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

Optimal. Leaf size=467 \[ \frac{2 (a-b) \sqrt{a+b} \left (12 a^2 b C+16 a^3 C+6 a b^2 (7 A+6 C)+21 b^3 (9 A+7 C)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{315 b^4 d}-\frac{2 \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^2 d}+\frac{2 a \left (8 a^2 C+21 A b^2+13 b^2 C\right ) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \left (6 a^2 b^2 (7 A+4 C)+16 a^4 C-21 b^4 (9 A+7 C)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{315 b^5 d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{9 d}+\frac{2 a C \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{63 b d} \]

[Out]

(2*(a - b)*Sqrt[a + b]*(16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Cot[c + d*x]*EllipticE[ArcSin[S
qrt[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^5*d) + (2*(a - b)*Sqrt[a + b]*(16*a^3*C + 12*a^2*b*C + 6*a*b^2*(7*A + 6*C) + 21*
b^3*(9*A + 7*C))*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^4*d) + (2*a*(21*A*b^2 + 8*a^2*C
+ 13*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^3*d) - (2*(6*a^2*C - 7*b^2*(9*A + 7*C))*Sec[c + d*x]
*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) + (2*a*C*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d
*x])/(63*b*d) + (2*C*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(9*d)

________________________________________________________________________________________

Rubi [A]  time = 1.30483, antiderivative size = 467, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4097, 4102, 4092, 4082, 4005, 3832, 4004} \[ -\frac{2 \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^2 d}+\frac{2 a \left (8 a^2 C+21 A b^2+13 b^2 C\right ) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \left (12 a^2 b C+16 a^3 C+6 a b^2 (7 A+6 C)+21 b^3 (9 A+7 C)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{315 b^4 d}+\frac{2 (a-b) \sqrt{a+b} \left (6 a^2 b^2 (7 A+4 C)+16 a^4 C-21 b^4 (9 A+7 C)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{315 b^5 d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{9 d}+\frac{2 a C \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{63 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a - b)*Sqrt[a + b]*(16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Cot[c + d*x]*EllipticE[ArcSin[S
qrt[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^5*d) + (2*(a - b)*Sqrt[a + b]*(16*a^3*C + 12*a^2*b*C + 6*a*b^2*(7*A + 6*C) + 21*
b^3*(9*A + 7*C))*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^4*d) + (2*a*(21*A*b^2 + 8*a^2*C
+ 13*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^3*d) - (2*(6*a^2*C - 7*b^2*(9*A + 7*C))*Sec[c + d*x]
*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) + (2*a*C*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d
*x])/(63*b*d) + (2*C*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(9*d)

Rule 4097

Int[((A_.) + 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 +
b*(A*(m + n + 1) + C*(m + n))*Csc[e + f*x] + a*C*m*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, 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 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 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) \sqrt{a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{2 C \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{9 d}+\frac{2}{9} \int \frac{\sec ^3(c+d x) \left (\frac{3}{2} a (3 A+2 C)+\frac{1}{2} b (9 A+7 C) \sec (c+d x)+\frac{1}{2} a C \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{2 a C \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{63 b d}+\frac{2 C \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{9 d}+\frac{4 \int \frac{\sec ^2(c+d x) \left (a^2 C+\frac{1}{4} a b (63 A+47 C) \sec (c+d x)-\frac{1}{4} \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{63 b}\\ &=-\frac{2 \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}+\frac{2 a C \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{63 b d}+\frac{2 C \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{9 d}+\frac{8 \int \frac{\sec (c+d x) \left (-\frac{1}{4} a \left (6 a^2 C-7 b^2 (9 A+7 C)\right )+\frac{1}{8} b \left (189 A b^2+2 a^2 C+147 b^2 C\right ) \sec (c+d x)+\frac{3}{8} a \left (21 A b^2+8 a^2 C+13 b^2 C\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{315 b^2}\\ &=\frac{2 a \left (21 A b^2+8 a^2 C+13 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^3 d}-\frac{2 \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}+\frac{2 a C \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{63 b d}+\frac{2 C \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{9 d}+\frac{16 \int \frac{\sec (c+d x) \left (\frac{3}{16} a b \left (147 A b^2-4 a^2 C+111 b^2 C\right )-\frac{3}{16} \left (16 a^4 C+6 a^2 b^2 (7 A+4 C)-21 b^4 (9 A+7 C)\right ) \sec (c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{945 b^3}\\ &=\frac{2 a \left (21 A b^2+8 a^2 C+13 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^3 d}-\frac{2 \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}+\frac{2 a C \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{63 b d}+\frac{2 C \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{9 d}+\frac{\left ((a-b) \left (16 a^3 C+12 a^2 b C+6 a b^2 (7 A+6 C)+21 b^3 (9 A+7 C)\right )\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{315 b^3}-\frac{\left (16 a^4 C+6 a^2 b^2 (7 A+4 C)-21 b^4 (9 A+7 C)\right ) \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx}{315 b^3}\\ &=\frac{2 (a-b) \sqrt{a+b} \left (16 a^4 C+6 a^2 b^2 (7 A+4 C)-21 b^4 (9 A+7 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^5 d}+\frac{2 (a-b) \sqrt{a+b} \left (16 a^3 C+12 a^2 b C+6 a b^2 (7 A+6 C)+21 b^3 (9 A+7 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^4 d}+\frac{2 a \left (21 A b^2+8 a^2 C+13 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^3 d}-\frac{2 \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}+\frac{2 a C \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{63 b d}+\frac{2 C \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{9 d}\\ \end{align*}

Mathematica [B]  time = 23.7268, size = 3518, normalized size = 7.53 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*((4*(-42*a^2*A*b^2 + 189*A*b^4 - 16*a^4*C - 24
*a^2*b^2*C + 147*b^4*C)*Sin[c + d*x])/(315*b^4) + (4*Sec[c + d*x]^2*(63*A*b^2*Sin[c + d*x] - 6*a^2*C*Sin[c + d
*x] + 49*b^2*C*Sin[c + d*x]))/(315*b^2) + (4*Sec[c + d*x]*(21*a*A*b^2*Sin[c + d*x] + 8*a^3*C*Sin[c + d*x] + 13
*a*b^2*C*Sin[c + d*x]))/(315*b^3) + (4*a*C*Sec[c + d*x]^2*Tan[c + d*x])/(63*b) + (4*C*Sec[c + d*x]^3*Tan[c + d
*x])/9))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (4*((4*a^2*A)/(15*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])
- (6*A*b)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (32*a^4*C)/(315*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt
[Sec[c + d*x]]) + (16*a^2*C)/(105*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (14*b*C)/(15*Sqrt[b + a*Cos
[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*a*A*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) + (4*a^3*A*Sqrt[Sec[
c + d*x]])/(15*b^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a*C*Sqrt[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]) + (32*
a^5*C*Sqrt[Sec[c + d*x]])/(315*b^4*Sqrt[b + a*Cos[c + d*x]]) + (8*a^3*C*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt[b + a
*Cos[c + d*x]]) - (6*a*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) + (4*a^3*A*Cos[2*(c
+ d*x)]*Sqrt[Sec[c + d*x]])/(15*b^2*Sqrt[b + a*Cos[c + d*x]]) - (14*a*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/
(15*Sqrt[b + a*Cos[c + d*x]]) + (32*a^5*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(315*b^4*Sqrt[b + a*Cos[c + d*x
]]) + (16*a^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*b^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]
^2*Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*((a + b)*((16*a^4*C + 6*a^2*b^2*(7*A + 4*C) -
21*b^4*(9*A + 7*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-16*a^3*C + 12*a^2*b*C - 6*a*b^
2*(7*A + 6*C) + 21*b^3*(9*A + 7*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] + (16*a^4*C + 6*a^2*b
^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(
315*b^4*d*(b + a*Cos[c + d*x])*(A + 2*C + A*Cos[2*c + 2*d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]^(5/2)*((
2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*((a + b)*((16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9
*A + 7*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-16*a^3*C + 12*a^2*b*C - 6*a*b^2*(7*A + 6
*C) + 21*b^3*(9*A + 7*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] + (16*a^4*C + 6*a^2*b^2*(7*A +
4*C) - 21*b^4*(9*A + 7*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(315*b^4*(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)*((16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a -
b)/(a + b)] + b*(-16*a^3*C + 12*a^2*b*C - 6*a*b^2*(7*A + 6*C) + 21*b^3*(9*A + 7*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] + (16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Cos[c + d*x]*(b + a*Cos[c
+ d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(105*b^4*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) +
(2*((a + b)*((16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a -
b)/(a + b)] + b*(-16*a^3*C + 12*a^2*b*C - 6*a*b^2*(7*A + 6*C) + 21*b^3*(9*A + 7*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] + (16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*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^4*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]]*(((16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21
*b^4*(9*A + 7*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^6)/2 - a*(16*a^4*C + 6*a^2*b^2*(7*A + 4*C
) - 21*b^4*(9*A + 7*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] - (16*a^4*C + 6*a^2*b^2*
(7*A + 4*C) - 21*b^4*(9*A + 7*C))*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] + 2*(1
6*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*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)*((16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*EllipticE[ArcSin[Tan[(c
+ d*x)/2]], (a - b)/(a + b)] + b*(-16*a^3*C + 12*a^2*b*C - 6*a*b^2*(7*A + 6*C) + 21*b^3*(9*A + 7*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)*((16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*EllipticE[ArcSin[Ta
n[(c + d*x)/2]], (a - b)/(a + b)] + b*(-16*a^3*C + 12*a^2*b*C - 6*a*b^2*(7*A + 6*C) + 21*b^3*(9*A + 7*C))*Elli
pticF[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*S
ec[(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*(-16*a^3*C + 12*a^2*b*C - 6*a*b^2*(
7*A + 6*C) + 21*b^3*(9*A + 7*C))*Sec[(c + d*x)/2]^2)/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c
+ d*x)/2]^2)/(a + b)]) + ((16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*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)*((16*a^4*C + 6*a^2*b^2*(7*A
+ 4*C) - 21*b^4*(9*A + 7*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-16*a^3*C + 12*a^2*b*C
- 6*a*b^2*(7*A + 6*C) + 21*b^3*(9*A + 7*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]*Tan[c + d*x]
))/(315*b^4*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2))))

________________________________________________________________________________________

Maple [B]  time = 1.671, size = 4131, normalized size = 8.9 \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+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2),x)

[Out]

2/315/d/b^4*(cos(d*x+c)+1)^2*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*(-1+cos(d*x+c))^2*(4*C*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^2-189*A*cos(d*x+c)^5*b^5+105*A*cos(d*x+c)^5*a*b^4+16*C*cos(d*x+c)^5*a^4*
b-26*C*cos(d*x+c)^5*a^3*b^2+24*C*cos(d*x+c)^5*a^2*b^3+85*C*cos(d*x+c)^5*a*b^4-21*A*cos(d*x+c)^4*a^2*b^3-8*C*co
s(d*x+c)^4*a^4*b-10*C*cos(d*x+c)^4*a^2*b^3+84*A*cos(d*x+c)^3*a*b^4+2*C*cos(d*x+c)^3*a^3*b^2+22*C*cos(d*x+c)^3*
a*b^4+42*A*cos(d*x+c)^6*a^3*b^2-21*A*cos(d*x+c)^6*a^2*b^3-189*A*cos(d*x+c)^6*a*b^4-8*C*cos(d*x+c)^6*a^4*b+24*C
*cos(d*x+c)^6*a^3*b^2-13*C*cos(d*x+c)^6*a^2*b^3-147*C*cos(d*x+c)^6*a*b^4-C*cos(d*x+c)^2*a^2*b^3+40*C*cos(d*x+c
)*a*b^4-42*A*cos(d*x+c)^5*a^3*b^2+42*A*cos(d*x+c)^5*a^2*b^3-16*C*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*C*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))*b^5-189*A*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)*EllipticF((-1+cos(d*x+c))/sin(d
*x+c),((a-b)/(a+b))^(1/2))*b^5+189*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))*b^5-147*C*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^5-16*C*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*C*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^5-189*A*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(c
os(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+189*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))*b^5-147*C*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^5-16*C*cos(d*x+c)^5*a^5-147*C*cos(d*x+c)^5*b^5+126*A*cos(d*x+c)^4*b^5+98*C
*cos(d*x+c)^4*b^5+63*A*cos(d*x+c)^2*b^5+14*C*cos(d*x+c)^2*b^5+16*C*cos(d*x+c)^6*a^5+24*C*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^3-111*C*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^4-16*
C*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)*El
lipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^4*b-24*C*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^2-24*C*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))/(c
os(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*C*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*b^4+42*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^3-1
47*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^4-42*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^3*b^2-42*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+189*A*cos(d*x+c)^4*si
n(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+16*C*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+
4*C*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+24*C*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(co
s(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^3-111*C*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^4-16*C*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-24*C*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^3*b^2
-24*C*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*C*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+42*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^3-147*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^4-42*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^2-42*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+189*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*b^4+16*C*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+35*C*b^5)/(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+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2),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 \sec \left (d x + c\right )^{5} + A \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)^3*(A+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral((C*sec(d*x + c)^5 + A*sec(d*x + c)^3)*sqrt(b*sec(d*x + c) + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sqrt{a + b \sec{\left (c + d x \right )}} \sec ^{3}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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} + A\right )} \sqrt{b \sec \left (d x + c\right ) + a} \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+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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