\(\int \frac {(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\) [604]

Optimal result

Integrand size = 35, antiderivative size = 730 \[ \int \frac {(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {(a-b) \sqrt {a+b} \left (24 a^2 A b+128 A b^3-9 a^3 B+156 a b^2 B\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{192 a b^2 d \sqrt {\sec (c+d x)}}-\frac {\sqrt {a+b} \left (9 a^3 B-6 a^2 b (4 A+B)-8 b^3 (16 A+9 B)-4 a b^2 (28 A+39 B)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{192 b^2 d \sqrt {\sec (c+d x)}}+\frac {\sqrt {a+b} \left (8 a^3 A b-96 a A b^3-3 a^4 B-24 a^2 b^2 B-48 b^4 B\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{64 b^3 d \sqrt {\sec (c+d x)}}+\frac {\left (8 a A b-3 a^2 B+12 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b d \sqrt {\sec (c+d x)}}+\frac {(8 A b-3 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d \sqrt {\sec (c+d x)}}+\frac {B (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d \sqrt {\sec (c+d x)}}+\frac {\left (24 a^2 A b+128 A b^3-9 a^3 B+156 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{192 b^2 d} \]

[Out]

1/24*(8*A*b-3*B*a)*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b/d/sec(d*x+c)^(1/2)+1/4*B*(a+b*cos(d*x+c))^(5/2)*sin(d*x
+c)/b/d/sec(d*x+c)^(1/2)+1/32*(8*A*a*b-3*B*a^2+12*B*b^2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b/d/sec(d*x+c)^(1/2
)+1/192*(24*A*a^2*b+128*A*b^3-9*B*a^3+156*B*a*b^2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^(1/2)/b^2/d-1/
192*(a-b)*(24*A*a^2*b+128*A*b^3-9*B*a^3+156*B*a*b^2)*csc(d*x+c)*EllipticE((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/c
os(d*x+c)^(1/2),((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*cos(d*x+c)^(1/2)*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*
x+c))/(a-b))^(1/2)/a/b^2/d/sec(d*x+c)^(1/2)-1/192*(9*B*a^3-6*a^2*b*(4*A+B)-8*b^3*(16*A+9*B)-4*a*b^2*(28*A+39*B
))*csc(d*x+c)*EllipticF((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*
cos(d*x+c)^(1/2)*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/b^2/d/sec(d*x+c)^(1/2)+1/64*(8*
A*a^3*b-96*A*a*b^3-3*B*a^4-24*B*a^2*b^2-48*B*b^4)*csc(d*x+c)*EllipticPi((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos
(d*x+c)^(1/2),(a+b)/b,((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*cos(d*x+c)^(1/2)*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+
sec(d*x+c))/(a-b))^(1/2)/b^3/d/sec(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 3.17 (sec) , antiderivative size = 730, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {3040, 3069, 3128, 3140, 3132, 2888, 3077, 2895, 3073} \[ \int \frac {(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\left (-3 a^2 B+8 a A b+12 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{32 b d \sqrt {\sec (c+d x)}}-\frac {\sqrt {a+b} \left (9 a^3 B-6 a^2 b (4 A+B)-4 a b^2 (28 A+39 B)-8 b^3 (16 A+9 B)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{192 b^2 d \sqrt {\sec (c+d x)}}-\frac {(a-b) \sqrt {a+b} \left (-9 a^3 B+24 a^2 A b+156 a b^2 B+128 A b^3\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{192 a b^2 d \sqrt {\sec (c+d x)}}+\frac {\left (-9 a^3 B+24 a^2 A b+156 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \cos (c+d x)}}{192 b^2 d}+\frac {\sqrt {a+b} \left (-3 a^4 B+8 a^3 A b-24 a^2 b^2 B-96 a A b^3-48 b^4 B\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{64 b^3 d \sqrt {\sec (c+d x)}}+\frac {(8 A b-3 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b d \sqrt {\sec (c+d x)}}+\frac {B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 b d \sqrt {\sec (c+d x)}} \]

[In]

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

[Out]

-1/192*((a - b)*Sqrt[a + b]*(24*a^2*A*b + 128*A*b^3 - 9*a^3*B + 156*a*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*E
llipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - S
ec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(9*a^
3*B - 6*a^2*b*(4*A + B) - 8*b^3*(16*A + 9*B) - 4*a*b^2*(28*A + 39*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*Elliptic
F[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)])/(192*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(8*a^3*A*b
- 96*a*A*b^3 - 3*a^4*B - 24*a^2*b^2*B - 48*b^4*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSi
n[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)])/(64*b^3*d*Sqrt[Sec[c + d*x]]) + ((8*a*A*b - 3*a^2*B + 12*b^2*B)
*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b*d*Sqrt[Sec[c + d*x]]) + ((8*A*b - 3*a*B)*(a + b*Cos[c + d*x])^(3
/2)*Sin[c + d*x])/(24*b*d*Sqrt[Sec[c + d*x]]) + (B*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*b*d*Sqrt[Sec[c
+ d*x]]) + ((24*a^2*A*b + 128*A*b^3 - 9*a^3*B + 156*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c
+ d*x])/(192*b^2*d)

Rule 2888

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

Rule 2895

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

Rule 3040

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(g*Csc[e + f*x])^p*(g*Sin[e + f*x])^p, Int[(a + b*Sin[e + f*x])^m*((
c + d*Sin[e + f*x])^n/(g*Sin[e + f*x])^p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d
, 0] &&  !IntegerQ[p] &&  !(IntegerQ[m] && IntegerQ[n])

Rule 3069

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

Rule 3073

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]/(f*b*c^2))*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)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ
[A, B] && PosQ[(c + d)/b]

Rule 3077

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 3128

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

Rule 3132

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

Rule 3140

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) +
(f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(Sqrt[c + d*Sin[
e + f*x]]/(d*f*Sqrt[a + b*Sin[e + f*x]])), x] + Dist[1/(2*d), Int[(1/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Si
n[e + f*x]]))*Simp[2*a*A*d - C*(b*c - a*d) - 2*(a*c*C - d*(A*b + a*B))*Sin[e + f*x] + (2*b*B*d - C*(b*c + a*d)
)*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]

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx \\ & = \frac {B (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d \sqrt {\sec (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^{3/2} \left (\frac {a B}{2}+3 b B \cos (c+d x)+\frac {1}{2} (8 A b-3 a B) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{4 b} \\ & = \frac {(8 A b-3 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d \sqrt {\sec (c+d x)}}+\frac {B (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d \sqrt {\sec (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{4} a (8 A b+3 a B)+\frac {1}{2} b (16 A b+15 a B) \cos (c+d x)+\frac {3}{4} \left (8 a A b-3 a^2 B+12 b^2 B\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{12 b} \\ & = \frac {\left (8 a A b-3 a^2 B+12 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b d \sqrt {\sec (c+d x)}}+\frac {(8 A b-3 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d \sqrt {\sec (c+d x)}}+\frac {B (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d \sqrt {\sec (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{8} a \left (56 a A b+3 a^2 B+36 b^2 B\right )+\frac {1}{4} b \left (104 a A b+57 a^2 B+36 b^2 B\right ) \cos (c+d x)+\frac {1}{8} \left (24 a^2 A b+128 A b^3-9 a^3 B+156 a b^2 B\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{24 b} \\ & = \frac {\left (8 a A b-3 a^2 B+12 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b d \sqrt {\sec (c+d x)}}+\frac {(8 A b-3 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d \sqrt {\sec (c+d x)}}+\frac {B (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d \sqrt {\sec (c+d x)}}+\frac {\left (24 a^2 A b+128 A b^3-9 a^3 B+156 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{192 b^2 d}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{8} a \left (24 a^2 A b+128 A b^3-9 a^3 B+156 a b^2 B\right )+\frac {1}{4} a b \left (56 a A b+3 a^2 B+36 b^2 B\right ) \cos (c+d x)-\frac {3}{8} \left (8 a^3 A b-96 a A b^3-3 a^4 B-24 a^2 b^2 B-48 b^4 B\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{48 b^2} \\ & = \frac {\left (8 a A b-3 a^2 B+12 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b d \sqrt {\sec (c+d x)}}+\frac {(8 A b-3 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d \sqrt {\sec (c+d x)}}+\frac {B (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d \sqrt {\sec (c+d x)}}+\frac {\left (24 a^2 A b+128 A b^3-9 a^3 B+156 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{192 b^2 d}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{8} a \left (24 a^2 A b+128 A b^3-9 a^3 B+156 a b^2 B\right )+\frac {1}{4} a b \left (56 a A b+3 a^2 B+36 b^2 B\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{48 b^2}-\frac {\left (\left (8 a^3 A b-96 a A b^3-3 a^4 B-24 a^2 b^2 B-48 b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}} \, dx}{128 b^2} \\ & = \frac {\sqrt {a+b} \left (8 a^3 A b-96 a A b^3-3 a^4 B-24 a^2 b^2 B-48 b^4 B\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \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}}}{64 b^3 d \sqrt {\sec (c+d x)}}+\frac {\left (8 a A b-3 a^2 B+12 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b d \sqrt {\sec (c+d x)}}+\frac {(8 A b-3 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d \sqrt {\sec (c+d x)}}+\frac {B (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d \sqrt {\sec (c+d x)}}+\frac {\left (24 a^2 A b+128 A b^3-9 a^3 B+156 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{192 b^2 d}-\frac {\left (a \left (24 a^2 A b+128 A b^3-9 a^3 B+156 a b^2 B\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}{384 b^2}-\frac {\left (a \left (9 a^3 B-6 a^2 b (4 A+B)-8 b^3 (16 A+9 B)-4 a b^2 (28 A+39 B)\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}{384 b^2} \\ & = -\frac {(a-b) \sqrt {a+b} \left (24 a^2 A b+128 A b^3-9 a^3 B+156 a b^2 B\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \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}}}{192 a b^2 d \sqrt {\sec (c+d x)}}-\frac {\sqrt {a+b} \left (9 a^3 B-6 a^2 b (4 A+B)-8 b^3 (16 A+9 B)-4 a b^2 (28 A+39 B)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \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}}}{192 b^2 d \sqrt {\sec (c+d x)}}+\frac {\sqrt {a+b} \left (8 a^3 A b-96 a A b^3-3 a^4 B-24 a^2 b^2 B-48 b^4 B\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \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}}}{64 b^3 d \sqrt {\sec (c+d x)}}+\frac {\left (8 a A b-3 a^2 B+12 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b d \sqrt {\sec (c+d x)}}+\frac {(8 A b-3 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d \sqrt {\sec (c+d x)}}+\frac {B (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d \sqrt {\sec (c+d x)}}+\frac {\left (24 a^2 A b+128 A b^3-9 a^3 B+156 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{192 b^2 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1888\) vs. \(2(730)=1460\).

Time = 19.45 (sec) , antiderivative size = 1888, normalized size of antiderivative = 2.59 \[ \int \frac {(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{96} (8 A b+9 a B) \sin (c+d x)+\frac {\left (56 a A b+3 a^2 B+48 b^2 B\right ) \sin (2 (c+d x))}{192 b}+\frac {1}{96} (8 A b+9 a B) \sin (3 (c+d x))+\frac {1}{32} b B \sin (4 (c+d x))\right )}{d}-\frac {\sqrt {\frac {1}{1-\tan ^2\left (\frac {1}{2} (c+d x)\right )}} \sqrt {\frac {a+b+a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}} \left (24 a^3 A b \tan \left (\frac {1}{2} (c+d x)\right )+24 a^2 A b^2 \tan \left (\frac {1}{2} (c+d x)\right )+128 a A b^3 \tan \left (\frac {1}{2} (c+d x)\right )+128 A b^4 \tan \left (\frac {1}{2} (c+d x)\right )-9 a^4 B \tan \left (\frac {1}{2} (c+d x)\right )-9 a^3 b B \tan \left (\frac {1}{2} (c+d x)\right )+156 a^2 b^2 B \tan \left (\frac {1}{2} (c+d x)\right )+156 a b^3 B \tan \left (\frac {1}{2} (c+d x)\right )-48 a^2 A b^2 \tan ^3\left (\frac {1}{2} (c+d x)\right )-256 A b^4 \tan ^3\left (\frac {1}{2} (c+d x)\right )+18 a^3 b B \tan ^3\left (\frac {1}{2} (c+d x)\right )-312 a b^3 B \tan ^3\left (\frac {1}{2} (c+d x)\right )-24 a^3 A b \tan ^5\left (\frac {1}{2} (c+d x)\right )+24 a^2 A b^2 \tan ^5\left (\frac {1}{2} (c+d x)\right )-128 a A b^3 \tan ^5\left (\frac {1}{2} (c+d x)\right )+128 A b^4 \tan ^5\left (\frac {1}{2} (c+d x)\right )+9 a^4 B \tan ^5\left (\frac {1}{2} (c+d x)\right )-9 a^3 b B \tan ^5\left (\frac {1}{2} (c+d x)\right )-156 a^2 b^2 B \tan ^5\left (\frac {1}{2} (c+d x)\right )+156 a b^3 B \tan ^5\left (\frac {1}{2} (c+d x)\right )-48 a^3 A b \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b+a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+576 a A b^3 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b+a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+18 a^4 B \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b+a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+144 a^2 b^2 B \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b+a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+288 b^4 B \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b+a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}-48 a^3 A b \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b+a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+576 a A b^3 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b+a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+18 a^4 B \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b+a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+144 a^2 b^2 B \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b+a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+288 b^4 B \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b+a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}-(a+b) \left (-24 a^2 A b-128 A b^3+9 a^3 B-156 a b^2 B\right ) E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {-a+b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\frac {a+b+a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+2 b \left (2 a^2 b (28 A-57 B)-4 a b^2 (52 A-9 B)+3 a^3 B-72 b^3 B\right ) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\frac {a+b+a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}\right )}{192 b^2 d \sqrt {1+\tan ^2\left (\frac {1}{2} (c+d x)\right )} \left (b \left (-1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-a \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )} \]

[In]

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

[Out]

(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(((8*A*b + 9*a*B)*Sin[c + d*x])/96 + ((56*a*A*b + 3*a^2*B + 48*b^
2*B)*Sin[2*(c + d*x)])/(192*b) + ((8*A*b + 9*a*B)*Sin[3*(c + d*x)])/96 + (b*B*Sin[4*(c + d*x)])/32))/d - (Sqrt
[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/
2]^2)]*(24*a^3*A*b*Tan[(c + d*x)/2] + 24*a^2*A*b^2*Tan[(c + d*x)/2] + 128*a*A*b^3*Tan[(c + d*x)/2] + 128*A*b^4
*Tan[(c + d*x)/2] - 9*a^4*B*Tan[(c + d*x)/2] - 9*a^3*b*B*Tan[(c + d*x)/2] + 156*a^2*b^2*B*Tan[(c + d*x)/2] + 1
56*a*b^3*B*Tan[(c + d*x)/2] - 48*a^2*A*b^2*Tan[(c + d*x)/2]^3 - 256*A*b^4*Tan[(c + d*x)/2]^3 + 18*a^3*b*B*Tan[
(c + d*x)/2]^3 - 312*a*b^3*B*Tan[(c + d*x)/2]^3 - 24*a^3*A*b*Tan[(c + d*x)/2]^5 + 24*a^2*A*b^2*Tan[(c + d*x)/2
]^5 - 128*a*A*b^3*Tan[(c + d*x)/2]^5 + 128*A*b^4*Tan[(c + d*x)/2]^5 + 9*a^4*B*Tan[(c + d*x)/2]^5 - 9*a^3*b*B*T
an[(c + d*x)/2]^5 - 156*a^2*b^2*B*Tan[(c + d*x)/2]^5 + 156*a*b^3*B*Tan[(c + d*x)/2]^5 - 48*a^3*A*b*EllipticPi[
-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^
2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 576*a*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sq
rt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 18*a^4*B*Elli
pticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d
*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 144*a^2*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a
+ b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 288*
b^4*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*
Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 48*a^3*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a
+ b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c +
d*x)/2]^2)/(a + b)] + 576*a*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^
2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 18*a^4*B*
EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqr
t[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 144*a^2*b^2*B*EllipticPi[-1, ArcSin[Tan[(c
+ d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]
^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 288*b^4*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan
[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b
)] - (a + b)*(-24*a^2*A*b - 128*A*b^3 + 9*a^3*B - 156*a*b^2*B)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a
+ b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d
*x)/2]^2)/(a + b)] + 2*b*(2*a^2*b*(28*A - 57*B) - 4*a*b^2*(52*A - 9*B) + 3*a^3*B - 72*b^3*B)*EllipticF[ArcSin[
Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan
[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(192*b^2*d*Sqrt[1 + Tan[(c + d*x)/2]^2]*(b*(-1 + Tan[(c + d
*x)/2]^2) - a*(1 + Tan[(c + d*x)/2]^2)))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(5424\) vs. \(2(664)=1328\).

Time = 21.10 (sec) , antiderivative size = 5425, normalized size of antiderivative = 7.43

method	result	size
parts	\(\text {Expression too large to display}\)	\(5425\)
default	\(\text {Expression too large to display}\)	\(5487\)

[In]

int((a+cos(d*x+c)*b)^(3/2)*(A+B*cos(d*x+c))/sec(d*x+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F(-1)]

Timed out. \[ \int \frac {(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

[In]

int(((A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(3/2))/(1/cos(c + d*x))^(3/2),x)

[Out]

int(((A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(3/2))/(1/cos(c + d*x))^(3/2), x)