\(\int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [990]

Optimal result

Integrand size = 43, antiderivative size = 343 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \left (18 a b B+3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (14 a A b+7 a^2 B+5 b^2 B+10 a b C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (18 a b B+3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (14 a A b+7 a^2 B+5 b^2 B+10 a b C\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 \left (9 A b^2+18 a b B+4 a^2 C+7 b^2 C\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 b (9 b B+4 a C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d} \]

[Out]

2/21*(14*A*a*b+7*B*a^2+5*B*b^2+10*C*a*b)*sec(d*x+c)^(3/2)*sin(d*x+c)/d+2/45*(9*A*b^2+18*B*a*b+4*C*a^2+7*C*b^2)
*sec(d*x+c)^(5/2)*sin(d*x+c)/d+2/63*b*(9*B*b+4*C*a)*sec(d*x+c)^(7/2)*sin(d*x+c)/d+2/9*C*sec(d*x+c)^(5/2)*(a+b*
sec(d*x+c))^2*sin(d*x+c)/d+2/15*(18*B*a*b+3*a^2*(5*A+3*C)+b^2*(9*A+7*C))*sin(d*x+c)*sec(d*x+c)^(1/2)/d-2/15*(1
8*B*a*b+3*a^2*(5*A+3*C)+b^2*(9*A+7*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+2/21*(14*A*a*b+7*B*a^2+5*B*b^2+10*C*a*b)*(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 = 0.65 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4181, 4161, 4132, 3853, 3856, 2720, 4131, 2719} \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (4 a^2 C+18 a b B+9 A b^2+7 b^2 C\right )}{45 d}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (7 a^2 B+14 a A b+10 a b C+5 b^2 B\right )}{21 d}+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (3 a^2 (5 A+3 C)+18 a b B+b^2 (9 A+7 C)\right )}{15 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (7 a^2 B+14 a A b+10 a b C+5 b^2 B\right )}{21 d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^2 (5 A+3 C)+18 a b B+b^2 (9 A+7 C)\right )}{15 d}+\frac {2 b (4 a C+9 b B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{63 d}+\frac {2 C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d} \]

[In]

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

[Out]

(-2*(18*a*b*B + 3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c +
d*x]])/(15*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqr
t[Sec[c + d*x]])/(21*d) + (2*(18*a*b*B + 3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])
/(15*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*(9*A*b^2 +
18*a*b*B + 4*a^2*C + 7*b^2*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(45*d) + (2*b*(9*b*B + 4*a*C)*Sec[c + d*x]^(7/
2)*Sin[c + d*x])/(63*d) + (2*C*Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(9*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 4131

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(-C)*Cot
[e + f*x]*((b*Csc[e + f*x])^m/(f*(m + 1))), x] + Dist[(C*m + A*(m + 1))/(m + 1), Int[(b*Csc[e + f*x])^m, x], x
] /; FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] &&  !LeQ[m, -1]

Rule 4132

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(
C_.)), x_Symbol] :> Dist[B/b, Int[(b*Csc[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x
]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, x]

Rule 4161

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

Rule 4181

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2}{9} \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x)) \left (\frac {3}{2} a (3 A+C)+\frac {1}{2} (9 A b+9 a B+7 b C) \sec (c+d x)+\frac {1}{2} (9 b B+4 a C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {2 b (9 b B+4 a C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {4}{63} \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {21}{4} a^2 (3 A+C)+\frac {9}{4} \left (14 a A b+7 a^2 B+5 b^2 B+10 a b C\right ) \sec (c+d x)+\frac {7}{4} \left (9 A b^2+18 a b B+4 a^2 C+7 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {2 b (9 b B+4 a C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {4}{63} \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {21}{4} a^2 (3 A+C)+\frac {7}{4} \left (9 A b^2+18 a b B+4 a^2 C+7 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx+\frac {1}{7} \left (14 a A b+7 a^2 B+5 b^2 B+10 a b C\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx \\ & = \frac {2 \left (14 a A b+7 a^2 B+5 b^2 B+10 a b C\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 \left (9 A b^2+18 a b B+4 a^2 C+7 b^2 C\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 b (9 b B+4 a C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {1}{21} \left (14 a A b+7 a^2 B+5 b^2 B+10 a b C\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{15} \left (18 a b B+3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {2 \left (18 a b B+3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (14 a A b+7 a^2 B+5 b^2 B+10 a b C\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 \left (9 A b^2+18 a b B+4 a^2 C+7 b^2 C\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 b (9 b B+4 a C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {1}{15} \left (-18 a b B-3 a^2 (5 A+3 C)-b^2 (9 A+7 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (\left (14 a A b+7 a^2 B+5 b^2 B+10 a b C\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 \left (14 a A b+7 a^2 B+5 b^2 B+10 a b C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (18 a b B+3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (14 a A b+7 a^2 B+5 b^2 B+10 a b C\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 \left (9 A b^2+18 a b B+4 a^2 C+7 b^2 C\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 b (9 b B+4 a C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {1}{15} \left (\left (-18 a b B-3 a^2 (5 A+3 C)-b^2 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = -\frac {2 \left (18 a b B+3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (14 a A b+7 a^2 B+5 b^2 B+10 a b C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (18 a b B+3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (14 a A b+7 a^2 B+5 b^2 B+10 a b C\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 \left (9 A b^2+18 a b B+4 a^2 C+7 b^2 C\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 b (9 b B+4 a C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 9.97 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.48 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \cos ^4(c+d x) \left (\frac {2 \left (105 a^2 A+63 A b^2+126 a b B+63 a^2 C+49 b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}+2 \left (-70 a A b-35 a^2 B-25 b^2 B-50 a b C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}\right ) (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{105 d (b+a \cos (c+d x))^2 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {4}{15} \left (15 a^2 A+9 A b^2+18 a b B+9 a^2 C+7 b^2 C\right ) \sin (c+d x)+\frac {4}{7} \sec ^3(c+d x) \left (b^2 B \sin (c+d x)+2 a b C \sin (c+d x)\right )+\frac {4}{21} \sec (c+d x) \left (14 a A b \sin (c+d x)+7 a^2 B \sin (c+d x)+5 b^2 B \sin (c+d x)+10 a b C \sin (c+d x)\right )+\frac {4}{45} \sec ^2(c+d x) \left (9 A b^2 \sin (c+d x)+18 a b B \sin (c+d x)+9 a^2 C \sin (c+d x)+7 b^2 C \sin (c+d x)\right )+\frac {4}{9} b^2 C \sec ^3(c+d x) \tan (c+d x)\right )}{d (b+a \cos (c+d x))^2 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {7}{2}}(c+d x)} \]

[In]

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

[Out]

(-2*Cos[c + d*x]^4*((2*(105*a^2*A + 63*A*b^2 + 126*a*b*B + 63*a^2*C + 49*b^2*C)*EllipticE[(c + d*x)/2, 2])/(Sq
rt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(-70*a*A*b - 35*a^2*B - 25*b^2*B - 50*a*b*C)*Sqrt[Cos[c + d*x]]*Ellip
ticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(105*
d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])^2*(A + B*S
ec[c + d*x] + C*Sec[c + d*x]^2)*((4*(15*a^2*A + 9*A*b^2 + 18*a*b*B + 9*a^2*C + 7*b^2*C)*Sin[c + d*x])/15 + (4*
Sec[c + d*x]^3*(b^2*B*Sin[c + d*x] + 2*a*b*C*Sin[c + d*x]))/7 + (4*Sec[c + d*x]*(14*a*A*b*Sin[c + d*x] + 7*a^2
*B*Sin[c + d*x] + 5*b^2*B*Sin[c + d*x] + 10*a*b*C*Sin[c + d*x]))/21 + (4*Sec[c + d*x]^2*(9*A*b^2*Sin[c + d*x]
+ 18*a*b*B*Sin[c + d*x] + 9*a^2*C*Sin[c + d*x] + 7*b^2*C*Sin[c + d*x]))/45 + (4*b^2*C*Sec[c + d*x]^3*Tan[c + d
*x])/9))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1168\) vs. \(2(363)=726\).

Time = 8.93 (sec) , antiderivative size = 1169, normalized size of antiderivative = 3.41

method	result	size
default	\(\text {Expression too large to display}\)	\(1169\)
parts	\(\text {Expression too large to display}\)	\(1422\)

[In]

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

[Out]

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*C*b^2*(-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(cos(1/2*d*x+1/2*c),2^(1/2))-7/1
5*(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*A*a^2/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*sin(1/2*d*x+1/2*c)^2*
cos(1/2*d*x+1/2*c)-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)^(1/2))+2*a*(2*A*b+B*a)*(-1/6*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+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)))+2*b*(B*b+2*C*a)*(-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)))+2/5*(A*b^2+2*B*a*b+C*a^2)/(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*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)^(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*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)^(1/2)*sin(1/2*d*x+1/2*c)^2+8*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-3*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)^(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 = 0.14 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.17 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {15 \, \sqrt {2} {\left (7 i \, B a^{2} + 2 i \, {\left (7 \, A + 5 \, C\right )} a b + 5 i \, B b^{2}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-7 i \, B a^{2} - 2 i \, {\left (7 \, A + 5 \, C\right )} a b - 5 i \, B b^{2}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (3 i \, {\left (5 \, A + 3 \, C\right )} a^{2} + 18 i \, B a b + i \, {\left (9 \, A + 7 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-3 i \, {\left (5 \, A + 3 \, C\right )} a^{2} - 18 i \, B a b - i \, {\left (9 \, A + 7 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (21 \, {\left (3 \, {\left (5 \, A + 3 \, C\right )} a^{2} + 18 \, B a b + {\left (9 \, A + 7 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left (7 \, B a^{2} + 2 \, {\left (7 \, A + 5 \, C\right )} a b + 5 \, B b^{2}\right )} \cos \left (d x + c\right )^{3} + 35 \, C b^{2} + 7 \, {\left (9 \, C a^{2} + 18 \, B a b + {\left (9 \, A + 7 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{2} + 45 \, {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d \cos \left (d x + c\right )^{4}} \]

[In]

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

[Out]

-1/315*(15*sqrt(2)*(7*I*B*a^2 + 2*I*(7*A + 5*C)*a*b + 5*I*B*b^2)*cos(d*x + c)^4*weierstrassPInverse(-4, 0, cos
(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-7*I*B*a^2 - 2*I*(7*A + 5*C)*a*b - 5*I*B*b^2)*cos(d*x + c)^4*weierst
rassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt(2)*(3*I*(5*A + 3*C)*a^2 + 18*I*B*a*b + I*(9*A + 7
*C)*b^2)*cos(d*x + c)^4*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21
*sqrt(2)*(-3*I*(5*A + 3*C)*a^2 - 18*I*B*a*b - I*(9*A + 7*C)*b^2)*cos(d*x + c)^4*weierstrassZeta(-4, 0, weierst
rassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(21*(3*(5*A + 3*C)*a^2 + 18*B*a*b + (9*A + 7*C)*b^2)*c
os(d*x + c)^4 + 15*(7*B*a^2 + 2*(7*A + 5*C)*a*b + 5*B*b^2)*cos(d*x + c)^3 + 35*C*b^2 + 7*(9*C*a^2 + 18*B*a*b +
(9*A + 7*C)*b^2)*cos(d*x + c)^2 + 45*(2*C*a*b + B*b^2)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(
d*x + c)^4)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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