (A b+3 a B) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 A (a+b \cos (c+d x))^{3/2} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{\left (\left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{315 a^2}+\frac{\left ((a-b) \left (8 A b^3+6 a b^2 (A-3 B)+3 a^2 b (13 A-57 B+21 C)-3 a^3 (49 A-25 B+63 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx}{315 a^2}\\ &=\frac{2 (a-b) \sqrt{a+b} \left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (1+\sec (c+d x))}{a-b}}}{315 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left (8 A b^3+6 a b^2 (A-3 B)+3 a^2 b (13 A-57 B+21 C)-3 a^3 (49 A-25 B+63 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (1+\sec (c+d x))}{a-b}}}{315 a^3 d \sqrt{\sec (c+d x)}}-\frac{2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 a^2 d}+\frac{2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{315 a d}+\frac{2 (A b+3 a B) \sqrt{a+b \cos (c+d x)} \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac{2 A (a+b \cos (c+d x))^{3/2} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}\\ \end{align*}

Mathematica [B]  time = 27.8527, size = 4669, normalized size = 7.91 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]

[Out]

(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*
B + 189*a^4*C + 63*a^2*b^2*C)*Sin[c + d*x])/(315*a^3) + (2*Sec[c + d*x]^3*(10*A*b*Sin[c + d*x] + 9*a*B*Sin[c +
d*x]))/63 + (2*Sec[c + d*x]^2*(49*a^2*A*Sin[c + d*x] + 3*A*b^2*Sin[c + d*x] + 72*a*b*B*Sin[c + d*x] + 63*a^2*
C*Sin[c + d*x]))/(315*a) + (2*Sec[c + d*x]*(88*a^2*A*b*Sin[c + d*x] - 4*A*b^3*Sin[c + d*x] + 75*a^3*B*Sin[c +
d*x] + 9*a*b^2*B*Sin[c + d*x] + 126*a^2*b*C*Sin[c + d*x]))/(315*a^2) + (2*a*A*Sec[c + d*x]^3*Tan[c + d*x])/9))
/d + (2*((-7*a^2*A)/(15*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (11*A*b^2)/(105*Sqrt[a + b*Cos[c + d*x]
]*Sqrt[Sec[c + d*x]]) - (8*A*b^4)/(315*a^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (82*a*b*B)/(105*Sqrt
[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^3*B)/(35*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*a
^2*C)/(5*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (b^2*C)/(5*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]
) + (13*a*A*b*Sqrt[Sec[c + d*x]])/(105*Sqrt[a + b*Cos[c + d*x]]) - (31*A*b^3*Sqrt[Sec[c + d*x]])/(315*a*Sqrt[a
+ b*Cos[c + d*x]]) - (8*A*b^5*Sqrt[Sec[c + d*x]])/(315*a^3*Sqrt[a + b*Cos[c + d*x]]) + (5*a^2*B*Sqrt[Sec[c +
d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) - (31*b^2*B*Sqrt[Sec[c + d*x]])/(105*Sqrt[a + b*Cos[c + d*x]]) + (2*b^4*B
*Sqrt[Sec[c + d*x]])/(35*a^2*Sqrt[a + b*Cos[c + d*x]]) + (a*b*C*Sqrt[Sec[c + d*x]])/(5*Sqrt[a + b*Cos[c + d*x]
]) - (b^3*C*Sqrt[Sec[c + d*x]])/(5*a*Sqrt[a + b*Cos[c + d*x]]) - (7*a*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])
/(15*Sqrt[a + b*Cos[c + d*x]]) - (11*A*b^3*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*a*Sqrt[a + b*Cos[c + d*x]
]) - (8*A*b^5*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(315*a^3*Sqrt[a + b*Cos[c + d*x]]) - (82*b^2*B*Cos[2*(c + d
*x)]*Sqrt[Sec[c + d*x]])/(105*Sqrt[a + b*Cos[c + d*x]]) + (2*b^4*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*a^
2*Sqrt[a + b*Cos[c + d*x]]) - (3*a*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[a + b*Cos[c + d*x]]) - (b^
3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*a*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]
*(-2*(a + b)*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqrt[Cos[c +
d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x
)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 + a*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^2*b*(13*A + 57*B +
21*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(Cos[c + d*x]*Sec[(c
+ d*x)/2]^2)^(3/2)*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] - (8*A*b^4 + 246*a^3*
b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*
x)/2]^4*Tan[(c + d*x)/2]))/(315*a^3*d*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)*((b*Sqrt[Cos[(c + d*
x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*(a + b)*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^
2*b^2*(11*A + 21*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x
]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 + a*(a + b)*(8*A*b^3 - 6*a*b^2*(
A + 3*B) + 3*a^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a
+ b)/(a + b)]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*
Sec[c + d*x] - (8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Cos[c + d*x
]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(315*a^3*(a + b*Cos[c + d*x])^(3/2)*(Sec[(c + d*x
)/2]^2)^(3/2)) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*(a + b)*(8*A*b^4 + 246*a^3*b*B -
18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*C
os[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)
/2]^2 + a*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*El
lipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((a + b*Cos[c
+ d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] - (8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C)
+ 3*a^2*b^2*(11*A + 21*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(105*a^3*Sq
rt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + ((-2*(a + b)*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4
*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a +
b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 + a*(a + b)*(
8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[ArcSin[Tan[
(c + d*x)/2]], (-a + b)/(a + b)]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d
*x)/2]^2)/(a + b)]*Sec[c + d*x] - (8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A +
21*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x
]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(315*a^3*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c
+ d*x)/2]^2)^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-((8*A*
b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Cos[c + d*x]*(a + b*Cos[c + d*x
])*Sec[(c + d*x)/2]^6)/2 - ((a + b)*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A
+ 21*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)
/(a + b)]*Sec[(c + d*x)/2]^2*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x
])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - ((a + b)*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C)
+ 3*a^2*b^2*(11*A + 21*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)
/(a + b)]*Sec[(c + d*x)/2]^2*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c +
d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - 2*(a + b)*(8
*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c
+ d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)
/(a + b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + b*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3
*a^2*b^2*(11*A + 21*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] + (8*A*b^4 + 246*a^3*b*B
- 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Sin[c +
d*x]*Tan[(c + d*x)/2] - 2*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*
Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]^2 + (3*a*(a + b)*(8*A*b^3 - 6*a*b^2*(A +
3*B) + 3*a^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b
)/(a + b)]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c
+ d*x]*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/2 + (a*(a + b
)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[ArcSin[T
an[(c + d*x)/2]], (-a + b)/(a + b)]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*(-((b*Sec[(c + d*x)/2
]^2*Sin[c + d*x])/(a + b)) + ((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((a
+ b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + (a*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^2*b*(13*A + 57
*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((a +
b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)
*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2
*(11*A + 21*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*
Sec[(c + d*x)/2]^4*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2] + a*(a + b)*(
8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[ArcSin[Tan[
(c + d*x)/2]], (-a + b)/(a + b)]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d
*x)/2]^2)/(a + b)]*Sec[c + d*x]*Tan[c + d*x]))/(315*a^3*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2))))

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Maple [B]  time = 0.645, size = 5965, normalized size = 10.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2),x, algorithm="giac")

[Out]

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