\(\int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} (A+C \sec ^2(c+d x)) \, dx\) [709]

Optimal result

Integrand size = 35, antiderivative size = 467 \[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 (a-b) \sqrt {a+b} \left (16 a^4 C+6 a^2 b^2 (7 A+4 C)-21 b^4 (9 A+7 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{315 b^5 d}+\frac {2 (a-b) \sqrt {a+b} \left (16 a^3 C+12 a^2 b C+6 a b^2 (7 A+6 C)+21 b^3 (9 A+7 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{315 b^4 d}+\frac {2 a \left (21 A b^2+8 a^2 C+13 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b^3 d}-\frac {2 \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \sec (c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}+\frac {2 a C \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{63 b d}+\frac {2 C \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{9 d} \]

[Out]

2/315*(a-b)*(16*a^4*C+6*a^2*b^2*(7*A+4*C)-21*b^4*(9*A+7*C))*cot(d*x+c)*EllipticE((a+b*sec(d*x+c))^(1/2)/(a+b)^
(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/b^5/d+2/
315*(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(d*x+c)*EllipticF((a+b*sec(d*x+c))^(1/2)
/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/b
^4/d+2/315*a*(21*A*b^2+8*C*a^2+13*C*b^2)*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b^3/d-2/315*(6*C*a^2-7*b^2*(9*A+7*C
))*sec(d*x+c)*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b^2/d+2/63*a*C*sec(d*x+c)^2*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/
b/d+2/9*C*sec(d*x+c)^3*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/d

Rubi [A] (verified)

Time = 1.52 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4182, 4187, 4177, 4167, 4090, 3917, 4089} \[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{315 b^2 d}+\frac {2 a \left (8 a^2 C+21 A b^2+13 b^2 C\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{315 b^3 d}+\frac {2 (a-b) \sqrt {a+b} \left (16 a^4 C+6 a^2 b^2 (7 A+4 C)-21 b^4 (9 A+7 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{315 b^5 d}+\frac {2 (a-b) \sqrt {a+b} \left (16 a^3 C+12 a^2 b C+6 a b^2 (7 A+6 C)+21 b^3 (9 A+7 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{315 b^4 d}+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}+\frac {2 a C \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{63 b d} \]

[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 3917

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

Rule 4089

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 + C
sc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a
*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]

Rule 4090

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 4167

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 4177

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 4182

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 4187

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]

Rubi steps \begin{align*} \text {integral}& = \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 (\arcsin \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) \operatorname {EllipticF}\left (\arcsin \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] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(3696\) vs. \(2(467)=934\).

Time = 29.39 (sec) , antiderivative size = 3696, normalized size of antiderivative = 7.91 \[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \]

[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)*(2*(a + b)*(16*a^4*C + 6*a^2*b^2*(7*A + 4*C)
- 21*b^4*(9*A + 7*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*C - 6*a*b^2*(7*
A + 6*C) + 21*b^3*(9*A + 7*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + C
os[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (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]^2*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])*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(5/2)*((2*a*Sqrt[Cos[(c + d*x)/
2]^2*Sec[c + d*x]]*Sin[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))*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*C - 6*a*b^2*(7*A + 6*C) + 21*b^3*(9*A + 7*C))*Sq
rt[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[T
an[(c + d*x)/2]], (a - b)/(a + b)] + (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]^2*Tan[(c + d*x)/2]))/(315*b^4*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/
2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*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))*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*C - 6*a*b^2*(
7*A + 6*C) + 21*b^3*(9*A + 7*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)] + (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]^2*Tan[(c + d*x)/2]))/(315*b^4*Sqrt[b + a*
Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (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]^4)/2 + ((a + b)*(16*a^4*C +
6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*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])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(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))*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)*(16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Sq
rt[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)*(-16*a^3*C + 12*a^2*b*C - 6*a*b^2*(7*A + 6*C) + 21*
b^3*(9*A + 7*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*(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]^2*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]^2*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))*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)*(-16*a^3*C + 12*a^2*b*C - 6*a*b^2*(7*A + 6*C) + 21*b^3*(9*A + 7*C))*Sqrt[Cos[c + d*x]/(1 + C
os[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)*(16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^
4*(9*A + 7*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 - ((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*(2*(a + b)*(16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A
+ 7*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]))]*Ellipti
cE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(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))*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)] + (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]^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*Sec[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*Sec[c + d*x]])))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(5220\) vs. \(2(429)=858\).

Time = 31.84 (sec) , antiderivative size = 5221, normalized size of antiderivative = 11.18

method	result	size
parts	\(\text {Expression too large to display}\)	\(5221\)
default	\(\text {Expression too large to display}\)	\(5289\)

[In]

int(sec(d*x+c)^3*(A+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F]

\[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{3} \,d x } \]

[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)

Sympy [F]

\[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sqrt {a + b \sec {\left (c + d x \right )}} \sec ^{3}{\left (c + d x \right )}\, dx \]

[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)

Maxima [F]

\[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{3} \,d x } \]

[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]

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

Giac [F]

\[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{3} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}}{{\cos \left (c+d\,x\right )}^3} \,d x \]

[In]

int(((A + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^(1/2))/cos(c + d*x)^3,x)

[Out]

int(((A + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^(1/2))/cos(c + d*x)^3, x)