\(\int \frac {\sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\) [1340]

Optimal result

Integrand size = 45, antiderivative size = 447 \[ \int \frac {\sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\left (24 A b^2+18 a b B-a^2 C+16 b^2 C\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{24 b d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (2 a^2 b B-8 b^3 B-a^3 C-4 a b^2 (2 A+C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{8 b^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (24 A b^2+6 a b B-3 a^2 C+16 b^2 C\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{24 b^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {C \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {(6 b B+a C) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{12 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (24 A b^2+6 a b B-3 a^2 C+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 b^2 d \sqrt {\cos (c+d x)}} \]

[Out]

1/24*(24*A*b^2+18*B*a*b-C*a^2+16*C*b^2)*(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)*(a/(a+b))^(1/2))*((b+a*cos(d*x+c))/(a+b))^(1/2)/b/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)-1/8
*(2*B*a^2*b-8*B*b^3-a^3*C-4*a*b^2*(2*A+C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*
d*x+1/2*c),2,2^(1/2)*(a/(a+b))^(1/2))*((b+a*cos(d*x+c))/(a+b))^(1/2)/b^2/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(
1/2)+1/3*C*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d/cos(d*x+c)^(5/2)+1/12*(6*B*b+C*a)*sin(d*x+c)*(a+b*sec(d*x+c))^(
1/2)/b/d/cos(d*x+c)^(3/2)+1/24*(24*A*b^2+6*B*a*b-3*C*a^2+16*C*b^2)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/b^2/d/cos
(d*x+c)^(1/2)-1/24*(24*A*b^2+6*B*a*b-3*C*a^2+16*C*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ellipti
cE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)/b^2/d/((b+a*cos(d*x+c))
/(a+b))^(1/2)

Rubi [A] (verified)

Time = 2.02 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.311, Rules used = {4350, 4181, 4187, 4193, 3944, 2886, 2884, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \frac {\sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sin (c+d x) \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{24 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (a^2 (-C)+18 a b B+24 A b^2+16 b^2 C\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{24 b d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\sqrt {\cos (c+d x)} \left (-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{24 b^2 d \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {\left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{8 b^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {(a C+6 b B) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{12 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d \cos ^{\frac {5}{2}}(c+d x)} \]

[In]

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

[Out]

((24*A*b^2 + 18*a*b*B - a^2*C + 16*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a +
b)])/(24*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((2*a^2*b*B - 8*b^3*B - a^3*C - 4*a*b^2*(2*A + C)
)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(8*b^2*d*Sqrt[Cos[c + d*x]]*Sq
rt[a + b*Sec[c + d*x]]) - ((24*A*b^2 + 6*a*b*B - 3*a^2*C + 16*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2,
(2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (C*Sqrt[a + b*Sec[c
+ d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(5/2)) + ((6*b*B + a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*b*d
*Cos[c + d*x]^(3/2)) + ((24*A*b^2 + 6*a*b*B - 3*a^2*C + 16*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*b
^2*d*Sqrt[Cos[c + d*x]])

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2884

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]

Rule 2886

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/
(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 3941

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)], x_Symbol] :> Dist[Sqrt[a +
b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*Sqrt[b + a*Sin[e + f*x]]), Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; Free
Q[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3943

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[Sqrt[d*C
sc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]), Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; Fr
eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3944

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[d*Sqrt
[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]), Int[1/(Sin[e + f*x]*Sqrt[b + a*Sin[e + f
*x]]), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4120

Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(
b_.) + (a_)]), x_Symbol] :> Dist[A/a, Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Dist[(A*b -
a*B)/(a*d), Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && Ne
Q[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]

Rule 4181

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(
(d*Csc[e + f*x])^n/(f*(m + n + 1))), x] + Dist[1/(m + n + 1), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x]
)^n*Simp[a*A*(m + n + 1) + a*C*n + ((A*b + a*B)*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) + a
*C*m)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] &&
!LeQ[n, -1]

Rule 4187

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

Rule 4193

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d
_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Dist[C/d^2, Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a +
b*Csc[e + f*x]], x], x] + Int[(A + B*Csc[e + f*x])/(Sqrt[d*Csc[e + f*x]]*Sqrt[a + b*Csc[e + f*x]]), x] /; Fre
eQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]

Rule 4350

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

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx \\ & = \frac {C \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{3} \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (\frac {3}{2} a (2 A+C)+(3 A b+3 a B+2 b C) \sec (c+d x)+\frac {1}{2} (6 b B+a C) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {C \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {(6 b B+a C) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{12 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)} \left (\frac {1}{4} a (6 b B+a C)+\frac {1}{2} b (12 a A+6 b B+7 a C) \sec (c+d x)+\frac {1}{4} \left (24 A b^2+6 a b B-3 a^2 C+16 b^2 C\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{6 b} \\ & = \frac {C \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {(6 b B+a C) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{12 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (24 A b^2+6 a b B-3 a^2 C+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{8} a \left (24 A b^2+6 a b B-3 a^2 C+16 b^2 C\right )+\frac {1}{4} a b (6 b B+a C) \sec (c+d x)-\frac {3}{8} \left (2 a^2 b B-8 b^3 B-a^3 C-4 a b^2 (2 A+C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{6 b^2} \\ & = \frac {C \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {(6 b B+a C) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{12 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (24 A b^2+6 a b B-3 a^2 C+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{8} a \left (24 A b^2+6 a b B-3 a^2 C+16 b^2 C\right )+\frac {1}{4} a b (6 b B+a C) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{6 b^2}-\frac {\left (\left (2 a^2 b B-8 b^3 B-a^3 C-4 a b^2 (2 A+C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{16 b^2} \\ & = \frac {C \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {(6 b B+a C) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{12 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (24 A b^2+6 a b B-3 a^2 C+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (\left (-24 A b^2-6 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 {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{48 b^2}+\frac {\left (\left (24 A b^2+18 a b B-a^2 C+16 b^2 C\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{48 b}-\frac {\left (\left (2 a^2 b B-8 b^3 B-a^3 C-4 a b^2 (2 A+C)\right ) \sqrt {b+a \cos (c+d x)}\right ) \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}} \, dx}{16 b^2 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}} \\ & = \frac {C \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {(6 b B+a C) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{12 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (24 A b^2+6 a b B-3 a^2 C+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (\left (24 A b^2+18 a b B-a^2 C+16 b^2 C\right ) \sqrt {b+a \cos (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{48 b \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (2 a^2 b B-8 b^3 B-a^3 C-4 a b^2 (2 A+C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}}\right ) \int \frac {\sec (c+d x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{16 b^2 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (-24 A b^2-6 a b B+3 a^2 C-16 b^2 C\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{48 b^2 \sqrt {b+a \cos (c+d x)}} \\ & = -\frac {\left (2 a^2 b B-8 b^3 B-a^3 C-4 a b^2 (2 A+C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{8 b^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {C \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {(6 b B+a C) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{12 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (24 A b^2+6 a b B-3 a^2 C+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (\left (24 A b^2+18 a b B-a^2 C+16 b^2 C\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{48 b \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (-24 A b^2-6 a b B+3 a^2 C-16 b^2 C\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{48 b^2 \sqrt {\frac {b+a \cos (c+d x)}{a+b}}} \\ & = \frac {\left (24 A b^2+18 a b B-a^2 C+16 b^2 C\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{24 b d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (2 a^2 b B-8 b^3 B-a^3 C-4 a b^2 (2 A+C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{8 b^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (24 A b^2+6 a b B-3 a^2 C+16 b^2 C\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{24 b^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {C \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {(6 b B+a C) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{12 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (24 A b^2+6 a b B-3 a^2 C+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 b^2 d \sqrt {\cos (c+d x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 37.28 (sec) , antiderivative size = 197409, normalized size of antiderivative = 441.63 \[ \int \frac {\sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 13.14 (sec) , antiderivative size = 4214, normalized size of antiderivative = 9.43

method	result	size
default	\(\text {Expression too large to display}\)	\(4214\)

[In]

int((A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2)/cos(d*x+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/24/d*(-48*A*cos(d*x+c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticPi(((a-b)/(a+b))^(1/2)*(cot
(d*x+c)-csc(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*a*b^2+6*B*cos(d*x+c)^3*sin(d*x+c)*((a-b)/(a+b))^(1/2)*a
^2*b*(1/(1+cos(d*x+c)))^(1/2)+12*B*cos(d*x+c)^3*sin(d*x+c)*((a-b)/(a+b))^(1/2)*a*b^2*(1/(1+cos(d*x+c)))^(1/2)+
2*C*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*a^2*b*sin(d*x+c)*(1/(1+cos(d*x+c)))^(1/2)+16*C*cos(d*x+c)^3*((a-b)/(a+b))
^(1/2)*a*b^2*sin(d*x+c)*(1/(1+cos(d*x+c)))^(1/2)+18*B*cos(d*x+c)^2*sin(d*x+c)*((a-b)/(a+b))^(1/2)*a*b^2*(1/(1+
cos(d*x+c)))^(1/2)-C*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*a^2*b*sin(d*x+c)*(1/(1+cos(d*x+c)))^(1/2)+10*C*cos(d*x+c
)^2*((a-b)/(a+b))^(1/2)*a*b^2*sin(d*x+c)*(1/(1+cos(d*x+c)))^(1/2)+10*C*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a*b^2*si
n(d*x+c)*(1/(1+cos(d*x+c)))^(1/2)-48*B*cos(d*x+c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticPi
(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*b^3+24*B*cos(d*x+c)^3*(1/(a+b)
*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(
1/2))*b^3-6*C*cos(d*x+c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticPi(((a-b)/(a+b))^(1/2)*(cot
(d*x+c)-csc(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*a^3+6*C*cos(d*x+c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d
*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^3-3*C*cos(d*x+c)^3
*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)
/(a-b))^(1/2))*a^3-16*C*cos(d*x+c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(
1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*b^3-48*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*El
lipticPi(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*b^3*cos(d*x+c)^4+24*B*
(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/
(a-b))^(1/2))*b^3*cos(d*x+c)^4-6*C*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticPi(((a-b)/(a+b))^(1
/2)*(cot(d*x+c)-csc(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*a^3*cos(d*x+c)^4+6*C*(1/(a+b)*(b+a*cos(d*x+c))/
(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^3*cos(d*x+
c)^4-3*C*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c))
,(-(a+b)/(a-b))^(1/2))*a^3*cos(d*x+c)^4-16*C*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/
(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*b^3*cos(d*x+c)^4+24*A*cos(d*x+c)^3*sin(d*x+c)*((a-b
)/(a+b))^(1/2)*a*b^2*(1/(1+cos(d*x+c)))^(1/2)+8*C*cos(d*x+c)*((a-b)/(a+b))^(1/2)*b^3*sin(d*x+c)*(1/(1+cos(d*x+
c)))^(1/2)-3*C*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*a^3*sin(d*x+c)*(1/(1+cos(d*x+c)))^(1/2)+24*A*cos(d*x+c)^2*sin(
d*x+c)*((a-b)/(a+b))^(1/2)*b^3*(1/(1+cos(d*x+c)))^(1/2)+12*B*(1/(1+cos(d*x+c)))^(1/2)*((a-b)/(a+b))^(1/2)*b^3*
cos(d*x+c)^2*sin(d*x+c)-6*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot
(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^2*cos(d*x+c)^4-24*C*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1
/2)*EllipticPi(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*a*b^2*cos(d*x+c)
^4-2*C*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(
-(a+b)/(a-b))^(1/2))*a^2*b*cos(d*x+c)^4-4*C*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(
a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^2*cos(d*x+c)^4+3*C*(1/(a+b)*(b+a*cos(d*x+c))/(1+
cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b*cos(d*x+c
)^4+16*C*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c))
,(-(a+b)/(a-b))^(1/2))*a*b^2*cos(d*x+c)^4-12*B*cos(d*x+c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*El
lipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^2+6*B*cos(d*x+c)^3*(1/(a+b)*(b+a
*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))
*a^2*b-6*B*cos(d*x+c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x
+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^2-24*C*cos(d*x+c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*
EllipticPi(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*a*b^2-2*C*cos(d*x+c)
^3*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+
b)/(a-b))^(1/2))*a^2*b-4*C*cos(d*x+c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b)
)^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^2+3*C*cos(d*x+c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(
d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b+16*C*cos(d*x+
c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(
a+b)/(a-b))^(1/2))*a*b^2+12*B*(1/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)*sin(d*x+c)*((a-b)/(a+b))^(1/2)*b^3+12*B*cos(
d*x+c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticPi(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)
),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*a^2*b-12*B*cos(d*x+c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*E
llipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b-48*A*cos(d*x+c)^4*(1/(a+b)*(b
+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticPi(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(a+b)/(a-b),I/((a-
b)/(a+b))^(1/2))*a*b^2+24*A*cos(d*x+c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b
))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^2+12*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1
/2)*EllipticPi(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*a^2*b*cos(d*x+c)
^4-12*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),
(-(a+b)/(a-b))^(1/2))*a^2*b*cos(d*x+c)^4-12*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)
/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^2*cos(d*x+c)^4+6*B*(1/(a+b)*(b+a*cos(d*x+c))/(
1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b*cos(d*x
+c)^4+8*C*((a-b)/(a+b))^(1/2)*b^3*sin(d*x+c)*(1/(1+cos(d*x+c)))^(1/2)-24*A*cos(d*x+c)^4*(1/(a+b)*(b+a*cos(d*x+
c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*b^3-24*A
*cos(d*x+c)^3*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*
x+c)),(-(a+b)/(a-b))^(1/2))*b^3+24*A*cos(d*x+c)^4*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((
a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^2+16*C*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*b^
3*sin(d*x+c)*(1/(1+cos(d*x+c)))^(1/2))*(a+b*sec(d*x+c))^(1/2)/b^2/((a-b)/(a+b))^(1/2)/(b+a*cos(d*x+c))/(1/(1+c
os(d*x+c)))^(1/2)/cos(d*x+c)^(5/2)/(1+cos(d*x+c))

Fricas [F(-1)]

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

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2)/cos(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

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

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)**2)*(a+b*sec(d*x+c))**(1/2)/cos(d*x+c)**(3/2),x)

[Out]

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

Maxima [F]

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

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2)/cos(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2)/cos(d*x+c)^(3/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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