\(\int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 (A+C \sec ^2(c+d x)) \, dx\) [222]

Optimal result

Integrand size = 35, antiderivative size = 319 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {4 a^3 (7 A+5 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (143 A+105 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {4 a^3 (7 A+5 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (143 A+105 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {8 a^3 (44 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {4 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac {2 (33 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d} \]

[Out]

4/231*a^3*(143*A+105*C)*sec(d*x+c)^(3/2)*sin(d*x+c)/d+8/385*a^3*(44*A+35*C)*sec(d*x+c)^(5/2)*sin(d*x+c)/d+2/11
*C*sec(d*x+c)^(5/2)*(a+a*sec(d*x+c))^3*sin(d*x+c)/d+4/33*C*sec(d*x+c)^(5/2)*(a^2+a^2*sec(d*x+c))^2*sin(d*x+c)/
a/d+2/231*(33*A+35*C)*sec(d*x+c)^(5/2)*(a^3+a^3*sec(d*x+c))*sin(d*x+c)/d+4/5*a^3*(7*A+5*C)*sin(d*x+c)*sec(d*x+
c)^(1/2)/d-4/5*a^3*(7*A+5*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*(143*A+105*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 = 0.77 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4174, 4103, 4082, 3872, 3853, 3856, 2719, 2720} \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {8 a^3 (44 A+35 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{385 d}+\frac {4 a^3 (143 A+105 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{231 d}+\frac {2 (33 A+35 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{231 d}+\frac {4 a^3 (7 A+5 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (143 A+105 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}-\frac {4 a^3 (7 A+5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {4 C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{33 a d}+\frac {2 C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)^3}{11 d} \]

[In]

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

[Out]

(-4*a^3*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(143*A + 1
05*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(7*A + 5*C)*Sqrt[Sec[c
+ d*x]]*Sin[c + d*x])/(5*d) + (4*a^3*(143*A + 105*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(231*d) + (8*a^3*(44*A
+ 35*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(385*d) + (2*C*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3*Sin[c + d*x]
)/(11*d) + (4*C*Sec[c + d*x]^(5/2)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(33*a*d) + (2*(33*A + 35*C)*Sec[c
+ d*x]^(5/2)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(231*d)

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 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[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 4082

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.
) + (A_)), x_Symbol] :> Simp[(-b)*B*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*(n + 1))), x] + Dist[1/(n + 1), Int[(d
*Csc[e + f*x])^n*Simp[A*a*(n + 1) + B*b*n + (A*b + B*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 4103

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

Rule 4174

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (\frac {1}{2} a (11 A+3 C)+3 a C \sec (c+d x)\right ) \, dx}{11 a} \\ & = \frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {4 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac {4 \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \left (\frac {9}{4} a^2 (11 A+5 C)+\frac {3}{4} a^2 (33 A+35 C) \sec (c+d x)\right ) \, dx}{99 a} \\ & = \frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {4 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac {2 (33 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d}+\frac {8 \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x)) \left (\frac {45}{4} a^3 (11 A+7 C)+\frac {9}{2} a^3 (44 A+35 C) \sec (c+d x)\right ) \, dx}{693 a} \\ & = \frac {8 a^3 (44 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {4 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac {2 (33 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d}+\frac {16 \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {693}{8} a^4 (7 A+5 C)+\frac {45}{8} a^4 (143 A+105 C) \sec (c+d x)\right ) \, dx}{3465 a} \\ & = \frac {8 a^3 (44 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {4 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac {2 (33 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d}+\frac {1}{5} \left (2 a^3 (7 A+5 C)\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{77} \left (2 a^3 (143 A+105 C)\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx \\ & = \frac {4 a^3 (7 A+5 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (143 A+105 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {8 a^3 (44 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {4 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac {2 (33 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d}-\frac {1}{5} \left (2 a^3 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{231} \left (2 a^3 (143 A+105 C)\right ) \int \sqrt {\sec (c+d x)} \, dx \\ & = \frac {4 a^3 (7 A+5 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (143 A+105 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {8 a^3 (44 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {4 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac {2 (33 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d}-\frac {1}{5} \left (2 a^3 (7 A+5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{231} \left (2 a^3 (143 A+105 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = -\frac {4 a^3 (7 A+5 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (143 A+105 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 {4 a^3 (7 A+5 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (143 A+105 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {8 a^3 (44 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {4 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac {2 (33 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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

Time = 12.43 (sec) , antiderivative size = 863, normalized size of antiderivative = 2.71 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {7 A e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{15 \sqrt {2} d (A+2 C+A \cos (2 c+2 d x))}+\frac {C e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{3 \sqrt {2} d (A+2 C+A \cos (2 c+2 d x))}+\frac {13 A \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{21 d (A+2 C+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)}+\frac {5 C \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{11 d (A+2 C+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)}+\frac {\sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \left (\frac {(7 A+5 C) \cos (d x) \csc (c)}{5 d}+\frac {C \sec (c) \sec ^5(c+d x) \sin (d x)}{22 d}+\frac {\sec (c) \sec ^4(c+d x) (3 C \sin (c)+11 C \sin (d x))}{66 d}+\frac {\sec (c) \sec ^3(c+d x) (77 C \sin (c)+33 A \sin (d x)+126 C \sin (d x))}{462 d}+\frac {\sec (c) \sec ^2(c+d x) (165 A \sin (c)+630 C \sin (c)+693 A \sin (d x)+770 C \sin (d x))}{2310 d}+\frac {\sec (c) \sec (c+d x) (693 A \sin (c)+770 C \sin (c)+1430 A \sin (d x)+1050 C \sin (d x))}{2310 d}+\frac {(143 A+105 C) \tan (c)}{231 d}\right )}{(A+2 C+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)} \]

[In]

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

[Out]

(7*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^5*Csc[c]*(-3*S
qrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c
+ d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(15*Sqrt[2]*d*E^(I*d*x)*(A + 2*C
+ A*Cos[2*c + 2*d*x])) + (C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos
[c + d*x]^5*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2,
3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(3*Sqrt[
2]*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])) + (13*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 +
(d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(21*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9
/2)) + (5*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Se
c[c + d*x]^2))/(11*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + (Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c +
d*x])^3*(A + C*Sec[c + d*x]^2)*(((7*A + 5*C)*Cos[d*x]*Csc[c])/(5*d) + (C*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(22*
d) + (Sec[c]*Sec[c + d*x]^4*(3*C*Sin[c] + 11*C*Sin[d*x]))/(66*d) + (Sec[c]*Sec[c + d*x]^3*(77*C*Sin[c] + 33*A*
Sin[d*x] + 126*C*Sin[d*x]))/(462*d) + (Sec[c]*Sec[c + d*x]^2*(165*A*Sin[c] + 630*C*Sin[c] + 693*A*Sin[d*x] + 7
70*C*Sin[d*x]))/(2310*d) + (Sec[c]*Sec[c + d*x]*(693*A*Sin[c] + 770*C*Sin[c] + 1430*A*Sin[d*x] + 1050*C*Sin[d*
x]))/(2310*d) + ((143*A + 105*C)*Tan[c])/(231*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1381\) vs. \(2(335)=670\).

Time = 6.34 (sec) , antiderivative size = 1382, normalized size of antiderivative = 4.33

method	result	size
default	\(\text {Expression too large to display}\)	\(1382\)
parts	\(\text {Expression too large to display}\)	\(1711\)

[In]

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

[Out]

-a^3*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*A/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)
^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2-(sin(1/2
*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2))+6*C*(-1/144*cos(1
/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^5-7/180*cos(1/2*
d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^3-14/15*sin(1/2*d*x
+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)+7/15*(sin(1/2*d*x+1/2*c
)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(co
s(1/2*d*x+1/2*c),2^(1/2))-7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+
1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/
2))))+2*C*(-1/352*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^
2-1/2)^6-9/616*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1
/2)^4-15/154*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2
)^2+15/77*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+
1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+6*A*(-1/6*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+s
in(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c
)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+16*(1
/8*A+3/8*C)*(-1/56*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)
^2-1/2)^4-5/42*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1
/2)^2+5/21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x
+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+16/5*(3/8*A+1/8*C)/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*
d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+1/2*c)^2*(24*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-12*(si
n(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1
/2*c)^4-24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+12*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c
),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^2+8*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2-3*(
sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2))*(-2*sin(1/
2*d*x+1/2*c)^4+sin(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 = 0.11 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.94 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (5 i \, \sqrt {2} {\left (143 \, A + 105 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (143 \, A + 105 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 i \, \sqrt {2} {\left (7 \, A + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 i \, \sqrt {2} {\left (7 \, A + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (462 \, {\left (7 \, A + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 10 \, {\left (143 \, A + 105 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 77 \, {\left (9 \, A + 10 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 15 \, {\left (11 \, A + 42 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 385 \, C a^{3} \cos \left (d x + c\right ) + 105 \, C a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{1155 \, d \cos \left (d x + c\right )^{5}} \]

[In]

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

[Out]

-2/1155*(5*I*sqrt(2)*(143*A + 105*C)*a^3*cos(d*x + c)^5*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x +
c)) - 5*I*sqrt(2)*(143*A + 105*C)*a^3*cos(d*x + c)^5*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))
+ 231*I*sqrt(2)*(7*A + 5*C)*a^3*cos(d*x + c)^5*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c)
+ I*sin(d*x + c))) - 231*I*sqrt(2)*(7*A + 5*C)*a^3*cos(d*x + c)^5*weierstrassZeta(-4, 0, weierstrassPInverse(
-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (462*(7*A + 5*C)*a^3*cos(d*x + c)^5 + 10*(143*A + 105*C)*a^3*cos(d*x
+ c)^4 + 77*(9*A + 10*C)*a^3*cos(d*x + c)^3 + 15*(11*A + 42*C)*a^3*cos(d*x + c)^2 + 385*C*a^3*cos(d*x + c) + 1
05*C*a^3)*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)^5)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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