\(\int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sec ^{\frac {13}{2}}(c+d x)} \, dx\) [557]

Optimal result

Integrand size = 43, antiderivative size = 343 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\frac {4 a^3 (175 A+195 B+221 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {4 a^3 (95 A+105 B+121 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (175 A+195 B+221 C) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (95 A+105 B+121 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)} \]

[Out]

20/9009*a^3*(236*A+273*B+286*C)*sin(d*x+c)/d/sec(d*x+c)^(5/2)+4/585*a^3*(175*A+195*B+221*C)*sin(d*x+c)/d/sec(d
*x+c)^(3/2)+2/13*A*(a+a*sec(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)^(11/2)+2/143*(6*A+13*B)*(a^2+a^2*sec(d*x+c))^2*s
in(d*x+c)/a/d/sec(d*x+c)^(9/2)+2/1287*(145*A+195*B+143*C)*(a^3+a^3*sec(d*x+c))*sin(d*x+c)/d/sec(d*x+c)^(7/2)+4
/231*a^3*(95*A+105*B+121*C)*sin(d*x+c)/d/sec(d*x+c)^(1/2)+4/195*a^3*(175*A+195*B+221*C)*(cos(1/2*d*x+1/2*c)^2)
^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+4/231*a^3*
(95*A+105*B+121*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d
*x+c)^(1/2)*sec(d*x+c)^(1/2)/d

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4171, 4102, 4081, 3872, 3854, 3856, 2719, 2720} \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\frac {4 a^3 (175 A+195 B+221 C) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (145 A+195 B+143 C) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {4 a^3 (95 A+105 B+121 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {4 a^3 (95 A+105 B+121 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {4 a^3 (175 A+195 B+221 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {2 (6 A+13 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{143 a d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac {11}{2}}(c+d x)} \]

[In]

Int[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(13/2),x]

[Out]

(4*a^3*(175*A + 195*B + 221*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (4*a
^3*(95*A + 105*B + 121*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (20*a^3*(
236*A + 273*B + 286*C)*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)) + (4*a^3*(175*A + 195*B + 221*C)*Sin[c + d*x]
)/(585*d*Sec[c + d*x]^(3/2)) + (4*a^3*(95*A + 105*B + 121*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*A*(
a + a*Sec[c + d*x])^3*Sin[c + d*x])/(13*d*Sec[c + d*x]^(11/2)) + (2*(6*A + 13*B)*(a^2 + a^2*Sec[c + d*x])^2*Si
n[c + d*x])/(143*a*d*Sec[c + d*x]^(9/2)) + (2*(145*A + 195*B + 143*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(
1287*d*Sec[c + d*x]^(7/2))

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 3854

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Csc[c + d*x])^(n + 1)/(b*d*n)), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[2*n]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 3872

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 4081

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.)
+ (A_)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Dist[1/(d*n), Int[(d*Csc[e + f*x
])^(n + 1)*Simp[n*(B*a + A*b) + (B*b*n + A*a*(n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B},
x] && NeQ[A*b - a*B, 0] && LeQ[n, -1]

Rule 4102

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

Rule 4171

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \sec (c+d x))^3 \left (\frac {1}{2} a (6 A+13 B)+\frac {1}{2} a (5 A+13 C) \sec (c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx}{13 a} \\ & = \frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {9}{2}}(c+d x)}+\frac {4 \int \frac {(a+a \sec (c+d x))^2 \left (\frac {1}{4} a^2 (145 A+195 B+143 C)+\frac {1}{4} a^2 (85 A+65 B+143 C) \sec (c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx}{143 a} \\ & = \frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {8 \int \frac {(a+a \sec (c+d x)) \left (\frac {5}{4} a^3 (236 A+273 B+286 C)+\frac {1}{4} a^3 (745 A+780 B+1001 C) \sec (c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx}{1287 a} \\ & = \frac {20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {16 \int \frac {-\frac {77}{8} a^4 (175 A+195 B+221 C)-\frac {117}{8} a^4 (95 A+105 B+121 C) \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{9009 a} \\ & = \frac {20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {1}{77} \left (2 a^3 (95 A+105 B+121 C)\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx+\frac {1}{117} \left (2 a^3 (175 A+195 B+221 C)\right ) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (175 A+195 B+221 C) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (95 A+105 B+121 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {1}{231} \left (2 a^3 (95 A+105 B+121 C)\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{195} \left (2 a^3 (175 A+195 B+221 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx \\ & = \frac {20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (175 A+195 B+221 C) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (95 A+105 B+121 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {1}{231} \left (2 a^3 (95 A+105 B+121 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{195} \left (2 a^3 (175 A+195 B+221 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = \frac {4 a^3 (175 A+195 B+221 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {4 a^3 (95 A+105 B+121 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (175 A+195 B+221 C) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (95 A+105 B+121 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 (6 A+13 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (145 A+195 B+143 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 9.13 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.87 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\frac {a^3 e^{-i d x} \sqrt {\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (12480 (95 A+105 B+121 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-4928 i (175 A+195 B+221 C) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+\cos (c+d x) (2587200 i A+2882880 i B+3267264 i C+780 (1811 A+1953 B+2134 C) \sin (c+d x)+77 (7825 A+7800 B+7592 C) \sin (2 (c+d x))+251550 A \sin (3 (c+d x))+221130 B \sin (3 (c+d x))+154440 C \sin (3 (c+d x))+90860 A \sin (4 (c+d x))+60060 B \sin (4 (c+d x))+20020 C \sin (4 (c+d x))+24570 A \sin (5 (c+d x))+8190 B \sin (5 (c+d x))+3465 A \sin (6 (c+d x)))\right )}{720720 d} \]

[In]

Integrate[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(13/2),x]

[Out]

(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(12480*(95*A + 105*B + 121*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c
+ d*x)/2, 2] - (4928*I)*(175*A + 195*B + 221*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F
1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((2587200*I)*A + (2882880*I)*B + (3267264*I)*C + 780*(18
11*A + 1953*B + 2134*C)*Sin[c + d*x] + 77*(7825*A + 7800*B + 7592*C)*Sin[2*(c + d*x)] + 251550*A*Sin[3*(c + d*
x)] + 221130*B*Sin[3*(c + d*x)] + 154440*C*Sin[3*(c + d*x)] + 90860*A*Sin[4*(c + d*x)] + 60060*B*Sin[4*(c + d*
x)] + 20020*C*Sin[4*(c + d*x)] + 24570*A*Sin[5*(c + d*x)] + 8190*B*Sin[5*(c + d*x)] + 3465*A*Sin[6*(c + d*x)])
))/(720720*d*E^(I*d*x))

Maple [A] (verified)

Time = 18.37 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.68

method	result	size
default	\(-\frac {4 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, a^{3} \left (-221760 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (1058400 A +131040 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-2122400 A -567840 B -80080 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (2331040 A +1004640 B +314600 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1535860 A -939120 B -487916 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (633710 A +510510 B +386386 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-121230 A -114660 B -105534 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+18525 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-40425 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+20475 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-45045 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+23595 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-51051 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{45045 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\)	\(576\)
parts	\(\text {Expression too large to display}\)	\(1319\)

[In]

int((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(13/2),x,method=_RETURNVERBOSE)

[Out]

-4/45045*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(-221760*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x
+1/2*c)^14+(1058400*A+131040*B)*sin(1/2*d*x+1/2*c)^12*cos(1/2*d*x+1/2*c)+(-2122400*A-567840*B-80080*C)*sin(1/2
*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(2331040*A+1004640*B+314600*C)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-153
5860*A-939120*B-487916*C)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(633710*A+510510*B+386386*C)*sin(1/2*d*x+1/2
*c)^4*cos(1/2*d*x+1/2*c)+(-121230*A-114660*B-105534*C)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+18525*A*(sin(1/
2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-40425*A*(sin(1/2*
d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+20475*B*(sin(1/2*d*
x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-45045*B*(sin(1/2*d*x+
1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+23595*C*(sin(1/2*d*x+1/
2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-51051*C*(sin(1/2*d*x+1/2*
c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))/(-2*sin(1/2*d*x+1/2*c)^4+s
in(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.13 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.86 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=-\frac {2 \, {\left (195 i \, \sqrt {2} {\left (95 \, A + 105 \, B + 121 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 195 i \, \sqrt {2} {\left (95 \, A + 105 \, B + 121 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 i \, \sqrt {2} {\left (175 \, A + 195 \, B + 221 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 i \, \sqrt {2} {\left (175 \, A + 195 \, B + 221 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (3465 \, A a^{3} \cos \left (d x + c\right )^{6} + 4095 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{5} + 385 \, {\left (50 \, A + 39 \, B + 13 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 585 \, {\left (38 \, A + 42 \, B + 33 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 154 \, {\left (175 \, A + 195 \, B + 221 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 390 \, {\left (95 \, A + 105 \, B + 121 \, C\right )} a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{45045 \, d} \]

[In]

integrate((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(13/2),x, algorithm="fricas")

[Out]

-2/45045*(195*I*sqrt(2)*(95*A + 105*B + 121*C)*a^3*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) -
195*I*sqrt(2)*(95*A + 105*B + 121*C)*a^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 231*I*sq
rt(2)*(175*A + 195*B + 221*C)*a^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x +
c))) + 231*I*sqrt(2)*(175*A + 195*B + 221*C)*a^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x +
c) - I*sin(d*x + c))) - (3465*A*a^3*cos(d*x + c)^6 + 4095*(3*A + B)*a^3*cos(d*x + c)^5 + 385*(50*A + 39*B + 13
*C)*a^3*cos(d*x + c)^4 + 585*(38*A + 42*B + 33*C)*a^3*cos(d*x + c)^3 + 154*(175*A + 195*B + 221*C)*a^3*cos(d*x
+ c)^2 + 390*(95*A + 105*B + 121*C)*a^3*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sec(d*x+c))**3*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/sec(d*x+c)**(13/2),x)

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(13/2),x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {13}{2}}} \,d x } \]

[In]

integrate((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(13/2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^3/sec(d*x + c)^(13/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{13/2}} \,d x \]

[In]

int(((a + a/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/cos(c + d*x))^(13/2),x)

[Out]

int(((a + a/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/cos(c + d*x))^(13/2), x)