\(\int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} (A+C \sec ^2(c+d x)) \, dx\) [726]

Optimal result

Integrand size = 35, antiderivative size = 534 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 a (a-b) \sqrt {a+b} \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{693 b^4 d}-\frac {2 (a-b) \sqrt {a+b} \left (8 a^4 C+6 a^3 b C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)-6 a b^3 (132 A+101 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{693 b^3 d}+\frac {2 \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{693 b^2 d}+\frac {2 a \left (99 A b^2+8 a^2 C+67 b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{693 b^2 d}+\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^2 d}-\frac {8 a C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d} \]

[Out]

-2/693*a*(a-b)*(8*a^4*C+3*a^2*b^2*(33*A+17*C)+3*b^4*(319*A+247*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
^4/d-2/693*(a-b)*(8*a^4*C+6*a^3*b*C+15*b^4*(11*A+9*C)+3*a^2*b^2*(33*A+19*C)-6*a*b^3*(132*A+101*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^3/d+2/693*a*(99*A*b^2+8*C*a^2+67*C*b^2)*(a+b*sec(d*x+c))^(3/2)*tan(d*x+c)/b^2
/d+2/693*(8*C*a^2+9*b^2*(11*A+9*C))*(a+b*sec(d*x+c))^(5/2)*tan(d*x+c)/b^2/d-8/99*a*C*(a+b*sec(d*x+c))^(7/2)*ta
n(d*x+c)/b^2/d+2/11*C*sec(d*x+c)*(a+b*sec(d*x+c))^(7/2)*tan(d*x+c)/b/d+2/693*(8*a^4*C+15*b^4*(11*A+9*C)+3*a^2*
b^2*(33*A+19*C))*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b^2/d

Rubi [A] (verified)

Time = 1.70 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4178, 4167, 4087, 4090, 3917, 4089} \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{693 b^2 d}+\frac {2 a \left (8 a^2 C+99 A b^2+67 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{693 b^2 d}-\frac {2 a (a-b) \sqrt {a+b} \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{693 b^4 d}+\frac {2 \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{693 b^2 d}-\frac {2 (a-b) \sqrt {a+b} \left (8 a^4 C+6 a^3 b C+3 a^2 b^2 (33 A+19 C)-6 a b^3 (132 A+101 C)+15 b^4 (11 A+9 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{693 b^3 d}-\frac {8 a C \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{99 b^2 d}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{7/2}}{11 b d} \]

[In]

Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]

[Out]

(-2*a*(a - b)*Sqrt[a + b]*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Cot[c + d*x]*EllipticE[A
rcSin[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))])/(693*b^4*d) - (2*(a - b)*Sqrt[a + b]*(8*a^4*C + 6*a^3*b*C + 15*b^4*(11*A + 9*C)
+ 3*a^2*b^2*(33*A + 19*C) - 6*a*b^3*(132*A + 101*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sq
rt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(69
3*b^3*d) + (2*(8*a^4*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])
/(693*b^2*d) + (2*a*(99*A*b^2 + 8*a^2*C + 67*b^2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(693*b^2*d) + (2*
(8*a^2*C + 9*b^2*(11*A + 9*C))*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(693*b^2*d) - (8*a*C*(a + b*Sec[c + d*
x])^(7/2)*Tan[c + d*x])/(99*b^2*d) + (2*C*Sec[c + d*x]*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(11*b*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 4087

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] /;
FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 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 4178

Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + 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] + Dis
t[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*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}+\frac {2 \int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left (a C+\frac {1}{2} b (11 A+9 C) \sec (c+d x)-2 a C \sec ^2(c+d x)\right ) \, dx}{11 b} \\ & = -\frac {8 a C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}+\frac {4 \int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left (-\frac {5}{2} a b C+\frac {1}{4} \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sec (c+d x)\right ) \, dx}{99 b^2} \\ & = \frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^2 d}-\frac {8 a C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}+\frac {8 \int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac {15}{8} b \left (33 A b^2-2 a^2 C+27 b^2 C\right )+\frac {5}{8} a \left (99 A b^2+8 a^2 C+67 b^2 C\right ) \sec (c+d x)\right ) \, dx}{693 b^2} \\ & = \frac {2 a \left (99 A b^2+8 a^2 C+67 b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{693 b^2 d}+\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^2 d}-\frac {8 a C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}+\frac {16 \int \sec (c+d x) \sqrt {a+b \sec (c+d x)} \left (\frac {15}{8} a b \left (132 A b^2-\left (a^2-101 b^2\right ) C\right )+\frac {15}{16} \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\right ) \sec (c+d x)\right ) \, dx}{3465 b^2} \\ & = \frac {2 \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{693 b^2 d}+\frac {2 a \left (99 A b^2+8 a^2 C+67 b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{693 b^2 d}+\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^2 d}-\frac {8 a C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}+\frac {32 \int \frac {\sec (c+d x) \left (\frac {15}{32} b \left (2 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (297 A+221 C)\right )+\frac {15}{32} a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{10395 b^2} \\ & = \frac {2 \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{693 b^2 d}+\frac {2 a \left (99 A b^2+8 a^2 C+67 b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{693 b^2 d}+\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^2 d}-\frac {8 a C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d}-\frac {\left ((a-b) \left (8 a^4 C+6 a^3 b C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)-6 a b^3 (132 A+101 C)\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{693 b^2}+\frac {\left (a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right )\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx}{693 b^2} \\ & = -\frac {2 a (a-b) \sqrt {a+b} \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 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}}}{693 b^4 d}-\frac {2 (a-b) \sqrt {a+b} \left (8 a^4 C+6 a^3 b C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)-6 a b^3 (132 A+101 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}}}{693 b^3 d}+\frac {2 \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{693 b^2 d}+\frac {2 a \left (99 A b^2+8 a^2 C+67 b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{693 b^2 d}+\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^2 d}-\frac {8 a C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{99 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{11 b d} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(3989\) vs. \(2(534)=1068\).

Time = 31.23 (sec) , antiderivative size = 3989, normalized size of antiderivative = 7.47 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \]

[In]

Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]

[Out]

(Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*((4*a*(99*a^2*A*b^2 + 957*A*b^4 + 8*a^4*C +
51*a^2*b^2*C + 741*b^4*C)*Sin[c + d*x])/(693*b^3) + (4*Sec[c + d*x]^3*(99*A*b^2*Sin[c + d*x] + 113*a^2*C*Sin[c
+ d*x] + 81*b^2*C*Sin[c + d*x]))/693 + (4*Sec[c + d*x]^2*(297*a*A*b^2*Sin[c + d*x] + 3*a^3*C*Sin[c + d*x] + 2
29*a*b^2*C*Sin[c + d*x]))/(693*b) + (4*Sec[c + d*x]*(297*a^2*A*b^2*Sin[c + d*x] + 165*A*b^4*Sin[c + d*x] - 4*a
^4*C*Sin[c + d*x] + 205*a^2*b^2*C*Sin[c + d*x] + 135*b^4*C*Sin[c + d*x]))/(693*b^2) + (92*a*b*C*Sec[c + d*x]^3
*Tan[c + d*x])/99 + (4*b^2*C*Sec[c + d*x]^4*Tan[c + d*x])/11))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c
+ 2*d*x])) - (4*((-2*a^3*A)/(7*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (58*a*A*b^2)/(21*Sqrt[b + a*Cos[
c + d*x]]*Sqrt[Sec[c + d*x]]) - (34*a^3*C)/(231*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (16*a^5*C)/(693
*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (494*a*b^2*C)/(231*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d
*x]]) - (2*a^4*A*Sqrt[Sec[c + d*x]])/(7*b*Sqrt[b + a*Cos[c + d*x]]) - (4*a^2*A*b*Sqrt[Sec[c + d*x]])/(21*Sqrt[
b + a*Cos[c + d*x]]) + (10*A*b^3*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (16*a^6*C*Sqrt[Sec[c + d*
x]])/(693*b^3*Sqrt[b + a*Cos[c + d*x]]) - (14*a^4*C*Sqrt[Sec[c + d*x]])/(99*b*Sqrt[b + a*Cos[c + d*x]]) - (52*
a^2*b*C*Sqrt[Sec[c + d*x]])/(231*Sqrt[b + a*Cos[c + d*x]]) + (30*b^3*C*Sqrt[Sec[c + d*x]])/(77*Sqrt[b + a*Cos[
c + d*x]]) - (2*a^4*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(7*b*Sqrt[b + a*Cos[c + d*x]]) - (58*a^2*A*b*Cos[2*
(c + d*x)]*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (16*a^6*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/
(693*b^3*Sqrt[b + a*Cos[c + d*x]]) - (34*a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(231*b*Sqrt[b + a*Cos[c +
d*x]]) - (494*a^2*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(231*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/
2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*(2*a*(a + b)*(8*a^4*C + 3*a^2*b^2*(33*A +
17*C) + 3*b^4*(319*A + 247*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)*(8*a^4*C - 6*a^3*b*C + 15*b
^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C) + 6*a*b^3*(132*A + 101*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqr
t[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*
(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]
^2*Tan[(c + d*x)/2]))/(693*b^3*d*(b + a*Cos[c + d*x])^3*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2
]*Sec[c + d*x]^(9/2)*((-2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*a*(a + b)*(8*a^4*C + 3*a^2*b
^2*(33*A + 17*C) + 3*b^4*(319*A + 247*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)*(8*a^4*C - 6*a^3*
b*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C) + 6*a*b^3*(132*A + 101*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)] + a*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c
+ d*x)/2]^2*Tan[(c + d*x)/2]))/(693*b^3*(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*(a + b)*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 2
47*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)*(8*a^4*C - 6*a^3*b*C + 15*b^4*(11*A + 9*C) + 3*a^2*b
^2*(33*A + 19*C) + 6*a*b^3*(132*A + 101*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)] + a*(8*a^4*C + 3*a^2*b^2*(33*
A + 17*C) + 3*b^4*(319*A + 247*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(69
3*b^3*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]]*((a*(8*a^4
*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2
+ (a*(a + b)*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*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)*(8*a^4*C -
6*a^3*b*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C) + 6*a*b^3*(132*A + 101*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*(a + b)
*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[A
rcSin[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^4*C - 6*a^3*b*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C) + 6*a*b^3*(132*A + 101*C))*Sq
rt[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^2*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C)
)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - a*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4
*(319*A + 247*C))*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + a*(8*a^4*C + 3*a^2*b
^2*(33*A + 17*C) + 3*b^4*(319*A + 247*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)*(8*a^4*C - 6*a^3*b*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C) + 6*a*b^3*(132*A + 101*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 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (a*(a + b)*(8*a^4*C + 3
*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*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 - Ta
n[(c + d*x)/2]^2]))/(693*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*(2*a*(a + b)*(8*a^4*C + 3
*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*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)*(8*a^4*C -
6*a^3*b*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C) + 6*a*b^3*(132*A + 101*C))*Sqrt[Cos[c + d*x]/(1 + Co
s[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)] + a*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*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*S
ec[c + d*x]*Tan[c + d*x]))/(693*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] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(5985\) vs. \(2(492)=984\).

Time = 102.83 (sec) , antiderivative size = 5986, normalized size of antiderivative = 11.21

method	result	size
parts	\(\text {Expression too large to display}\)	\(5986\)
default	\(\text {Expression too large to display}\)	\(6071\)

[In]

int(sec(d*x+c)^2*(a+b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F]

\[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(sec(d*x+c)^2*(a+b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

integral((C*b^2*sec(d*x + c)^6 + 2*C*a*b*sec(d*x + c)^5 + 2*A*a*b*sec(d*x + c)^3 + A*a^2*sec(d*x + c)^2 + (C*a
^2 + A*b^2)*sec(d*x + c)^4)*sqrt(b*sec(d*x + c) + a), x)

Sympy [F(-1)]

Timed out. \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**2*(a+b*sec(d*x+c))**(5/2)*(A+C*sec(d*x+c)**2),x)

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)^2*(a+b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(sec(d*x+c)^2*(a+b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + A)*(b*sec(d*x + c) + a)^(5/2)*sec(d*x + c)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^2} \,d x \]

[In]

int(((A + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^(5/2))/cos(c + d*x)^2,x)

[Out]

int(((A + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^(5/2))/cos(c + d*x)^2, x)