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

Optimal. Leaf size=475 \[ -\frac{2 (a-b) \sqrt{a+b} \left (-6 a^2 b (3 A-B)+8 a^3 B-3 a b^2 (57 A-13 B)+3 b^3 (25 A-49 B)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{315 b^3 d}-\frac{2 \left (-8 a^2 B+18 a A b-49 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{315 b^2 d}-\frac{2 \left (18 a^2 A b-8 a^3 B-39 a b^2 B-75 A b^3\right ) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^2 d}+\frac{2 (a-b) \sqrt{a+b} \left (18 a^3 A b-33 a^2 b^2 B-8 a^4 B-246 a A b^3-147 b^4 B\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{315 b^4 d}+\frac{2 (9 A b-4 a B) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b^2 d}+\frac{2 B \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d} \]

[Out]

(2*(a - b)*Sqrt[a + b]*(18*a^3*A*b - 246*a*A*b^3 - 8*a^4*B - 33*a^2*b^2*B - 147*b^4*B)*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]*(3*b^3*(25*A - 49*B) - 3*a*b^2*(57*A - 13*
B) - 6*a^2*b*(3*A - B) + 8*a^3*B)*Cot[c + d*x]*EllipticF[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^3*d) - (2*(18*a
^2*A*b - 75*A*b^3 - 8*a^3*B - 39*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) - (2*(18*a*A*b -
8*a^2*B - 49*b^2*B)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b^2*d) + (2*(9*A*b - 4*a*B)*(a + b*Sec[c + d
*x])^(5/2)*Tan[c + d*x])/(63*b^2*d) + (2*B*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(9*b*d)

________________________________________________________________________________________

Rubi [A]  time = 1.22462, antiderivative size = 475, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4033, 4082, 4002, 4005, 3832, 4004} \[ -\frac{2 \left (-8 a^2 B+18 a A b-49 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{315 b^2 d}-\frac{2 \left (18 a^2 A b-8 a^3 B-39 a b^2 B-75 A b^3\right ) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^2 d}-\frac{2 (a-b) \sqrt{a+b} \left (-6 a^2 b (3 A-B)+8 a^3 B-3 a b^2 (57 A-13 B)+3 b^3 (25 A-49 B)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{315 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \left (18 a^3 A b-33 a^2 b^2 B-8 a^4 B-246 a A b^3-147 b^4 B\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{315 b^4 d}+\frac{2 (9 A b-4 a B) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b^2 d}+\frac{2 B \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a - b)*Sqrt[a + b]*(18*a^3*A*b - 246*a*A*b^3 - 8*a^4*B - 33*a^2*b^2*B - 147*b^4*B)*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]*(3*b^3*(25*A - 49*B) - 3*a*b^2*(57*A - 13*
B) - 6*a^2*b*(3*A - B) + 8*a^3*B)*Cot[c + d*x]*EllipticF[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^3*d) - (2*(18*a
^2*A*b - 75*A*b^3 - 8*a^3*B - 39*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) - (2*(18*a*A*b -
8*a^2*B - 49*b^2*B)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b^2*d) + (2*(9*A*b - 4*a*B)*(a + b*Sec[c + d
*x])^(5/2)*Tan[c + d*x])/(63*b^2*d) + (2*B*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(9*b*d)

Rule 4033

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> -Simp[(B*d^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2))/(
b*f*(m + n)), x] + Dist[d^2/(b*(m + n)), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 2)*Simp[a*B*(n - 2)
+ B*b*(m + n - 1)*Csc[e + f*x] + (A*b*(m + n) - a*B*(n - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e,
f, A, B, m}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && NeQ[m + n, 0] &&  !IGtQ[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 ^3(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx &=\frac{2 B \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 B+\frac{7}{2} b B \sec (c+d x)+\frac{1}{2} (9 A b-4 a B) \sec ^2(c+d x)\right ) \, dx}{9 b}\\ &=\frac{2 (9 A b-4 a B) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b^2 d}+\frac{2 B \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 A b-2 a B)-\frac{1}{4} \left (18 a A b-8 a^2 B-49 b^2 B\right ) \sec (c+d x)\right ) \, dx}{63 b^2}\\ &=-\frac{2 \left (18 a A b-8 a^2 B-49 b^2 B\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b^2 d}+\frac{2 (9 A b-4 a B) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b^2 d}+\frac{2 B \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 (57 a A b-2 a^2 B+49 b^2 B\right )-\frac{3}{8} \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \sec (c+d x)\right ) \, dx}{315 b^2}\\ &=-\frac{2 \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}-\frac{2 \left (18 a A b-8 a^2 B-49 b^2 B\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b^2 d}+\frac{2 (9 A b-4 a B) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b^2 d}+\frac{2 B \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 A b+75 A b^3+2 a^3 B+186 a b^2 B\right )-\frac{3}{16} \left (18 a^3 A b-246 a A b^3-8 a^4 B-33 a^2 b^2 B-147 b^4 B\right ) \sec (c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{945 b^2}\\ &=-\frac{2 \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}-\frac{2 \left (18 a A b-8 a^2 B-49 b^2 B\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b^2 d}+\frac{2 (9 A b-4 a B) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b^2 d}+\frac{2 B \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{9 b d}-\frac{\left ((a-b) \left (3 b^3 (25 A-49 B)-3 a b^2 (57 A-13 B)-6 a^2 b (3 A-B)+8 a^3 B\right )\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{315 b^2}-\frac{\left (18 a^3 A b-246 a A b^3-8 a^4 B-33 a^2 b^2 B-147 b^4 B\right ) \int \frac{\sec (c+d x) (1+\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 A b-246 a A b^3-8 a^4 B-33 a^2 b^2 B-147 b^4 B\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 (3 b^3 (25 A-49 B)-3 a b^2 (57 A-13 B)-6 a^2 b (3 A-B)+8 a^3 B\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 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}-\frac{2 \left (18 a A b-8 a^2 B-49 b^2 B\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b^2 d}+\frac{2 (9 A b-4 a B) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b^2 d}+\frac{2 B \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{9 b d}\\ \end{align*}

Mathematica [B]  time = 25.7647, size = 3766, normalized size = 7.93 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*((2*(-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*
Sin[c + d*x])/(315*b^3) + (2*Sec[c + d*x]^3*(9*A*b*Sin[c + d*x] + 10*a*B*Sin[c + d*x]))/63 + (2*Sec[c + d*x]^2
*(72*a*A*b*Sin[c + d*x] + 3*a^2*B*Sin[c + d*x] + 49*b^2*B*Sin[c + d*x]))/(315*b) + (2*Sec[c + d*x]*(9*a^2*A*b*
Sin[c + d*x] + 75*A*b^3*Sin[c + d*x] - 4*a^3*B*Sin[c + d*x] + 88*a*b^2*B*Sin[c + d*x]))/(315*b^2) + (2*b*B*Sec
[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x])) - (2*((2*a^3*A)/(35*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec
[c + d*x]]) - (82*a*A*b)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (11*a^2*B)/(105*Sqrt[b + a*Cos[c
+ d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^4*B)/(315*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (7*b^2*B)/(15*
Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (31*a^2*A*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) +
(2*a^4*A*Sqrt[Sec[c + d*x]])/(35*b^2*Sqrt[b + a*Cos[c + d*x]]) + (5*A*b^2*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*C
os[c + d*x]]) - (8*a^5*B*Sqrt[Sec[c + d*x]])/(315*b^3*Sqrt[b + a*Cos[c + d*x]]) - (31*a^3*B*Sqrt[Sec[c + d*x]]
)/(315*b*Sqrt[b + a*Cos[c + d*x]]) + (13*a*b*B*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) - (82*a^2*A*
Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (2*a^4*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d
*x]])/(35*b^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^5*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(315*b^3*Sqrt[b + a*Co
s[c + d*x]]) - (11*a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*b*Sqrt[b + a*Cos[c + d*x]]) - (7*a*b*B*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*S
ec[c + d*x])^(3/2)*(2*(a + b)*(-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Sqrt[Cos[c + d*
x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/
2]], (a - b)/(a + b)] - 2*b*(a + b)*(8*a^3*B - 6*a^2*b*(3*A + B) + 3*a*b^2*(57*A + 13*B) + 3*b^3*(25*A + 49*B)
)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcS
in[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Cos[
c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^3*d*(b + a*Cos[c + d*x])^2*Sqrt[Sec
[(c + d*x)/2]^2]*Sec[c + d*x]^(3/2)*(-(a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-18*a^
3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Co
s[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(
8*a^3*B - 6*a^2*b*(3*A + B) + 3*a*b^2*(57*A + 13*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x]
)]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)
] + (-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c
+ d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^3*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[Cos[(c +
d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*
b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE
[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(8*a^3*B - 6*a^2*b*(3*A + B) + 3*a*b^2*(57*A + 13*B)
+ 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c +
d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b
^2*B + 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^3*Sqrt[b + a*
Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-18*a^3*A*b + 246*a*A*b^
3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-1
8*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c +
d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2
- Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(a + b)*(8*a^3*B - 6*a^2*b*(3*
A + B) + 3*a*b^2*(57*A + 13*B) + 3*b^3*(25*A + 49*B))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*
EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c
+ d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-18*a^3*A*b + 246*a*A*b^3 + 8*a
^4*B + 33*a^2*b^2*B + 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a
- b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)
*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(a + b)*(8*a^3*B - 6*a^2
*b*(3*A + B) + 3*a*b^2*(57*A + 13*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[Ar
cSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d
*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] -
a*(-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x
]*Tan[(c + d*x)/2] - (-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*(b + a*Cos[c + d*x])*Sec
[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*
B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(a + b)*(8*a^3*B - 6*a^2*b*(3*
A + B) + 3*a*b^2*(57*A + 13*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c
+ d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[
(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Sqrt[Cos
[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[
1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(315*b^3*Sqrt[b + a*Cos[c + d*x]]*Sq
rt[Sec[(c + d*x)/2]^2]) - ((2*(a + b)*(-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Sqrt[Co
s[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c
+ d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(8*a^3*B - 6*a^2*b*(3*A + B) + 3*a*b^2*(57*A + 13*B) + 3*b^3*(25*A
+ 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Ellipt
icF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4
*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Si
n[(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]]*Sqrt[Sec[(c
+ d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))

________________________________________________________________________________________

Maple [B]  time = 1.639, size = 4394, normalized size = 9.3 \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)^3*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)),x)

[Out]

-2/315/d/b^3*(cos(d*x+c)+1)^2*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*(-1+cos(d*x+c))^2*(-35*B*b^5-18*A*cos(d*x+c)
^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+
cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b^2+8*B*cos(d*x+c)^6*a^5-246*A*cos(d*x+c)^5*sin(d*x+c)*(cos(d*
x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c
),((a-b)/(a+b))^(1/2))*a^2*b^3-246*A*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*c
os(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^4+8*B*cos(d*x+c
)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1
+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^4*b+2*B*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/
2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a
^3*b^2+33*B*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1)
)^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^3-8*B*cos(d*x+c)^5*a^5+147*B*cos(d*x+c
)^5*b^5-98*B*cos(d*x+c)^4*b^5-14*B*cos(d*x+c)^2*b^5+75*A*cos(d*x+c)^5*b^5-30*A*cos(d*x+c)^3*b^5-45*A*cos(d*x+c
)*b^5-8*B*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^
(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^5-147*B*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/
(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a
-b)/(a+b))^(1/2))*b^5+75*A*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))
/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*b^5+147*B*cos(d*x+c)^4*sin(d*
x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c
))/sin(d*x+c),((a-b)/(a+b))^(1/2))*b^5-8*B*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*
(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^5-147*B*cos
(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Ellipti
cE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*b^5+75*A*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1)
)^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/
2))*b^5+147*B*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+
1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*b^5-85*B*cos(d*x+c)*a*b^4+246*A*cos(d*x+c)
^6*a^2*b^3+75*A*cos(d*x+c)^6*a*b^4-4*B*cos(d*x+c)^6*a^4*b+33*B*cos(d*x+c)^6*a^3*b^2+88*B*cos(d*x+c)^6*a^2*b^3+
147*B*cos(d*x+c)^6*a*b^4+18*A*cos(d*x+c)^5*a^4*b-18*A*cos(d*x+c)^5*a^3*b^2-165*A*cos(d*x+c)^5*a^2*b^3+246*A*co
s(d*x+c)^5*a*b^4+8*B*cos(d*x+c)^5*a^4*b-34*B*cos(d*x+c)^5*a^3*b^2+33*B*cos(d*x+c)^5*a^2*b^3-10*B*cos(d*x+c)^5*
a*b^4+9*A*cos(d*x+c)^4*a^3*b^2-204*A*cos(d*x+c)^4*a*b^4-4*B*cos(d*x+c)^4*a^4*b-68*B*cos(d*x+c)^4*a^2*b^3-81*A*
cos(d*x+c)^3*a^2*b^3+B*cos(d*x+c)^3*a^3*b^2-52*B*cos(d*x+c)^3*a*b^4-117*A*cos(d*x+c)^2*a*b^4-53*B*cos(d*x+c)^2
*a^2*b^3-18*A*cos(d*x+c)^6*a^4*b+9*A*cos(d*x+c)^6*a^3*b^2+186*B*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c
)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))
^(1/2))*a*b^4-8*B*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x
+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^4*b-33*B*cos(d*x+c)^5*sin(d*x+c)*(co
s(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d
*x+c),((a-b)/(a+b))^(1/2))*a^3*b^2-33*B*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+
a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^3-147*B*co
s(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Ellipt
icE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^4-18*A*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)
+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^
(1/2))*a^3*b^2+153*A*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(
d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^3+246*A*cos(d*x+c)^4*sin(d*x+
c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))
/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^4+18*A*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)
*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^4*b+18*A*c
os(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Ellip
ticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b^2-246*A*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*
x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+
b))^(1/2))*a^2*b^3-246*A*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(
cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^4+8*B*cos(d*x+c)^4*sin(d*x+
c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))
/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^4*b+2*B*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*
(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b^2+33*B*
cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Elli
pticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^3+186*B*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d
*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a
+b))^(1/2))*a*b^4-8*B*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos
(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^4*b-33*B*cos(d*x+c)^4*sin(d*x+c)
*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/s
in(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b^2-33*B*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)
*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^3-147*
B*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*El
lipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^4+153*A*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d
*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a
+b))^(1/2))*a^2*b^3+246*A*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/
(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^4+18*A*cos(d*x+c)^5*sin(d*
x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c
))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^4*b+18*A*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+
b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b^2)/(
b+a*cos(d*x+c))/cos(d*x+c)^4/sin(d*x+c)^5

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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)^3*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)),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 (B b \sec \left (d x + c\right )^{5} + A a \sec \left (d x + c\right )^{3} +{\left (B a + A b\right )} \sec \left (d x + c\right )^{4}\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)^3*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)),x, algorithm="fricas")

[Out]

integral((B*b*sec(d*x + c)^5 + A*a*sec(d*x + c)^3 + (B*a + A*b)*sec(d*x + c)^4)*sqrt(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)**3*(a+b*sec(d*x+c))**(3/2)*(A+B*sec(d*x+c)),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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