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\(\int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [483]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [A] (verified)
   Fricas [A] (verification not implemented)
   Sympy [F]
   Maxima [F]
   Giac [F]
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 43, antiderivative size = 193 \[ \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 a (21 A+18 B+16 C) \tan (c+d x)}{45 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (9 B+C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}-\frac {4 (21 A+18 B+16 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (21 A+18 B+16 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 a d} \]

[Out]

2/105*(21*A+18*B+16*C)*(a+a*sec(d*x+c))^(3/2)*tan(d*x+c)/a/d+2/45*a*(21*A+18*B+16*C)*tan(d*x+c)/d/(a+a*sec(d*x
+c))^(1/2)+2/63*a*(9*B+C)*sec(d*x+c)^3*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)-4/315*(21*A+18*B+16*C)*(a+a*sec(d*x
+c))^(1/2)*tan(d*x+c)/d+2/9*C*sec(d*x+c)^3*(a+a*sec(d*x+c))^(1/2)*tan(d*x+c)/d

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {4173, 4101, 3885, 4086, 3877} \[ \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 (21 A+18 B+16 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 a d}-\frac {4 (21 A+18 B+16 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{315 d}+\frac {2 a (21 A+18 B+16 C) \tan (c+d x)}{45 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a (9 B+C) \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt {a \sec (c+d x)+a}}+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{9 d} \]

[In]

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

[Out]

(2*a*(21*A + 18*B + 16*C)*Tan[c + d*x])/(45*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(9*B + C)*Sec[c + d*x]^3*Tan[c
+ d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) - (4*(21*A + 18*B + 16*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*
d) + (2*C*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(9*d) + (2*(21*A + 18*B + 16*C)*(a + a*Sec[c +
 d*x])^(3/2)*Tan[c + d*x])/(105*a*d)

Rule 3877

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

Rule 3885

Int[csc[(e_.) + (f_.)*(x_)]^3*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-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*
(b*(m + 1) - a*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] &&  !LtQ[m, -2^(-1)]

Rule 4086

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

Rule 4101

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

Rule 4173

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/(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 + b*B*(m + n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A
, B, 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 ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {3}{2} a (3 A+2 C)+\frac {1}{2} a (9 B+C) \sec (c+d x)\right ) \, dx}{9 a} \\ & = \frac {2 a (9 B+C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}+\frac {2 C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {1}{21} (21 A+18 B+16 C) \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {2 a (9 B+C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}+\frac {2 C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (21 A+18 B+16 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 a d}+\frac {(2 (21 A+18 B+16 C)) \int \sec (c+d x) \left (\frac {3 a}{2}-a \sec (c+d x)\right ) \sqrt {a+a \sec (c+d x)} \, dx}{105 a} \\ & = \frac {2 a (9 B+C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}-\frac {4 (21 A+18 B+16 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (21 A+18 B+16 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 a d}+\frac {1}{45} (21 A+18 B+16 C) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {2 a (21 A+18 B+16 C) \tan (c+d x)}{45 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (9 B+C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}-\frac {4 (21 A+18 B+16 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (21 A+18 B+16 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.75 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.78 \[ \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a (189 A+162 B+214 C+2 (63 A+99 B+88 C) \cos (c+d x)+11 (21 A+18 B+16 C) \cos (2 (c+d x))+42 A \cos (3 (c+d x))+36 B \cos (3 (c+d x))+32 C \cos (3 (c+d x))+42 A \cos (4 (c+d x))+36 B \cos (4 (c+d x))+32 C \cos (4 (c+d x))) \sec ^4(c+d x) \tan (c+d x)}{315 d \sqrt {a (1+\sec (c+d x))}} \]

[In]

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

[Out]

(a*(189*A + 162*B + 214*C + 2*(63*A + 99*B + 88*C)*Cos[c + d*x] + 11*(21*A + 18*B + 16*C)*Cos[2*(c + d*x)] + 4
2*A*Cos[3*(c + d*x)] + 36*B*Cos[3*(c + d*x)] + 32*C*Cos[3*(c + d*x)] + 42*A*Cos[4*(c + d*x)] + 36*B*Cos[4*(c +
 d*x)] + 32*C*Cos[4*(c + d*x)])*Sec[c + d*x]^4*Tan[c + d*x])/(315*d*Sqrt[a*(1 + Sec[c + d*x])])

Maple [A] (verified)

Time = 0.85 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.84

method result size
default \(\frac {2 \left (168 A \cos \left (d x +c \right )^{4}+144 B \cos \left (d x +c \right )^{4}+128 C \cos \left (d x +c \right )^{4}+84 A \cos \left (d x +c \right )^{3}+72 B \cos \left (d x +c \right )^{3}+64 C \cos \left (d x +c \right )^{3}+63 A \cos \left (d x +c \right )^{2}+54 B \cos \left (d x +c \right )^{2}+48 C \cos \left (d x +c \right )^{2}+45 B \cos \left (d x +c \right )+40 C \cos \left (d x +c \right )+35 C \right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \tan \left (d x +c \right ) \sec \left (d x +c \right )^{3}}{315 d \left (\cos \left (d x +c \right )+1\right )}\) \(163\)
parts \(\frac {2 A \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (8 \sin \left (d x +c \right )+4 \tan \left (d x +c \right )+3 \sec \left (d x +c \right ) \tan \left (d x +c \right )\right )}{15 d \left (\cos \left (d x +c \right )+1\right )}+\frac {2 B \left (16 \cos \left (d x +c \right )^{3}+8 \cos \left (d x +c \right )^{2}+6 \cos \left (d x +c \right )+5\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \tan \left (d x +c \right ) \sec \left (d x +c \right )^{2}}{35 d \left (\cos \left (d x +c \right )+1\right )}+\frac {2 C \left (128 \cos \left (d x +c \right )^{4}+64 \cos \left (d x +c \right )^{3}+48 \cos \left (d x +c \right )^{2}+40 \cos \left (d x +c \right )+35\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \tan \left (d x +c \right ) \sec \left (d x +c \right )^{3}}{315 d \left (\cos \left (d x +c \right )+1\right )}\) \(215\)

[In]

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

[Out]

2/315/d*(168*A*cos(d*x+c)^4+144*B*cos(d*x+c)^4+128*C*cos(d*x+c)^4+84*A*cos(d*x+c)^3+72*B*cos(d*x+c)^3+64*C*cos
(d*x+c)^3+63*A*cos(d*x+c)^2+54*B*cos(d*x+c)^2+48*C*cos(d*x+c)^2+45*B*cos(d*x+c)+40*C*cos(d*x+c)+35*C)*(a*(1+se
c(d*x+c)))^(1/2)/(cos(d*x+c)+1)*tan(d*x+c)*sec(d*x+c)^3

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.68 \[ \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (8 \, {\left (21 \, A + 18 \, B + 16 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (21 \, A + 18 \, B + 16 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (21 \, A + 18 \, B + 16 \, C\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left (9 \, B + 8 \, C\right )} \cos \left (d x + c\right ) + 35 \, C\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\right )}} \]

[In]

integrate(sec(d*x+c)^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+a*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

2/315*(8*(21*A + 18*B + 16*C)*cos(d*x + c)^4 + 4*(21*A + 18*B + 16*C)*cos(d*x + c)^3 + 3*(21*A + 18*B + 16*C)*
cos(d*x + c)^2 + 5*(9*B + 8*C)*cos(d*x + c) + 35*C)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c)/(d*co
s(d*x + c)^5 + d*cos(d*x + c)^4)

Sympy [F]

\[ \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}\, dx \]

[In]

integrate(sec(d*x+c)**3*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)*(a+a*sec(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(a*(sec(c + d*x) + 1))*(A + B*sec(c + d*x) + C*sec(c + d*x)**2)*sec(c + d*x)**3, x)

Maxima [F]

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

[In]

integrate(sec(d*x+c)^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+a*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

8/315*(315*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*(((A + 2*B)*d*cos(2*d*x +
2*c)^4 + (A + 2*B)*d*sin(2*d*x + 2*c)^4 + 4*(A + 2*B)*d*cos(2*d*x + 2*c)^3 + 6*(A + 2*B)*d*cos(2*d*x + 2*c)^2
+ 4*(A + 2*B)*d*cos(2*d*x + 2*c) + 2*((A + 2*B)*d*cos(2*d*x + 2*c)^2 + 2*(A + 2*B)*d*cos(2*d*x + 2*c) + (A + 2
*B)*d)*sin(2*d*x + 2*c)^2 + (A + 2*B)*d)*integrate((((cos(12*d*x + 12*c)*cos(2*d*x + 2*c) + 5*cos(10*d*x + 10*
c)*cos(2*d*x + 2*c) + 10*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 10*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 5*cos(4*d*
x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 5*sin(10*d*x + 10*c)*si
n(2*d*x + 2*c) + 10*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 10*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 5*sin(4*d*x + 4
*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + (cos(2*d*x +
 2*c)*sin(12*d*x + 12*c) + 5*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 10*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 10*c
os(2*d*x + 2*c)*sin(6*d*x + 6*c) + 5*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(12*d*x + 12*c)*sin(2*d*x + 2*c) -
 5*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 10*cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 10*cos(6*d*x + 6*c)*sin(2*d*x
+ 2*c) - 5*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*sin(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*cos(1/2*ar
ctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - ((cos(2*d*x + 2*c)*sin(12*d*x + 12*c) + 5*cos(2*d*x + 2*c)*si
n(10*d*x + 10*c) + 10*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 10*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 5*cos(2*d*x +
 2*c)*sin(4*d*x + 4*c) - cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 5*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 10*cos(
8*d*x + 8*c)*sin(2*d*x + 2*c) - 10*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 5*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*co
s(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - (cos(12*d*x + 12*c)*cos(2*d*x + 2*c) + 5*cos(10*d*x + 10*
c)*cos(2*d*x + 2*c) + 10*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 10*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 5*cos(4*d*
x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 5*sin(10*d*x + 10*c)*si
n(2*d*x + 2*c) + 10*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 10*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 5*sin(4*d*x + 4
*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*sin(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*sin(1/2*arcta
n2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))/(((2*(5*cos(10*d*x + 10*c) + 10*cos(8*d*x + 8*c) + 10*cos(6*d*x +
 6*c) + 5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + cos(12*d*x + 12*c)^2 + 10*(10*cos(8*d*x +
8*c) + 10*cos(6*d*x + 6*c) + 5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 25*cos(10*d*x + 10*c)
^2 + 20*(10*cos(6*d*x + 6*c) + 5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 100*cos(8*d*x + 8*c)^
2 + 20*(5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 100*cos(6*d*x + 6*c)^2 + 25*cos(4*d*x + 4*c)
^2 + 10*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(5*sin(10*d*x + 10*c) + 10*sin(8*d*x + 8*c)
 + 10*sin(6*d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + sin(12*d*x + 12*c)^2 + 10
*(10*sin(8*d*x + 8*c) + 10*sin(6*d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 25*s
in(10*d*x + 10*c)^2 + 20*(10*sin(6*d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*
sin(8*d*x + 8*c)^2 + 20*(5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 100*sin(6*d*x + 6*c)^2 + 25
*sin(4*d*x + 4*c)^2 + 10*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(1/2*arctan2(sin(2*d*x + 2
*c), cos(2*d*x + 2*c) + 1))^2 + (2*(5*cos(10*d*x + 10*c) + 10*cos(8*d*x + 8*c) + 10*cos(6*d*x + 6*c) + 5*cos(4
*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + cos(12*d*x + 12*c)^2 + 10*(10*cos(8*d*x + 8*c) + 10*cos(6
*d*x + 6*c) + 5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 25*cos(10*d*x + 10*c)^2 + 20*(10*cos
(6*d*x + 6*c) + 5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 100*cos(8*d*x + 8*c)^2 + 20*(5*cos(4
*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 100*cos(6*d*x + 6*c)^2 + 25*cos(4*d*x + 4*c)^2 + 10*cos(4*d
*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(5*sin(10*d*x + 10*c) + 10*sin(8*d*x + 8*c) + 10*sin(6*d*x
 + 6*c) + 5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + sin(12*d*x + 12*c)^2 + 10*(10*sin(8*d*x
+ 8*c) + 10*sin(6*d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 25*sin(10*d*x + 10*
c)^2 + 20*(10*sin(6*d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*sin(8*d*x + 8*c
)^2 + 20*(5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 100*sin(6*d*x + 6*c)^2 + 25*sin(4*d*x + 4*
c)^2 + 10*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x
+ 2*c) + 1))^2)*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)), x) + 2*((A + B + 2*
C)*d*cos(2*d*x + 2*c)^4 + (A + B + 2*C)*d*sin(2*d*x + 2*c)^4 + 4*(A + B + 2*C)*d*cos(2*d*x + 2*c)^3 + 6*(A + B
 + 2*C)*d*cos(2*d*x + 2*c)^2 + 4*(A + B + 2*C)*d*cos(2*d*x + 2*c) + 2*((A + B + 2*C)*d*cos(2*d*x + 2*c)^2 + 2*
(A + B + 2*C)*d*cos(2*d*x + 2*c) + (A + B + 2*C)*d)*sin(2*d*x + 2*c)^2 + (A + B + 2*C)*d)*integrate((((cos(12*
d*x + 12*c)*cos(2*d*x + 2*c) + 5*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 10*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) +
10*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 5*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(12*d*x +
 12*c)*sin(2*d*x + 2*c) + 5*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 10*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 10*si
n(6*d*x + 6*c)*sin(2*d*x + 2*c) + 5*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(5/2*arctan2(si
n(2*d*x + 2*c), cos(2*d*x + 2*c))) + (cos(2*d*x + 2*c)*sin(12*d*x + 12*c) + 5*cos(2*d*x + 2*c)*sin(10*d*x + 10
*c) + 10*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 10*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 5*cos(2*d*x + 2*c)*sin(4*d
*x + 4*c) - cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 5*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 10*cos(8*d*x + 8*c)*
sin(2*d*x + 2*c) - 10*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 5*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*sin(5/2*arctan2
(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - ((cos(2*d*x
+ 2*c)*sin(12*d*x + 12*c) + 5*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 10*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 10*
cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 5*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(12*d*x + 12*c)*sin(2*d*x + 2*c)
- 5*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 10*cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 10*cos(6*d*x + 6*c)*sin(2*d*x
 + 2*c) - 5*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*cos(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - (cos(12*
d*x + 12*c)*cos(2*d*x + 2*c) + 5*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 10*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) +
10*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 5*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(12*d*x +
 12*c)*sin(2*d*x + 2*c) + 5*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 10*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 10*si
n(6*d*x + 6*c)*sin(2*d*x + 2*c) + 5*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*sin(5/2*arctan2(si
n(2*d*x + 2*c), cos(2*d*x + 2*c))))*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))/(((2*(5*cos(10*d
*x + 10*c) + 10*cos(8*d*x + 8*c) + 10*cos(6*d*x + 6*c) + 5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*x + 1
2*c) + cos(12*d*x + 12*c)^2 + 10*(10*cos(8*d*x + 8*c) + 10*cos(6*d*x + 6*c) + 5*cos(4*d*x + 4*c) + cos(2*d*x +
 2*c))*cos(10*d*x + 10*c) + 25*cos(10*d*x + 10*c)^2 + 20*(10*cos(6*d*x + 6*c) + 5*cos(4*d*x + 4*c) + cos(2*d*x
 + 2*c))*cos(8*d*x + 8*c) + 100*cos(8*d*x + 8*c)^2 + 20*(5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*
c) + 100*cos(6*d*x + 6*c)^2 + 25*cos(4*d*x + 4*c)^2 + 10*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^
2 + 2*(5*sin(10*d*x + 10*c) + 10*sin(8*d*x + 8*c) + 10*sin(6*d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c
))*sin(12*d*x + 12*c) + sin(12*d*x + 12*c)^2 + 10*(10*sin(8*d*x + 8*c) + 10*sin(6*d*x + 6*c) + 5*sin(4*d*x + 4
*c) + sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 25*sin(10*d*x + 10*c)^2 + 20*(10*sin(6*d*x + 6*c) + 5*sin(4*d*x +
 4*c) + sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*sin(8*d*x + 8*c)^2 + 20*(5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c
))*sin(6*d*x + 6*c) + 100*sin(6*d*x + 6*c)^2 + 25*sin(4*d*x + 4*c)^2 + 10*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) +
sin(2*d*x + 2*c)^2)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2 + (2*(5*cos(10*d*x + 10*c) + 10
*cos(8*d*x + 8*c) + 10*cos(6*d*x + 6*c) + 5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + cos(12*d
*x + 12*c)^2 + 10*(10*cos(8*d*x + 8*c) + 10*cos(6*d*x + 6*c) + 5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(10*d
*x + 10*c) + 25*cos(10*d*x + 10*c)^2 + 20*(10*cos(6*d*x + 6*c) + 5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*
d*x + 8*c) + 100*cos(8*d*x + 8*c)^2 + 20*(5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 100*cos(6*
d*x + 6*c)^2 + 25*cos(4*d*x + 4*c)^2 + 10*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(5*sin(10
*d*x + 10*c) + 10*sin(8*d*x + 8*c) + 10*sin(6*d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x +
 12*c) + sin(12*d*x + 12*c)^2 + 10*(10*sin(8*d*x + 8*c) + 10*sin(6*d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin(2*d*x
 + 2*c))*sin(10*d*x + 10*c) + 25*sin(10*d*x + 10*c)^2 + 20*(10*sin(6*d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin(2*d
*x + 2*c))*sin(8*d*x + 8*c) + 100*sin(8*d*x + 8*c)^2 + 20*(5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x +
6*c) + 100*sin(6*d*x + 6*c)^2 + 25*sin(4*d*x + 4*c)^2 + 10*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c
)^2)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))^2)*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*
cos(2*d*x + 2*c) + 1)^(1/4)), x) + (A*d*cos(2*d*x + 2*c)^4 + A*d*sin(2*d*x + 2*c)^4 + 4*A*d*cos(2*d*x + 2*c)^3
 + 6*A*d*cos(2*d*x + 2*c)^2 + 4*A*d*cos(2*d*x + 2*c) + 2*(A*d*cos(2*d*x + 2*c)^2 + 2*A*d*cos(2*d*x + 2*c) + A*
d)*sin(2*d*x + 2*c)^2 + A*d)*integrate((((cos(12*d*x + 12*c)*cos(2*d*x + 2*c) + 5*cos(10*d*x + 10*c)*cos(2*d*x
 + 2*c) + 10*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 10*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 5*cos(4*d*x + 4*c)*cos
(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 5*sin(10*d*x + 10*c)*sin(2*d*x + 2*
c) + 10*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 10*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 5*sin(4*d*x + 4*c)*sin(2*d*
x + 2*c) + sin(2*d*x + 2*c)^2)*cos(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + (cos(2*d*x + 2*c)*sin(12
*d*x + 12*c) + 5*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 10*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 10*cos(2*d*x + 2
*c)*sin(6*d*x + 6*c) + 5*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 5*cos(10*d*
x + 10*c)*sin(2*d*x + 2*c) - 10*cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 10*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 5*c
os(4*d*x + 4*c)*sin(2*d*x + 2*c))*sin(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*cos(1/2*arctan2(sin(2*
d*x + 2*c), cos(2*d*x + 2*c) + 1)) - ((cos(2*d*x + 2*c)*sin(12*d*x + 12*c) + 5*cos(2*d*x + 2*c)*sin(10*d*x + 1
0*c) + 10*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 10*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 5*cos(2*d*x + 2*c)*sin(4*
d*x + 4*c) - cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 5*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 10*cos(8*d*x + 8*c)
*sin(2*d*x + 2*c) - 10*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 5*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*cos(3/2*arctan
2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - (cos(12*d*x + 12*c)*cos(2*d*x + 2*c) + 5*cos(10*d*x + 10*c)*cos(2*d*x
 + 2*c) + 10*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 10*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 5*cos(4*d*x + 4*c)*cos
(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 5*sin(10*d*x + 10*c)*sin(2*d*x + 2*
c) + 10*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 10*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 5*sin(4*d*x + 4*c)*sin(2*d*
x + 2*c) + sin(2*d*x + 2*c)^2)*sin(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*sin(1/2*arctan2(sin(2*d*x
 + 2*c), cos(2*d*x + 2*c) + 1)))/(((2*(5*cos(10*d*x + 10*c) + 10*cos(8*d*x + 8*c) + 10*cos(6*d*x + 6*c) + 5*co
s(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + cos(12*d*x + 12*c)^2 + 10*(10*cos(8*d*x + 8*c) + 10*co
s(6*d*x + 6*c) + 5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 25*cos(10*d*x + 10*c)^2 + 20*(10*
cos(6*d*x + 6*c) + 5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 100*cos(8*d*x + 8*c)^2 + 20*(5*co
s(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 100*cos(6*d*x + 6*c)^2 + 25*cos(4*d*x + 4*c)^2 + 10*cos(
4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(5*sin(10*d*x + 10*c) + 10*sin(8*d*x + 8*c) + 10*sin(6*
d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + sin(12*d*x + 12*c)^2 + 10*(10*sin(8*d
*x + 8*c) + 10*sin(6*d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 25*sin(10*d*x +
10*c)^2 + 20*(10*sin(6*d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*sin(8*d*x +
8*c)^2 + 20*(5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 100*sin(6*d*x + 6*c)^2 + 25*sin(4*d*x +
 4*c)^2 + 10*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d
*x + 2*c) + 1))^2 + (2*(5*cos(10*d*x + 10*c) + 10*cos(8*d*x + 8*c) + 10*cos(6*d*x + 6*c) + 5*cos(4*d*x + 4*c)
+ cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + cos(12*d*x + 12*c)^2 + 10*(10*cos(8*d*x + 8*c) + 10*cos(6*d*x + 6*c)
+ 5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 25*cos(10*d*x + 10*c)^2 + 20*(10*cos(6*d*x + 6*c
) + 5*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 100*cos(8*d*x + 8*c)^2 + 20*(5*cos(4*d*x + 4*c)
+ cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 100*cos(6*d*x + 6*c)^2 + 25*cos(4*d*x + 4*c)^2 + 10*cos(4*d*x + 4*c)*co
s(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(5*sin(10*d*x + 10*c) + 10*sin(8*d*x + 8*c) + 10*sin(6*d*x + 6*c) + 5*
sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + sin(12*d*x + 12*c)^2 + 10*(10*sin(8*d*x + 8*c) + 10*
sin(6*d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 25*sin(10*d*x + 10*c)^2 + 20*(1
0*sin(6*d*x + 6*c) + 5*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*sin(8*d*x + 8*c)^2 + 20*(5*
sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 100*sin(6*d*x + 6*c)^2 + 25*sin(4*d*x + 4*c)^2 + 10*si
n(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))
^2)*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)), x))*sqrt(a) - (3*(35*A*sin(6*d*
x + 6*c) + 42*(2*A + B + 2*C)*sin(4*d*x + 4*c) + 3*(21*A + 18*B + 16*C)*sin(2*d*x + 2*c))*cos(9/2*arctan2(sin(
2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - (105*A*cos(6*d*x + 6*c) + 126*(2*A + B + 2*C)*cos(4*d*x + 4*c) + 9*(21*
A + 18*B + 16*C)*cos(2*d*x + 2*c) + 42*A + 36*B + 32*C)*sin(9/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1
)))*sqrt(a))/((d*cos(2*d*x + 2*c)^4 + d*sin(2*d*x + 2*c)^4 + 4*d*cos(2*d*x + 2*c)^3 + 6*d*cos(2*d*x + 2*c)^2 +
 2*(d*cos(2*d*x + 2*c)^2 + 2*d*cos(2*d*x + 2*c) + d)*sin(2*d*x + 2*c)^2 + 4*d*cos(2*d*x + 2*c) + d)*(cos(2*d*x
 + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4))

Giac [F]

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

[In]

integrate(sec(d*x+c)^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+a*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 29.31 (sec) , antiderivative size = 600, normalized size of antiderivative = 3.11 \[ \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {A\,8{}\mathrm {i}}{3\,d}-\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (168\,A+144\,B+128\,C\right )\,1{}\mathrm {i}}{315\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}+\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {A\,8{}\mathrm {i}}{9\,d}-\frac {\left (16\,A+16\,B+32\,C\right )\,1{}\mathrm {i}}{9\,d}+\frac {\left (8\,A+16\,B\right )\,1{}\mathrm {i}}{9\,d}\right )-\frac {A\,8{}\mathrm {i}}{9\,d}+\frac {\left (16\,A+16\,B+32\,C\right )\,1{}\mathrm {i}}{9\,d}-\frac {\left (8\,A+16\,B\right )\,1{}\mathrm {i}}{9\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}+\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {A\,8{}\mathrm {i}}{7\,d}-\frac {C\,32{}\mathrm {i}}{7\,d}-\frac {\left (72\,A+144\,B+288\,C\right )\,1{}\mathrm {i}}{63\,d}+{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {C\,32{}\mathrm {i}}{63\,d}-\frac {\left (72\,A+144\,B\right )\,1{}\mathrm {i}}{63\,d}+\frac {\left (72\,A+288\,C\right )\,1{}\mathrm {i}}{63\,d}\right )\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}+\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {\left (168\,A+336\,B\right )\,1{}\mathrm {i}}{105\,d}+\frac {\left (48\,B-32\,C\right )\,1{}\mathrm {i}}{105\,d}\right )-\frac {A\,8{}\mathrm {i}}{5\,d}+\frac {\left (336\,B+672\,C\right )\,1{}\mathrm {i}}{105\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}-\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left (336\,A+288\,B+256\,C\right )\,1{}\mathrm {i}}{315\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )} \]

[In]

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

[Out]

((a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*((A*8i)/(3*d) - (exp(c*1i + d*x*1i)*(168*A + 14
4*B + 128*C)*1i)/(315*d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)) + ((a + a/(exp(- c*1i - d*x*1i)
/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(exp(c*1i + d*x*1i)*((A*8i)/(9*d) - ((16*A + 16*B + 32*C)*1i)/(9*d) + ((8*A
+ 16*B)*1i)/(9*d)) - (A*8i)/(9*d) + ((16*A + 16*B + 32*C)*1i)/(9*d) - ((8*A + 16*B)*1i)/(9*d)))/((exp(c*1i + d
*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)^4) + ((a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*((A*8
i)/(7*d) - (C*32i)/(7*d) - ((72*A + 144*B + 288*C)*1i)/(63*d) + exp(c*1i + d*x*1i)*((C*32i)/(63*d) - ((72*A +
144*B)*1i)/(63*d) + ((72*A + 288*C)*1i)/(63*d))))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)^3) + ((a
+ a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(exp(c*1i + d*x*1i)*(((168*A + 336*B)*1i)/(105*d) +
 ((48*B - 32*C)*1i)/(105*d)) - (A*8i)/(5*d) + ((336*B + 672*C)*1i)/(105*d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*
2i + d*x*2i) + 1)^2) - (exp(c*1i + d*x*1i)*(a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(336*
A + 288*B + 256*C)*1i)/(315*d*(exp(c*1i + d*x*1i) + 1))

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