3.14.54 \(\int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sqrt {\cos (c+d x)}} \, dx\) [1354]

Optimal. Leaf size=550 \[ \frac {\left (472 a^2 b B+128 b^3 B+4 a b^2 (132 A+89 C)+a^3 (384 A+133 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{192 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{64 b d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{192 b d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d \sqrt {\cos (c+d x)}}+\frac {(8 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)} \]

[Out]

1/24*(8*B*b+5*C*a)*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d/cos(d*x+c)^(3/2)+1/4*C*(a+b*sec(d*x+c))^(5/2)*sin(d*x+c
)/d/cos(d*x+c)^(3/2)+1/192*(472*a^2*b*B+128*b^3*B+4*a*b^2*(132*A+89*C)+a^3*(384*A+133*C))*(cos(1/2*d*x+1/2*c)^
2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*((b+a*cos(d*x+c))/(a+b))^(1/
2)/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)+1/64*(40*a^3*b*B+160*a*b^3*B-5*a^4*C+120*a^2*b^2*(2*A+C)+16*b^4*(
4*A+3*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/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)+1/32*(16*A*b^2+24*B*a*b+5*C*a^
2+12*C*b^2)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d/cos(d*x+c)^(3/2)+1/192*(264*a^2*b*B+128*b^3*B+15*a^3*C+4*a*b^2
*(108*A+71*C))*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/b/d/cos(d*x+c)^(1/2)-1/192*(264*a^2*b*B+128*b^3*B+15*a^3*C+4*
a*b^2*(108*A+71*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a
+b))^(1/2))*cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)/b/d/((b+a*cos(d*x+c))/(a+b))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.61, antiderivative size = 550, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 14, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.311, Rules used = {4350, 4181, 4187, 4193, 3944, 2886, 2884, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \begin {gather*} \frac {\sin (c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{32 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\sin (c+d x) \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{192 b d \sqrt {\cos (c+d x)}}+\frac {\left (a^3 (384 A+133 C)+472 a^2 b B+4 a b^2 (132 A+89 C)+128 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{192 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\sqrt {\cos (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{192 b d \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {\left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{64 b d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {(5 a C+8 b B) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{4 d \cos ^{\frac {3}{2}}(c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]

[Out]

((472*a^2*b*B + 128*b^3*B + 4*a*b^2*(132*A + 89*C) + a^3*(384*A + 133*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*E
llipticF[(c + d*x)/2, (2*a)/(a + b)])/(192*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((40*a^3*b*B + 160
*a*b^3*B - 5*a^4*C + 120*a^2*b^2*(2*A + C) + 16*b^4*(4*A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi
[2, (c + d*x)/2, (2*a)/(a + b)])/(64*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((264*a^2*b*B + 128*b^
3*B + 15*a^3*C + 4*a*b^2*(108*A + 71*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*S
ec[c + d*x]])/(192*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + ((16*A*b^2 + 24*a*b*B + 5*a^2*C + 12*b^2*C)*Sqrt[
a + b*Sec[c + d*x]]*Sin[c + d*x])/(32*d*Cos[c + d*x]^(3/2)) + ((264*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(
108*A + 71*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*b*d*Sqrt[Cos[c + d*x]]) + ((8*b*B + 5*a*C)*(a + b*S
ec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d*Cos[c + d*x]^(3/2)) + (C*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d*
Cos[c + d*x]^(3/2))

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*} \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {1}{4} \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \left (\frac {1}{2} a (8 A+C)+(4 A b+4 a B+3 b C) \sec (c+d x)+\frac {1}{2} (8 b B+5 a C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {(8 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {1}{12} \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \left (\frac {1}{4} a (48 a A+8 b B+11 a C)+\frac {1}{2} \left (24 a^2 B+16 b^2 B+a b (48 A+31 C)\right ) \sec (c+d x)+\frac {3}{4} \left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {(8 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {1}{24} \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)} \left (\frac {1}{8} a \left (192 a^2 A+48 A b^2+104 a b B+59 a^2 C+36 b^2 C\right )+\frac {1}{4} \left (96 a^3 B+152 a b^2 B+12 b^3 (4 A+3 C)+a^2 b (288 A+161 C)\right ) \sec (c+d x)+\frac {1}{8} \left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx\\ &=\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d \sqrt {\cos (c+d x)}}+\frac {(8 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{16} a \left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right )+\frac {1}{8} a b \left (104 a b B+12 b^2 (4 A+3 C)+a^2 (192 A+59 C)\right ) \sec (c+d x)+\frac {3}{16} \left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{24 b}\\ &=\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d \sqrt {\cos (c+d x)}}+\frac {(8 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{16} a \left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right )+\frac {1}{8} a b \left (104 a b B+12 b^2 (4 A+3 C)+a^2 (192 A+59 C)\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{24 b}+\frac {\left (\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 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}{128 b}\\ &=\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d \sqrt {\cos (c+d x)}}+\frac {(8 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (\left (-264 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (108 A+71 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}{384 b}+\frac {1}{384} \left (\left (472 a^2 b B+128 b^3 B+4 a b^2 (132 A+89 C)+a^3 (384 A+133 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+\frac {\left (\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {b+a \cos (c+d x)}\right ) \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}} \, dx}{128 b \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}\\ &=\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d \sqrt {\cos (c+d x)}}+\frac {(8 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (\left (472 a^2 b B+128 b^3 B+4 a b^2 (132 A+89 C)+a^3 (384 A+133 C)\right ) \sqrt {b+a \cos (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{384 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 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}{128 b \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (-264 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (108 A+71 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{384 b \sqrt {b+a \cos (c+d x)}}\\ &=\frac {\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{64 b d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d \sqrt {\cos (c+d x)}}+\frac {(8 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (\left (472 a^2 b B+128 b^3 B+4 a b^2 (132 A+89 C)+a^3 (384 A+133 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}{384 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (-264 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (108 A+71 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}{384 b \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}\\ &=\frac {\left (472 a^2 b B+128 b^3 B+4 a b^2 (132 A+89 C)+a^3 (384 A+133 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{192 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{64 b d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 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)}}{192 b d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d \sqrt {\cos (c+d x)}}+\frac {(8 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 36.85, size = 180789, normalized size = 328.71 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]

[Out]

Result too large to show

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Maple [C] Result contains complex when optimal does not.
time = 0.51, size = 4031, normalized size = 7.33

method	result	size
default	\(\text {Expression too large to display}\)	\(4031\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/192/d*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*(1+cos(d*x+c))^2*(-1+cos(d*x+c))^3*(-384*A*((b+a*cos(d*x+c))/(1+c
os(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x
+c)^4*a^3*b+128*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^3*b^4*(1/(1+cos(d*x+c)))^(3/2)+118*C*((a-b)/(a+b))
^(1/2)*cos(d*x+c)^4*a^3*b*(1/(1+cos(d*x+c)))^(3/2)*sin(d*x+c)+96*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^4
*a*b^3*(1/(1+cos(d*x+c)))^(3/2)+272*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^2*a*b^3*(1/(1+cos(d*x+c)))^(3/
2)+284*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)^4*a^2*b^2*(1/(1+cos(d*x+c)))^(3/2)*sin(d*x+c)+72*C*((a-b)/(a+b))^(1/2)
*cos(d*x+c)^4*a*b^3*(1/(1+cos(d*x+c)))^(3/2)*sin(d*x+c)+133*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)^3*a^3*b*(1/(1+cos
(d*x+c)))^(3/2)*sin(d*x+c)+254*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)^3*a^2*b^2*(1/(1+cos(d*x+c)))^(3/2)*sin(d*x+c)+
356*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)^3*a*b^3*(1/(1+cos(d*x+c)))^(3/2)*sin(d*x+c)+254*C*((a-b)/(a+b))^(1/2)*cos
(d*x+c)^2*a^2*b^2*(1/(1+cos(d*x+c)))^(3/2)*sin(d*x+c)+184*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)^2*a*b^3*(1/(1+cos(d
*x+c)))^(3/2)*sin(d*x+c)+184*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)*a*b^3*(1/(1+cos(d*x+c)))^(3/2)*sin(d*x+c)+264*B*
((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^4*a^3*b*(1/(1+cos(d*x+c)))^(3/2)+208*B*((a-b)/(a+b))^(1/2)*sin(d*x+c
)*cos(d*x+c)^4*a^2*b^2*(1/(1+cos(d*x+c)))^(3/2)+128*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^4*a*b^3*(1/(1+
cos(d*x+c)))^(3/2)+528*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^3*a*b^3*(1/(1+cos(d*x+c)))^(3/2)+432*A*((a-
b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^4*a^2*b^2*(1/(1+cos(d*x+c)))^(3/2)+472*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*
cos(d*x+c)^3*a^2*b^2*(1/(1+cos(d*x+c)))^(3/2)+272*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^3*a*b^3*(1/(1+co
s(d*x+c)))^(3/2)-128*B*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+
c)^4*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*b^4+15*C*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*Elli
pticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^4*a^4+30*C*((b+a*cos(d*x
+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticPi((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(a+b)/(a-b),I/((a-b
)/(a+b))^(1/2))*cos(d*x+c)^4*a^4-288*C*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticPi((-1+cos(d*x+c)
)*((a-b)/(a+b))^(1/2)/sin(d*x+c),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*cos(d*x+c)^4*b^4-30*C*((b+a*cos(d*x+c))/(1
+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d
*x+c)^4*a^4+144*C*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/
sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^4*b^4+48*C*((a-b)/(a+b))^(1/2)*sin(d*x+c)*b^4*(1/(1+cos(d*x+c)))^(
3/2)+192*A*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x
+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^4*b^4-384*A*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticPi((-1+
cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*cos(d*x+c)^4*b^4+64*B*((a-b)/(a+
b))^(1/2)*sin(d*x+c)*cos(d*x+c)*b^4*(1/(1+cos(d*x+c)))^(3/2)+96*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^3*
b^4*(1/(1+cos(d*x+c)))^(3/2)+72*C*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^3*b^4*(1/(1+cos(d*x+c)))^(3/2)+96*
A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^2*b^4*(1/(1+cos(d*x+c)))^(3/2)+64*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)
*cos(d*x+c)^2*b^4*(1/(1+cos(d*x+c)))^(3/2)+288*A*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+c
os(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^4*a^2*b^2-96*A*((b+a*cos(d*x+c))/(1
+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d
*x+c)^4*a*b^3+432*A*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2
)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^4*a^2*b^2-432*A*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*El
lipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^4*a*b^3-1440*A*((b+a*c
os(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticPi((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(a+b)/(a-b),I
/((a-b)/(a+b))^(1/2))*cos(d*x+c)^4*a^2*b^2+15*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)^4*a^4*(1/(1+cos(d*x+c)))^(3/2)*
sin(d*x+c)+72*C*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^2*b^4*(1/(1+cos(d*x+c)))^(3/2)+264*B*EllipticE((-1+c
os(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^4*((b+a*cos(d*x+c))/(1+cos(d*x+c))/
(a+b))^(1/2)*a^3*b-264*B*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*
x+c)^4*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*a^2*b^2+128*B*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/
2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^4*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*a*b^3-240*B*((b
+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticPi((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(a+b)/(a-
b),I/((a-b)/(a+b))^(1/2))*cos(d*x+c)^4*a^3*b-96...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))**(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/cos(d*x+c)**(1/2),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{\sqrt {\cos \left (c+d\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/cos(c + d*x)^(1/2),x)

[Out]

int(((a + b/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/cos(c + d*x)^(1/2), x)

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