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

Optimal. Leaf size=485 \[ -\frac{2 (a-b) \sqrt{a+b} \left (12 a^2 b (2 B-C)-16 a^3 C+18 a b^2 (B-2 C)+3 b^3 (25 B-49 C)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{315 b^4 d}+\frac{2 \left (-6 a^2 C+9 a b B+49 b^2 C\right ) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^2 d}-\frac{2 \left (12 a^2 b B-8 a^3 C-13 a b^2 C-75 b^3 B\right ) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \left (-24 a^2 b^2 C+24 a^3 b B-16 a^4 C+57 a b^3 B+147 b^4 C\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{315 b^5 d}+\frac{2 (a C+9 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{63 b d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{9 d} \]

[Out]

(-2*(a - b)*Sqrt[a + b]*(24*a^3*b*B + 57*a*b^3*B - 16*a^4*C - 24*a^2*b^2*C + 147*b^4*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^5*d) - (2*(a - b)*Sqrt[a + b]*(3*b^3*(25*B - 49*C) + 18*a*b^2*(B - 2*C)
+ 12*a^2*b*(2*B - C) - 16*a^3*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*(12*a
^2*b*B - 75*b^3*B - 8*a^3*C - 13*a*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^3*d) + (2*(9*a*b*B - 6
*a^2*C + 49*b^2*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) + (2*(9*b*B + 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 = 1.45366, antiderivative size = 485, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {4072, 4031, 4102, 4092, 4082, 4005, 3832, 4004} \[ \frac{2 \left (-6 a^2 C+9 a b B+49 b^2 C\right ) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^2 d}-\frac{2 \left (12 a^2 b B-8 a^3 C-13 a b^2 C-75 b^3 B\right ) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \left (12 a^2 b (2 B-C)-16 a^3 C+18 a b^2 (B-2 C)+3 b^3 (25 B-49 C)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{315 b^4 d}-\frac{2 (a-b) \sqrt{a+b} \left (-24 a^2 b^2 C+24 a^3 b B-16 a^4 C+57 a b^3 B+147 b^4 C\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{315 b^5 d}+\frac{2 (a C+9 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{63 b d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{9 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a - b)*Sqrt[a + b]*(24*a^3*b*B + 57*a*b^3*B - 16*a^4*C - 24*a^2*b^2*C + 147*b^4*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^5*d) - (2*(a - b)*Sqrt[a + b]*(3*b^3*(25*B - 49*C) + 18*a*b^2*(B - 2*C)
+ 12*a^2*b*(2*B - C) - 16*a^3*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*(12*a
^2*b*B - 75*b^3*B - 8*a^3*C - 13*a*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^3*d) + (2*(9*a*b*B - 6
*a^2*C + 49*b^2*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) + (2*(9*b*B + 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 4072

Int[((a_.) + csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(
x_)]^2*(C_.))*((c_.) + csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.), x_Symbol] :> Dist[1/b^2, Int[(a + b*Csc[e + f*x])
^(m + 1)*(c + d*Csc[e + f*x])^n*(b*B - a*C + b*C*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m,
n}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]

Rule 4031

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*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1))/(f*(m + n
)), x] + Dist[d/(m + n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n - 1)*Simp[a*B*(n - 1) + (b*B*(m
+ n - 1) + a*A*(m + n))*Csc[e + f*x] + (a*B*m + A*b*(m + n))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e,
f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[0, m, 1] && GtQ[n, 0]

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 (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \sec ^4(c+d x) \sqrt{a+b \sec (c+d x)} (B+C \sec (c+d x)) \, 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 (3 a C+\frac{1}{2} (9 a B+7 b C) \sec (c+d x)+\frac{1}{2} (9 b B+a C) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{2 (9 b B+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 (9 b B+a C)+\frac{1}{4} b (45 b B+47 a C) \sec (c+d x)+\frac{1}{4} \left (9 a b B-6 a^2 C+49 b^2 C\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{63 b}\\ &=\frac{2 \left (9 a b B-6 a^2 C+49 b^2 C\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}+\frac{2 (9 b B+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 (9 a b B-6 a^2 C+49 b^2 C\right )+\frac{1}{8} b \left (207 a b B+2 a^2 C+147 b^2 C\right ) \sec (c+d x)-\frac{3}{8} \left (12 a^2 b B-75 b^3 B-8 a^3 C-13 a b^2 C\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{315 b^2}\\ &=-\frac{2 \left (12 a^2 b B-75 b^3 B-8 a^3 C-13 a b^2 C\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^3 d}+\frac{2 \left (9 a b B-6 a^2 C+49 b^2 C\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}+\frac{2 (9 b B+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} b \left (6 a^2 b B+75 b^3 B-4 a^3 C+111 a b^2 C\right )+\frac{3}{16} \left (24 a^3 b B+57 a b^3 B-16 a^4 C-24 a^2 b^2 C+147 b^4 C\right ) \sec (c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{945 b^3}\\ &=-\frac{2 \left (12 a^2 b B-75 b^3 B-8 a^3 C-13 a b^2 C\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^3 d}+\frac{2 \left (9 a b B-6 a^2 C+49 b^2 C\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}+\frac{2 (9 b B+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 (3 b^3 (25 B-49 C)+18 a b^2 (B-2 C)+12 a^2 b (2 B-C)-16 a^3 C\right )\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{315 b^3}+\frac{\left (24 a^3 b B+57 a b^3 B-16 a^4 C-24 a^2 b^2 C+147 b^4 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 (24 a^3 b B+57 a b^3 B-16 a^4 C-24 a^2 b^2 C+147 b^4 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 (3 b^3 (25 B-49 C)+18 a b^2 (B-2 C)+12 a^2 b (2 B-C)-16 a^3 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 \left (12 a^2 b B-75 b^3 B-8 a^3 C-13 a b^2 C\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^3 d}+\frac{2 \left (9 a b B-6 a^2 C+49 b^2 C\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}+\frac{2 (9 b B+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 = 25.5508, size = 3734, normalized size = 7.7 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[a + b*Sec[c + d*x]]*((2*(24*a^3*b*B + 57*a*b^3*B - 16*a^4*C - 24*a^2*b^2*C + 147*b^4*C)*Sin[c + d*x])/(3
15*b^4) + (2*Sec[c + d*x]^3*(9*b*B*Sin[c + d*x] + a*C*Sin[c + d*x]))/(63*b) + (2*Sec[c + d*x]^2*(9*a*b*B*Sin[c
+ d*x] - 6*a^2*C*Sin[c + d*x] + 49*b^2*C*Sin[c + d*x]))/(315*b^2) + (2*Sec[c + d*x]*(-12*a^2*b*B*Sin[c + d*x]
+ 75*b^3*B*Sin[c + d*x] + 8*a^3*C*Sin[c + d*x] + 13*a*b^2*C*Sin[c + d*x]))/(315*b^3) + (2*C*Sec[c + d*x]^3*Ta
n[c + d*x])/9))/d + (2*((-19*a*B)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^3*B)/(105*b^2*Sqrt[
b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*a^4*C)/(315*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (
8*a^2*C)/(105*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (7*b*C)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c
+ d*x]]) - (8*a^4*B*Sqrt[Sec[c + d*x]])/(105*b^3*Sqrt[b + a*Cos[c + d*x]]) - (17*a^2*B*Sqrt[Sec[c + d*x]])/(1
05*b*Sqrt[b + a*Cos[c + d*x]]) + (5*b*B*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (4*a*C*Sqrt[Sec[c
+ d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]) + (16*a^5*C*Sqrt[Sec[c + d*x]])/(315*b^4*Sqrt[b + a*Cos[c + d*x]]) + (4
*a^3*C*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^4*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(
105*b^3*Sqrt[b + a*Cos[c + d*x]]) - (19*a^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*b*Sqrt[b + a*Cos[c + d
*x]]) - (7*a*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) + (16*a^5*C*Cos[2*(c + d*x)]
*Sqrt[Sec[c + d*x]])/(315*b^4*Sqrt[b + a*Cos[c + d*x]]) + (8*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]]*(2*(a + b)*(-24*a
^3*b*B - 57*a*b^3*B + 16*a^4*C + 24*a^2*b^2*C - 147*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*C
os[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)*
(-16*a^3*C + 12*a^2*b*(2*B + C) - 18*a*b^2*(B + 2*C) + 3*b^3*(25*B + 49*C))*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
)] + (-24*a^3*b*B - 57*a*b^3*B + 16*a^4*C + 24*a^2*b^2*C - 147*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c
+ d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^4*d*(b + a*Cos[c + d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Sec[c + d*x]]*((
a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-24*a^3*b*B - 57*a*b^3*B + 16*a^4*C + 24*a^2*
b^2*C - 147*b^4*C)*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)*(-16*a^3*C + 12*a^2*b*(2*B + C) - 18*a*b
^2*(B + 2*C) + 3*b^3*(25*B + 49*C))*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)] + (-24*a^3*b*B - 57*a*b^3*B + 16*a^4*
C + 24*a^2*b^2*C - 147*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^4
*(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)*(-24*a^3*b*B - 57*a*b^3*B + 16*a^4*C + 24*a^2*b^2*C - 147*b^4*C)*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)*(-16*a^3*C + 12*a^2*b*(2*B + C) - 18*a*b^2*(B + 2*C) + 3*b^3*(25*B + 49*C))*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)] + (-24*a^3*b*B - 57*a*b^3*B + 16*a^4*C + 24*a^2*b^2*C - 147*b^4*C)*Cos[c + d*x]*(b + a*C
os[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^4*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]]*(((-24*a^3*b*B - 57*a*b^3*B + 16*a^4*C + 24*a^2*b^2*C - 147*b^4*C)
*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-24*a^3*b*B - 57*a*b^3*B + 16*a^4*C + 24*
a^2*b^2*C - 147*b^4*C)*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])))/Sq
rt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(a + b)*(-16*a^3*C + 12*a^2*b*(2*B + C) - 18*a*b^2*(B + 2*C) + 3*b^3*
(25*B + 49*C))*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)*(-24*a^3*b*B - 57*a*b^3*B + 16*a^4*C + 24*a^2*b^2*C - 147*b^4*C)*Sqrt[C
os[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*Co
s[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(a + b)*(-16*a^3*C + 12*a^2*b*(2*B + C) - 18*a*b^2*(B + 2*C) +
3*b^3*(25*B + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[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]))] - a*(-24*a^3*b*B - 57*a*b^3*B + 16*a^4*C
+ 24*a^2*b^2*C - 147*b^4*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-24*a^3*b*B - 57*
a*b^3*B + 16*a^4*C + 24*a^2*b^2*C - 147*b^4*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d
*x)/2] + (-24*a^3*b*B - 57*a*b^3*B + 16*a^4*C + 24*a^2*b^2*C - 147*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Se
c[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(a + b)*(-16*a^3*C + 12*a^2*b*(2*B + C) - 18*a*b^2*(B + 2*C) + 3*b^3*
(25*B + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*S
ec[(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)*(-
24*a^3*b*B - 57*a*b^3*B + 16*a^4*C + 24*a^2*b^2*C - 147*b^4*C)*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^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((2*(a + b)*(
-24*a^3*b*B - 57*a*b^3*B + 16*a^4*C + 24*a^2*b^2*C - 147*b^4*C)*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)*(-16*a^3*C + 12*a^2*b*(2*B + C) - 18*a*b^2*(B + 2*C) + 3*b^3*(25*B + 49*C))*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)] + (-24*a^3*b*B - 57*a*b^3*B + 16*a^4*C + 24*a^2*b^2*C - 147*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*S
ec[(c + d*x)/2]^2*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*Se
c[c + d*x]*Tan[c + d*x]))/(315*b^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*S
ec[c + d*x]])))

________________________________________________________________________________________

Maple [B]  time = 1.737, size = 4395, normalized size = 9.1 \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*(B*sec(d*x+c)+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*(-75*B*cos(d*x+c)^5*b^5+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)*Ell
ipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b^2-24*B*sin(d*x+c)*cos(d*x+c)^5*(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+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^4-8*C*cos(d*x+c)^4*a^4*b-10*C*cos(d*x+c)^4*a^2*b^3+2*C*cos(d*x+c)^3*a^3*b^2+22*C*cos(d*x+c)^3*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+24*B*sin(d*x+c)*cos(d*x+c)^5*(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*B*sin(d*x
+c)*cos(d*x+c)^5*(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+57*B*sin(d*x+c)*cos(d*x+c)^5*(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+57*B*sin(d*x+c)*cos(d*x+c)^5*(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^4-6*B*sin(d*x+c)*cos(d*x+c)^5*(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^3-57*B*sin(d*x+c)*cos(d*x+c)^5*(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+24*B*sin(d*x
+c)*cos(d*x+c)^4*(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*B*sin(d*x+c)*cos(d*x+c)^4*(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^2-16*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^5-147*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*s
in(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^5-14
7*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+98*
C*cos(d*x+c)^4*b^5+14*C*cos(d*x+c)^2*b^5+16*C*cos(d*x+c)^6*a^5+57*B*sin(d*x+c)*cos(d*x+c)^4*(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+57*B*sin(d*x+c)*cos(d*x+c)^4*(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^4-24*B*sin(d*x+c)*cos(d*x+c
)^4*(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^2-6*B*sin(d*x+c)*cos(d*x+c)^4*(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^3-57
*B*sin(d*x+c)*cos(d*x+c)^4*(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)*E
llipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^4+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)*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)^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))/(cos(d*x+c)+1))^(1/2)*E
llipticE((-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)/(co
s(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)/(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
11*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+
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+30*B*cos(d*x+c)^3*b^5+45*B*cos(d*x+c
)*b^5+12*B*cos(d*x+c)^4*a^3*b^2+78*B*cos(d*x+c)^4*a*b^4-3*B*cos(d*x+c)^3*a^2*b^3+54*B*cos(d*x+c)^2*a*b^4-24*B*
cos(d*x+c)^6*a^4*b+12*B*cos(d*x+c)^6*a^3*b^2-57*B*cos(d*x+c)^6*a^2*b^3-75*B*cos(d*x+c)^6*a*b^4+24*B*cos(d*x+c)
^5*a^4*b-24*B*cos(d*x+c)^5*a^3*b^2+60*B*cos(d*x+c)^5*a^2*b^3-57*B*cos(d*x+c)^5*a*b^4-75*B*sin(d*x+c)*cos(d*x+c
)^4*(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^5-75*B*sin(d*x+c)*cos(d*x+c)^5*(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)/(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*(B*sec(d*x+c)+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} + B \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*(B*sec(d*x+c)+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 + B*sec(d*x + c)^4)*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 (B + C \sec{\left (c + d x \right )}\right ) \sqrt{a + b \sec{\left (c + d x \right )}} \sec ^{4}{\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*(B*sec(d*x+c)+C*sec(d*x+c)**2)*(a+b*sec(d*x+c))**(1/2),x)

[Out]

Integral((B + C*sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sec(c + d*x)**4, 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} + B \sec \left (d x + c\right )\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*(B*sec(d*x+c)+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 + B*sec(d*x + c))*sqrt(b*sec(d*x + c) + a)*sec(d*x + c)^3, x)