3.943 \(\int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=505 \[ \frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left (6 a^2 b (3 B-C)-8 a^3 C-3 a b^2 (21 A-57 B+13 C)+3 b^3 (63 A-25 B+49 C)\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{315 b^3 d}+\frac{2 \tan (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^{3/2}}{315 b^2 d}-\frac{2 \tan (c+d x) \left (18 a^2 b B-8 a^3 C-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt{a+b \sec (c+d x)}}{315 b^2 d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left (-3 a^2 b^2 (21 A+11 C)+18 a^3 b B-8 a^4 C-246 a b^3 B-21 b^4 (9 A+7 C)\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{315 b^4 d}+\frac{2 (9 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b^2 d}+\frac{2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d} \]

[Out]

(2*(a - b)*Sqrt[a + b]*(18*a^3*b*B - 246*a*b^3*B - 8*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(21*A + 11*C))*Cot
[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))
/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) + (2*(a - b)*Sqrt[a + b]*(6*a^2*b*(3*B - C) - 8
*a^3*C - 3*a*b^2*(21*A - 57*B + 13*C) + 3*b^3*(63*A - 25*B + 49*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*S
ec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))
/(a - b))])/(315*b^3*d) - (2*(18*a^2*b*B - 75*b^3*B - 8*a^3*C - 3*a*b^2*(21*A + 13*C))*Sqrt[a + b*Sec[c + d*x]
]*Tan[c + d*x])/(315*b^2*d) + (2*(63*A*b^2 - 18*a*b*B + 8*a^2*C + 49*b^2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c +
d*x])/(315*b^2*d) + (2*(9*b*B - 4*a*C)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b^2*d) + (2*C*Sec[c + d*x
]*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(9*b*d)

________________________________________________________________________________________

Rubi [A]  time = 1.32093, antiderivative size = 505, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used = {4092, 4082, 4002, 4005, 3832, 4004} \[ \frac{2 \tan (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^{3/2}}{315 b^2 d}-\frac{2 \tan (c+d x) \left (18 a^2 b B-8 a^3 C-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt{a+b \sec (c+d x)}}{315 b^2 d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left (6 a^2 b (3 B-C)-8 a^3 C-3 a b^2 (21 A-57 B+13 C)+3 b^3 (63 A-25 B+49 C)\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{315 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left (-3 a^2 b^2 (21 A+11 C)+18 a^3 b B-8 a^4 C-246 a b^3 B-21 b^4 (9 A+7 C)\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{315 b^4 d}+\frac{2 (9 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b^2 d}+\frac{2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a - b)*Sqrt[a + b]*(18*a^3*b*B - 246*a*b^3*B - 8*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(21*A + 11*C))*Cot
[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))
/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) + (2*(a - b)*Sqrt[a + b]*(6*a^2*b*(3*B - C) - 8
*a^3*C - 3*a*b^2*(21*A - 57*B + 13*C) + 3*b^3*(63*A - 25*B + 49*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*S
ec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))
/(a - b))])/(315*b^3*d) - (2*(18*a^2*b*B - 75*b^3*B - 8*a^3*C - 3*a*b^2*(21*A + 13*C))*Sqrt[a + b*Sec[c + d*x]
]*Tan[c + d*x])/(315*b^2*d) + (2*(63*A*b^2 - 18*a*b*B + 8*a^2*C + 49*b^2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c +
d*x])/(315*b^2*d) + (2*(9*b*B - 4*a*C)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b^2*d) + (2*C*Sec[c + d*x
]*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(9*b*d)

Rule 4092

Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(
e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Simp[(C*Csc[e + f*x]*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m
+ 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[a*C + b*(C*(m + 2)
+ A*(m + 3))*Csc[e + f*x] - (2*a*C - b*B*(m + 3))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m
}, x] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1]

Rule 4082

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

Rule 4002

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

Rule 4005

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

Rule 3832

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(-2*Rt[a + b, 2]*Sqr
t[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Csc[e + f*x]))/(a - b))]*EllipticF[ArcSin[Sqrt[a + b*Csc[e +
f*x]]/Rt[a + b, 2]], (a + b)/(a - b)])/(b*f*Cot[e + f*x]), x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4004

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)
], x_Symbol] :> Simp[(-2*(A*b - a*B)*Rt[a + (b*B)/A, 2]*Sqrt[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Cs
c[e + f*x]))/(a - b))]*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + (b*B)/A, 2]], (a*A + b*B)/(a*A - b*B)]
)/(b^2*f*Cot[e + f*x]), x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]

Rubi steps

\begin{align*} \int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{2 C \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{9 b d}+\frac{2 \int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left (a C+\frac{1}{2} b (9 A+7 C) \sec (c+d x)+\frac{1}{2} (9 b B-4 a C) \sec ^2(c+d x)\right ) \, dx}{9 b}\\ &=\frac{2 (9 b B-4 a C) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b^2 d}+\frac{2 C \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{9 b d}+\frac{4 \int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac{3}{4} b (15 b B-2 a C)+\frac{1}{4} \left (63 A b^2-18 a b B+8 a^2 C+49 b^2 C\right ) \sec (c+d x)\right ) \, dx}{63 b^2}\\ &=\frac{2 \left (63 A b^2-18 a b B+8 a^2 C+49 b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b^2 d}+\frac{2 (9 b B-4 a C) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b^2 d}+\frac{2 C \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{9 b d}+\frac{8 \int \sec (c+d x) \sqrt{a+b \sec (c+d x)} \left (\frac{3}{8} b \left (63 A b^2+57 a b B-2 a^2 C+49 b^2 C\right )-\frac{3}{8} \left (18 a^2 b B-75 b^3 B-8 a^3 C-3 a b^2 (21 A+13 C)\right ) \sec (c+d x)\right ) \, dx}{315 b^2}\\ &=-\frac{2 \left (18 a^2 b B-75 b^3 B-8 a^3 C-3 a b^2 (21 A+13 C)\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}+\frac{2 \left (63 A b^2-18 a b B+8 a^2 C+49 b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b^2 d}+\frac{2 (9 b B-4 a C) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b^2 d}+\frac{2 C \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{9 b d}+\frac{16 \int \frac{\sec (c+d x) \left (\frac{3}{16} b \left (153 a^2 b B+75 b^3 B+2 a^3 C+6 a b^2 (42 A+31 C)\right )-\frac{3}{16} \left (18 a^3 b B-246 a b^3 B-8 a^4 C-21 b^4 (9 A+7 C)-3 a^2 b^2 (21 A+11 C)\right ) \sec (c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{945 b^2}\\ &=-\frac{2 \left (18 a^2 b B-75 b^3 B-8 a^3 C-3 a b^2 (21 A+13 C)\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}+\frac{2 \left (63 A b^2-18 a b B+8 a^2 C+49 b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b^2 d}+\frac{2 (9 b B-4 a C) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b^2 d}+\frac{2 C \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{9 b d}-\frac{\left (18 a^3 b B-246 a b^3 B-8 a^4 C-21 b^4 (9 A+7 C)-3 a^2 b^2 (21 A+11 C)\right ) \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx}{315 b^2}+\frac{\left ((a-b) \left (6 a^2 b (3 B-C)-8 a^3 C-3 a b^2 (21 A-57 B+13 C)+3 b^3 (63 A-25 B+49 C)\right )\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{315 b^2}\\ &=\frac{2 (a-b) \sqrt{a+b} \left (18 a^3 b B-246 a b^3 B-8 a^4 C-21 b^4 (9 A+7 C)-3 a^2 b^2 (21 A+11 C)\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{315 b^4 d}+\frac{2 (a-b) \sqrt{a+b} \left (6 a^2 b (3 B-C)-8 a^3 C-3 a b^2 (21 A-57 B+13 C)+3 b^3 (63 A-25 B+49 C)\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{315 b^3 d}-\frac{2 \left (18 a^2 b B-75 b^3 B-8 a^3 C-3 a b^2 (21 A+13 C)\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}+\frac{2 \left (63 A b^2-18 a b B+8 a^2 C+49 b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b^2 d}+\frac{2 (9 b B-4 a C) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b^2 d}+\frac{2 C \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{9 b d}\\ \end{align*}

Mathematica [B]  time = 26.4849, size = 4186, normalized size = 8.29 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(63*a^2*A*b^2 + 189*A*b
^4 - 18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Sin[c + d*x])/(315*b^3) + (4*Sec[c + d*x]^
3*(9*b*B*Sin[c + d*x] + 10*a*C*Sin[c + d*x]))/63 + (4*Sec[c + d*x]^2*(63*A*b^2*Sin[c + d*x] + 72*a*b*B*Sin[c +
d*x] + 3*a^2*C*Sin[c + d*x] + 49*b^2*C*Sin[c + d*x]))/(315*b) + (4*Sec[c + d*x]*(126*a*A*b^2*Sin[c + d*x] + 9
*a^2*b*B*Sin[c + d*x] + 75*b^3*B*Sin[c + d*x] - 4*a^3*C*Sin[c + d*x] + 88*a*b^2*C*Sin[c + d*x]))/(315*b^2) + (
4*b*C*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x
])) - (4*((-2*a^2*A)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (6*A*b^2)/(5*Sqrt[b + a*Cos[c + d*x]]*S
qrt[Sec[c + d*x]]) + (4*a^3*B)/(35*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (164*a*b*B)/(105*Sqrt[b +
a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (22*a^2*C)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (16*a^4*C
)/(315*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (14*b^2*C)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c +
d*x]]) - (2*a^3*A*Sqrt[Sec[c + d*x]])/(5*b*Sqrt[b + a*Cos[c + d*x]]) + (2*a*A*b*Sqrt[Sec[c + d*x]])/(5*Sqrt[b
+ a*Cos[c + d*x]]) - (62*a^2*B*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (4*a^4*B*Sqrt[Sec[c + d*x
]])/(35*b^2*Sqrt[b + a*Cos[c + d*x]]) + (10*b^2*B*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (16*a^5*
C*Sqrt[Sec[c + d*x]])/(315*b^3*Sqrt[b + a*Cos[c + d*x]]) - (62*a^3*C*Sqrt[Sec[c + d*x]])/(315*b*Sqrt[b + a*Cos
[c + d*x]]) + (26*a*b*C*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) - (2*a^3*A*Cos[2*(c + d*x)]*Sqrt[Se
c[c + d*x]])/(5*b*Sqrt[b + a*Cos[c + d*x]]) - (6*a*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[
c + d*x]]) - (164*a^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (4*a^4*B*Cos[2*(
c + d*x)]*Sqrt[Sec[c + d*x]])/(35*b^2*Sqrt[b + a*Cos[c + d*x]]) - (16*a^5*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]
])/(315*b^3*Sqrt[b + a*Cos[c + d*x]]) - (22*a^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*b*Sqrt[b + a*Cos[c
+ d*x]]) - (14*a*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2
]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((a + b)*((-18*a^3*b*B +
246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a -
b)/(a + b)] - b*(8*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(21*A + 57*B + 13*C) + 3*b^3*(63*A + 25*B + 49*C))*Ell
ipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c
+ d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C)
+ 3*a^2*b^2*(21*A + 11*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(315*b^3*d*
(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*
x]^(7/2)*((-2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*((a + b)*((-18*a^3*b*B + 246*a*b^3*B + 8*a^
4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(
8*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(21*A + 57*B + 13*C) + 3*b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[
(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d
*x)/2]^2)/(a + b)]*Sec[c + d*x] + (-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A
+ 11*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(315*b^3*(b + a*Cos[c + d*x])
^(3/2)*(Sec[(c + d*x)/2]^2)^(3/2)) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*((a + b)*((-18*
a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)
/2]], (a - b)/(a + b)] - b*(8*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(21*A + 57*B + 13*C) + 3*b^3*(63*A + 25*B +
49*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b
+ a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9
*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(
105*b^3*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) - (2*((a + b)*((-18*a^3*b*B + 246*a*b^3*B + 8*a^4
*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(8
*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(21*A + 57*B + 13*C) + 3*b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[(
c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*
x)/2]^2)/(a + b)]*Sec[c + d*x] + (-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A +
11*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*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]))/(315*b^3*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c
+ d*x)/2]^2)^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]) - (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-18*a
^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Cos[c + d*x]*(b + a*Cos[c + d*x
])*Sec[(c + d*x)/2]^6)/2 - a*(-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*
C))*Cos[c + d*x]*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] - (-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*
b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/
2] + 2*(-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Cos[c + d*x]*(b +
a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]^2 + (3*(a + b)*((-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*
b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(8*a^3*C -
6*a^2*b*(3*B + C) + 3*a*b^2*(21*A + 57*B + 13*C) + 3*b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[(c + d*x)
/2]], (a - b)/(a + b)])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(
a + b)]*Sec[c + d*x]*(-(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]))/
2 + ((a + b)*((-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[A
rcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(8*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(21*A + 57*B + 13*C) + 3*
b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]
^2)^(3/2)*Sec[c + d*x]*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2
]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + (a + b)*(Cos[c +
d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]*(-(b*(8*a
^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(21*A + 57*B + 13*C) + 3*b^3*(63*A + 25*B + 49*C))*Sec[(c + d*x)/2]^2)/(2*S
qrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((-18*a^3*b*B + 246*a*b^3*B + 8*
a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)
/(a + b)])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2])) + (a + b)*((-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7
*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(8*a^3*C - 6*a^2*b*(3*
B + C) + 3*a*b^2*(21*A + 57*B + 13*C) + 3*b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b
)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Se
c[c + d*x]*Tan[c + d*x]))/(315*b^3*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2))))

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Maple [B]  time = 2.063, size = 5945, normalized size = 11.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C b \sec \left (d x + c\right )^{5} +{\left (C a + B b\right )} \sec \left (d x + c\right )^{4} + A a \sec \left (d x + c\right )^{2} +{\left (B a + A b\right )} \sec \left (d x + c\right )^{3}\right )} \sqrt{b \sec \left (d x + c\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((C*b*sec(d*x + c)^5 + (C*a + B*b)*sec(d*x + c)^4 + A*a*sec(d*x + c)^2 + (B*a + A*b)*sec(d*x + c)^3)*s
qrt(b*sec(d*x + c) + a), x)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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{3}{2}} \sec \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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