\(\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sqrt {\sec (c+d x)} \, dx\) [1515]
Optimal result
Integrand size = 45, antiderivative size = 647 \[
\int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=-\frac {(a-b) \sqrt {a+b} \left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{24 a b d \sqrt {\sec (c+d x)}}+\frac {\sqrt {a+b} \left (3 a^2 C+4 b^2 (6 A+3 B+4 C)+2 a b (24 A+15 B+7 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{24 b d \sqrt {\sec (c+d x)}}-\frac {\sqrt {a+b} \left (6 a^2 b B+8 b^3 B-a^3 C+12 a b^2 (2 A+C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{8 b^2 d \sqrt {\sec (c+d x)}}+\frac {(2 b B+a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{4 d \sqrt {\sec (c+d x)}}+\frac {C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {\left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{24 b d}
\]
[Out]
1/3*C*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/d/sec(d*x+c)^(1/2)+1/4*(2*B*b+C*a)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/d
/sec(d*x+c)^(1/2)+1/24*(24*A*b^2+30*B*a*b+3*C*a^2+16*C*b^2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^(1/2)
/b/d-1/24*(a-b)*(24*A*b^2+30*B*a*b+3*C*a^2+16*C*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/d/sec(d*x+c)^(1/2)+1/24*(3*a^2*C+4*b^2*(6*A+3*B+4*C)+2*a*b*(24*A+15*B+7*C))*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/d/sec(d*x+c)^(1/2)-1/8*(6*B*a^2*b+8*B*b^3-a
^3*C+12*a*b^2*(2*A+C))*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^
2/d/sec(d*x+c)^(1/2)
Rubi [A] (verified)
Time = 2.26 (sec) , antiderivative size = 647, normalized size of antiderivative = 1.00, number of
steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.178, Rules used = {4306, 3128, 3140, 3132, 2888,
3077, 2895, 3073} \[
\int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\frac {\sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (3 a^2 C+2 a b (24 A+15 B+7 C)+4 b^2 (6 A+3 B+4 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{24 b d \sqrt {\sec (c+d x)}}-\frac {(a-b) \sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{24 a b d \sqrt {\sec (c+d x)}}+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{24 b d}-\frac {\sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (a^3 (-C)+6 a^2 b B+12 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{8 b^2 d \sqrt {\sec (c+d x)}}+\frac {(a C+2 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{4 d \sqrt {\sec (c+d x)}}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}
\]
[In]
Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]
[Out]
-1/24*((a - b)*Sqrt[a + b]*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*Elliptic
E[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)])/(a*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(3*a^2*C + 4*b
^2*(6*A + 3*B + 4*C) + 2*a*b*(24*A + 15*B + 7*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[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)])/(24*b*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(6*a^2*b*B + 8*b^3*B - a^3*C + 12
*a*b^2*(2*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, 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)])/(8*b^2*d*Sqrt[Sec[c + d*x]]) + ((2*b*B + a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[S
ec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + ((24*A*b^2 + 30*a*b*B +
3*a^2*C + 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*b*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 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]
Rule 4306
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]
Rubi steps \begin{align*}
\text {integral}& = \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 )}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {1}{3} \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{2} a (6 A+C)+(3 A b+3 a B+2 b C) \cos (c+d x)+\frac {3}{2} (2 b B+a C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {(2 b B+a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{4 d \sqrt {\sec (c+d x)}}+\frac {C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {1}{6} \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{4} a (24 a A+6 b B+7 a C)+\frac {1}{2} \left (12 a^2 B+6 b^2 B+a b (24 A+13 C)\right ) \cos (c+d x)+\frac {1}{4} \left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx \\ & = \frac {(2 b B+a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{4 d \sqrt {\sec (c+d x)}}+\frac {C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {\left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{24 b d}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{4} a \left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\right )+\frac {1}{2} a b (24 a A+6 b B+7 a C) \cos (c+d x)+\frac {3}{4} \left (6 a^2 b B+8 b^3 B-a^3 C+12 a b^2 (2 A+C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{12 b} \\ & = \frac {(2 b B+a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{4 d \sqrt {\sec (c+d x)}}+\frac {C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {\left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{24 b d}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{4} a \left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\right )+\frac {1}{2} a b (24 a A+6 b B+7 a C) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{12 b}+\frac {\left (\left (6 a^2 b B+8 b^3 B-a^3 C+12 a b^2 (2 A+C)\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}{16 b} \\ & = -\frac {\sqrt {a+b} \left (6 a^2 b B+8 b^3 B-a^3 C+12 a b^2 (2 A+C)\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}}}{8 b^2 d \sqrt {\sec (c+d x)}}+\frac {(2 b B+a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{4 d \sqrt {\sec (c+d x)}}+\frac {C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {\left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{24 b d}-\frac {\left (a \left (24 A b^2+30 a b B+3 a^2 C+16 b^2 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}{48 b}+\frac {\left (a \left (3 a^2 C+4 b^2 (6 A+3 B+4 C)+2 a b (24 A+15 B+7 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}{48 b} \\ & = -\frac {(a-b) \sqrt {a+b} \left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\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}}}{24 a b d \sqrt {\sec (c+d x)}}+\frac {\sqrt {a+b} \left (3 a^2 C+4 b^2 (6 A+3 B+4 C)+2 a b (24 A+15 B+7 C)\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}}}{24 b d \sqrt {\sec (c+d x)}}-\frac {\sqrt {a+b} \left (6 a^2 b B+8 b^3 B-a^3 C+12 a b^2 (2 A+C)\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}}}{8 b^2 d \sqrt {\sec (c+d x)}}+\frac {(2 b B+a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{4 d \sqrt {\sec (c+d x)}}+\frac {C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {\left (24 A b^2+30 a b B+3 a^2 C+16 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{24 b d} \\
\end{align*}
Mathematica [B] (warning: unable to verify)
Leaf count is larger than twice the leaf count of optimal. \(4952\) vs. \(2(647)=1294\).
Time = 28.59 (sec) , antiderivative size = 4952, normalized size of antiderivative = 7.65
\[
\int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\text {Result too large to show}
\]
[In]
Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]
[Out]
(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((b*C*Sin[c + d*x])/12 + ((6*b*B + 7*a*C)*Sin[2*(c + d*x)])/24 +
(b*C*Sin[3*(c + d*x)])/12))/d + (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*((2*a*A*b)/(Sqrt[a + b*Cos[c + d*x]]*Sq
rt[Sec[c + d*x]]) + (a^2*B)/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (b^2*B)/(2*Sqrt[a + b*Cos[c + d*x]
]*Sqrt[Sec[c + d*x]]) + (13*a*b*C)/(12*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*A*Sqrt[Sec[c + d*x]
])/Sqrt[a + b*Cos[c + d*x]] + (A*b^2*Sqrt[Sec[c + d*x]])/(2*Sqrt[a + b*Cos[c + d*x]]) + (7*a*b*B*Sqrt[Sec[c +
d*x]])/(8*Sqrt[a + b*Cos[c + d*x]]) + (17*a^2*C*Sqrt[Sec[c + d*x]])/(48*Sqrt[a + b*Cos[c + d*x]]) + (b^2*C*Sqr
t[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]) + (A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(2*Sqrt[a + b*Cos[
c + d*x]]) + (5*a*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(8*Sqrt[a + b*Cos[c + d*x]]) + (a^2*C*Cos[2*(c + d*
x)]*Sqrt[Sec[c + d*x]])/(16*Sqrt[a + b*Cos[c + d*x]]) + (b^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*Sqrt[a
+ b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(b*(a + b)*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C
)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c +
d*x)/2]^2)/(a + b)] + a*(a + b)*(-24*A*b^2 + 3*a^2*C - 6*a*b*(3*B + C) - 4*b^2*(3*B + 4*C))*EllipticF[ArcSin[
Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)
] + 3*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)
/(a + b)] + 2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Co
s[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + b*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])*(
Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(24*b^2*d*Sqrt[a + b*Cos[c + d*x]]*(Sec
[(c + d*x)/2]^2)^(3/2)*((Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d
*x]*(b*(a + b)*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)
]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + a*(a + b)*(-24*A*b^2 + 3*a^2*C
- 6*a*b*(3*B + C) - 4*b^2*(3*B + 4*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^
2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 3*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C
))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]
], (-a + b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + b*(24*A*b^2
+ 30*a*b*B + 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Ta
n[(c + d*x)/2]))/(48*b*(a + b*Cos[c + d*x])^(3/2)*(Sec[(c + d*x)/2]^2)^(3/2)) - (Sqrt[Cos[c + d*x]*Sec[(c + d*
x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(b*(a + b)*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16
*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Se
c[(c + d*x)/2]^2)/(a + b)] + a*(a + b)*(-24*A*b^2 + 3*a^2*C - 6*a*b*(3*B + C) - 4*b^2*(3*B + 4*C))*EllipticF[A
rcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(
a + b)] + 3*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-
a + b)/(a + b)] + 2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((a
+ b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + b*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d
*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(16*b^2*Sqrt[a + b*Cos[c + d*x]]*
(Sec[(c + d*x)/2]^2)^(3/2)) + (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(
-(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])*(b*(a + b)*(24*A*b^2 +
30*a*b*B + 3*a^2*C + 16*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[(
(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + a*(a + b)*(-24*A*b^2 + 3*a^2*C - 6*a*b*(3*B + C) - 4*b^2*(
3*B + 4*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x]
)*Sec[(c + d*x)/2]^2)/(a + b)] + 3*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*((a - b)*EllipticF[ArcSi
n[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*Sec[(
c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + b*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*
b^2*C)*(a + b*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(48*b^2*Sq
rt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(b*(a + b)*(24*A*b
^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*S
qrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + a*(a + b)*(-24*A*b^2 + 3*a^2*C - 6*a*b*(3*B + C) - 4*
b^2*(3*B + 4*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c +
d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 3*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*((a - b)*EllipticF[
ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*
Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + b*(24*A*b^2 + 30*a*b*B + 3*a^2*C
+ 16*b^2*C)*(a + b*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*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]))/(48*b^2*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]]) + (Sqrt[Cos[c + d*x]*Sec[(c
+ d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((b*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*(a + b*Cos[
c + d*x])*Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x])/2 + b*(a + b)*(24*A*b^2 + 3
0*a*b*B + 3*a^2*C + 16*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((
a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + a*(a + b)*(-24*A*b^2 + 3*a^2*C - 6*a*b*(3*
B + C) - 4*b^2*(3*B + 4*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a
+ b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + 3*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2
*A + C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*b*EllipticPi[-1, ArcSin[Tan[(c + d
*x)/2]], (-a + b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c
+ d*x)/2] + (3*b*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])*Sqrt[Cos[c + d*x]*Sec[(c + d*
x)/2]^2]*Sec[c + d*x]*Tan[(c + d*x)/2]*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*T
an[(c + d*x)/2]))/2 + (b*(a + b)*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]]
, (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((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)*(-24*A*b^2 + 3*a^2*C - 6*a*b*(3*B + C) - 4*b^2*(3*B + 4*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a
+ b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((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)]) + (3*(6
*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)
] + 2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*Sec[(c + d*x)/2]^2*(-((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)*(-24*A*b^2 + 3*a^2*C - 6*a*b*(3*B + C) - 4*b^2*(3*B
+ 4*C))*Sec[(c + d*x)/2]^4*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 - Tan[(c + d*x)
/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (b*(a + b)*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C
)*Sec[(c + d*x)/2]^4*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/
2]^2)/(a + b)])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]) + 3*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*Sec[(c
+ d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*(((a - b)*Sec[(c + d*x)/2]^2)/(2*Sqrt[1 -
Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (b*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(
c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])) - b^2*(24*A*b^2 + 30
*a*b*B + 3*a^2*C + 16*b^2*C)*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]*Tan[c + d*x] + b*(24*A*b
^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*
Tan[(c + d*x)/2]*Tan[c + d*x]))/(24*b^2*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2))))
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(5621\) vs. \(2(587)=1174\).
Time = 13.83 (sec) , antiderivative size = 5622, normalized size of antiderivative =
8.69
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| method | result | size |
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| parts |
\(\text {Expression too large to display}\) |
\(5622\) |
| default |
\(\text {Expression too large to display}\) |
\(5674\) |
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[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)^(1/2),x,method=_RETURNVERBOSE)
[Out]
result too large to display
Fricas [F]
\[
\int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sqrt {\sec \left (d x + c\right )} \,d x }
\]
[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)^(1/2),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)*sqrt(sec(d*x + c)), x)
Sympy [F(-1)]
Timed out. \[
\int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\text {Timed out}
\]
[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)**(1/2),x)
[Out]
Timed out
Maxima [F]
\[
\int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sqrt {\sec \left (d x + c\right )} \,d x }
\]
[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)^(1/2),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)*sqrt(sec(d*x + c)), x)
Giac [F]
\[
\int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sqrt {\sec \left (d x + c\right )} \,d x }
\]
[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)^(1/2),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)*sqrt(sec(d*x + c)), x)
Mupad [F(-1)]
Timed out. \[
\int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\int \sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x
\]
[In]
int((1/cos(c + d*x))^(1/2)*(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)
[Out]
int((1/cos(c + d*x))^(1/2)*(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2), x)