3.1354 \(\int \frac{(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sqrt{\cos (c+d x)}} \, dx\)

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

[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))

________________________________________________________________________________________

Rubi [A]  time = 2.38615, antiderivative size = 550, normalized size of antiderivative = 1., 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 = {4265, 4096, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661} \[ \frac{\sin (c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt{a+b \sec (c+d x)}}{32 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left (264 a^2 b B+15 a^3 C+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt{a+b \sec (c+d x)}}{192 b d \sqrt{\cos (c+d x)}}+\frac{\left (a^3 (384 A+133 C)+472 a^2 b B+4 a b^2 (132 A+89 C)+128 b^3 B\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{192 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \left (264 a^2 b B+15 a^3 C+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{192 b d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\left (120 a^2 b^2 (2 A+C)+40 a^3 b B-5 a^4 C+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{64 b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(5 a C+8 b B) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{24 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{4 d \cos ^{\frac{3}{2}}(c+d x)} \]

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 4265

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]

Rule 4096

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 4102

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 4108

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 3859

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(d*Sqr
t[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 2807

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*
Sin[e + f*x])/(c + d)]), 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 2805

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[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 4035

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 3856

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 2655

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*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

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

Rule 3858

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(Sqrt[d*
Csc[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 2663

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*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

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

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*}

Mathematica [C]  time = 35.9669, size = 180789, normalized size = 328.71 \[ \text{Result too large to show} \]

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

________________________________________________________________________________________

Maple [C]  time = 0.953, size = 4031, normalized size = 7.3 \begin{align*} \text{output too large to display} \end{align*}

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)

[Out]

-1/192/d*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*(cos(d*x+c)+1)^2*(-1+cos(d*x+c))^3*(72*C*((a-b)/(a+b))^(1/2)*(1/(
cos(d*x+c)+1))^(3/2)*cos(d*x+c)^2*sin(d*x+c)*b^4-384*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Ellipti
cF((-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+432*A*((a-b)/(a+b))
^(1/2)*cos(d*x+c)^4*sin(d*x+c)*a^2*b^2*(1/(cos(d*x+c)+1))^(3/2)+272*B*((a-b)/(a+b))^(1/2)*cos(d*x+c)^2*sin(d*x
+c)*a*b^3*(1/(cos(d*x+c)+1))^(3/2)+472*B*((a-b)/(a+b))^(1/2)*cos(d*x+c)^3*sin(d*x+c)*a^2*b^2*(1/(cos(d*x+c)+1)
)^(3/2)+254*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)^2*a^2*b^2*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3/2)+184*C*((a-b)/(a+b))
^(1/2)*cos(d*x+c)^2*a*b^3*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3/2)+264*B*((a-b)/(a+b))^(1/2)*cos(d*x+c)^4*sin(d*x+c
)*a^3*b*(1/(cos(d*x+c)+1))^(3/2)+208*B*((a-b)/(a+b))^(1/2)*cos(d*x+c)^4*sin(d*x+c)*a^2*b^2*(1/(cos(d*x+c)+1))^
(3/2)+128*B*((a-b)/(a+b))^(1/2)*cos(d*x+c)^4*sin(d*x+c)*a*b^3*(1/(cos(d*x+c)+1))^(3/2)+356*C*((a-b)/(a+b))^(1/
2)*cos(d*x+c)^3*a*b^3*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3/2)+133*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)^3*a^3*b*sin(d*x
+c)*(1/(cos(d*x+c)+1))^(3/2)+254*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)^3*a^2*b^2*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3/2
)+118*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)^4*a^3*b*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3/2)+284*C*((a-b)/(a+b))^(1/2)*c
os(d*x+c)^4*a^2*b^2*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3/2)+72*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)^4*a*b^3*sin(d*x+c)
*(1/(cos(d*x+c)+1))^(3/2)+528*A*((a-b)/(a+b))^(1/2)*cos(d*x+c)^3*sin(d*x+c)*a*b^3*(1/(cos(d*x+c)+1))^(3/2)+272
*B*((a-b)/(a+b))^(1/2)*cos(d*x+c)^3*sin(d*x+c)*a*b^3*(1/(cos(d*x+c)+1))^(3/2)+184*C*((a-b)/(a+b))^(1/2)*cos(d*
x+c)*a*b^3*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3/2)+96*A*((a-b)/(a+b))^(1/2)*cos(d*x+c)^4*sin(d*x+c)*a*b^3*(1/(cos(
d*x+c)+1))^(3/2)-240*B*cos(d*x+c)^4*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(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))*a^3*b-960*B*cos(d*x+c)^4*(1/(a+b)*(b+a*cos(
d*x+c))/(cos(d*x+c)+1))^(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))*a*b^3-144*B*cos(d*x+c)^4*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^
(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^3*b-208*B*cos(d*x+c)^4*EllipticF((-1+cos(d*x+c))*((a-
b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b^2+352*B
*cos(d*x+c)^4*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos
(d*x+c))/(cos(d*x+c)+1))^(1/2)*a*b^3+264*B*cos(d*x+c)^4*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Ellipt
icE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^3*b-264*B*cos(d*x+c)^4*(1/(a+b)*(b+
a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1
/2))*a^2*b^2+128*B*cos(d*x+c)^4*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-
b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b^3+48*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)*sin(d*x+c)*b^4*(1/(
cos(d*x+c)+1))^(3/2)-720*C*cos(d*x+c)^4*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(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))*a^2*b^2-118*C*cos(d*x+c)^4*EllipticF((-
1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(
1/2)*a^3*b+76*C*cos(d*x+c)^4*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1
/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b^2-72*C*cos(d*x+c)^4*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b)
)^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a*b^3-15*C*cos(d*x+c)
^4*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-
(a+b)/(a-b))^(1/2))*a^3*b+284*C*cos(d*x+c)^4*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos
(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^2*b^2-284*C*cos(d*x+c)^4*(1/(a+b)*(b+a*cos(d*x
+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b^
3+15*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)^4*a^4*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3/2)-96*A*cos(d*x+c)^4*EllipticF((-
1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(
1/2)*a*b^3+432*A*cos(d*x+c)^4*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)
/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^2*b^2-432*A*cos(d*x+c)^4*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c
)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b^3+64*B*((a-b)/(
a+b))^(1/2)*cos(d*x+c)*sin(d*x+c)*b^4*(1/(cos(d*x+c)+1))^(3/2)+96*A*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(3/
2)*cos(d*x+c)^3*sin(d*x+c)*b^4+72*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)^3*sin(d*x+c)*b^4*(1/(cos(d*x+c)+1))^(3/2)+9
6*A*((a-b)/(a+b))^(1/2)*cos(d*x+c)^2*sin(d*x+c)*b^4*(1/(cos(d*x+c)+1))^(3/2)+128*B*((a-b)/(a+b))^(1/2)*cos(d*x
+c)^3*sin(d*x+c)*b^4*(1/(cos(d*x+c)+1))^(3/2)+288*A*cos(d*x+c)^4*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)
/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b^2+64*B*((a-b)/(a+b))^(
1/2)*cos(d*x+c)^2*sin(d*x+c)*b^4*(1/(cos(d*x+c)+1))^(3/2)-1440*A*cos(d*x+c)^4*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d
*x+c)+1))^(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))*a
^2*b^2-384*A*cos(d*x+c)^4*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(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))*b^4-128*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(
1/2)*cos(d*x+c)^4*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*b^4+30*C*cos(
d*x+c)^4*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(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))*a^4-288*C*cos(d*x+c)^4*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(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))*b^4-30*C*cos(d*x
+c)^4*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))
/(cos(d*x+c)+1))^(1/2)*a^4+144*C*cos(d*x+c)^4*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)
/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*b^4+15*C*cos(d*x+c)^4*(1/(a+b)*(b+a*cos(d*x+c))
/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^4+48*C
*((a-b)/(a+b))^(1/2)*sin(d*x+c)*b^4*(1/(cos(d*x+c)+1))^(3/2)+192*A*cos(d*x+c)^4*EllipticF((-1+cos(d*x+c))*((a-
b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*b^4)/b/((a-b)
/(a+b))^(1/2)/(b+a*cos(d*x+c))/cos(d*x+c)^(7/2)/sin(d*x+c)^6/(1/(cos(d*x+c)+1))^(3/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*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]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\sqrt{\cos \left (d x + c\right )}}\,{d x} \end{align*}

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)