\(\int \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} (B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [356]

Optimal result

Integrand size = 42, antiderivative size = 230 \[ \int \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {32 a (11 B+10 C) \tan (c+d x)}{495 d \sqrt {a+a \sec (c+d x)}}+\frac {16 a (11 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (11 B+10 C) \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a C \sec ^5(c+d x) \tan (c+d x)}{11 d \sqrt {a+a \sec (c+d x)}}-\frac {64 (11 B+10 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac {32 (11 B+10 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 a d} \]

[Out]

32/1155*(11*B+10*C)*(a+a*sec(d*x+c))^(3/2)*tan(d*x+c)/a/d+32/495*a*(11*B+10*C)*tan(d*x+c)/d/(a+a*sec(d*x+c))^(
1/2)+16/693*a*(11*B+10*C)*sec(d*x+c)^3*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+2/99*a*(11*B+10*C)*sec(d*x+c)^4*tan
(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+2/11*a*C*sec(d*x+c)^5*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)-64/3465*(11*B+10*C)
*(a+a*sec(d*x+c))^(1/2)*tan(d*x+c)/d

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4157, 4101, 3888, 3885, 4086, 3877} \[ \int \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 a (11 B+10 C) \tan (c+d x) \sec ^4(c+d x)}{99 d \sqrt {a \sec (c+d x)+a}}+\frac {16 a (11 B+10 C) \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt {a \sec (c+d x)+a}}+\frac {32 (11 B+10 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{1155 a d}-\frac {64 (11 B+10 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{3465 d}+\frac {32 a (11 B+10 C) \tan (c+d x)}{495 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a C \tan (c+d x) \sec ^5(c+d x)}{11 d \sqrt {a \sec (c+d x)+a}} \]

[In]

Int[Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(32*a*(11*B + 10*C)*Tan[c + d*x])/(495*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a*(11*B + 10*C)*Sec[c + d*x]^3*Tan[c
+ d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(11*B + 10*C)*Sec[c + d*x]^4*Tan[c + d*x])/(99*d*Sqrt[a + a*Se
c[c + d*x]]) + (2*a*C*Sec[c + d*x]^5*Tan[c + d*x])/(11*d*Sqrt[a + a*Sec[c + d*x]]) - (64*(11*B + 10*C)*Sqrt[a
+ a*Sec[c + d*x]]*Tan[c + d*x])/(3465*d) + (32*(11*B + 10*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(1155*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 3888

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

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 4157

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

Rubi steps \begin{align*} \text {integral}& = \int \sec ^5(c+d x) \sqrt {a+a \sec (c+d x)} (B+C \sec (c+d x)) \, dx \\ & = \frac {2 a C \sec ^5(c+d x) \tan (c+d x)}{11 d \sqrt {a+a \sec (c+d x)}}+\frac {1}{11} (11 B+10 C) \int \sec ^5(c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {2 a (11 B+10 C) \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a C \sec ^5(c+d x) \tan (c+d x)}{11 d \sqrt {a+a \sec (c+d x)}}+\frac {1}{99} (8 (11 B+10 C)) \int \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {16 a (11 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (11 B+10 C) \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a C \sec ^5(c+d x) \tan (c+d x)}{11 d \sqrt {a+a \sec (c+d x)}}+\frac {1}{231} (16 (11 B+10 C)) \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {16 a (11 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (11 B+10 C) \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a C \sec ^5(c+d x) \tan (c+d x)}{11 d \sqrt {a+a \sec (c+d x)}}+\frac {32 (11 B+10 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 a d}+\frac {(32 (11 B+10 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}{1155 a} \\ & = \frac {16 a (11 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (11 B+10 C) \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a C \sec ^5(c+d x) \tan (c+d x)}{11 d \sqrt {a+a \sec (c+d x)}}-\frac {64 (11 B+10 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac {32 (11 B+10 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 a d}+\frac {1}{495} (16 (11 B+10 C)) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {32 a (11 B+10 C) \tan (c+d x)}{495 d \sqrt {a+a \sec (c+d x)}}+\frac {16 a (11 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (11 B+10 C) \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a C \sec ^5(c+d x) \tan (c+d x)}{11 d \sqrt {a+a \sec (c+d x)}}-\frac {64 (11 B+10 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac {32 (11 B+10 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.29 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.50 \[ \int \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 a \left (128 (11 B+10 C)+64 (11 B+10 C) \sec (c+d x)+48 (11 B+10 C) \sec ^2(c+d x)+40 (11 B+10 C) \sec ^3(c+d x)+35 (11 B+10 C) \sec ^4(c+d x)+315 C \sec ^5(c+d x)\right ) \tan (c+d x)}{3465 d \sqrt {a (1+\sec (c+d x))}} \]

[In]

Integrate[Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(2*a*(128*(11*B + 10*C) + 64*(11*B + 10*C)*Sec[c + d*x] + 48*(11*B + 10*C)*Sec[c + d*x]^2 + 40*(11*B + 10*C)*S
ec[c + d*x]^3 + 35*(11*B + 10*C)*Sec[c + d*x]^4 + 315*C*Sec[c + d*x]^5)*Tan[c + d*x])/(3465*d*Sqrt[a*(1 + Sec[
c + d*x])])

Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.66

method	result	size
default	\(\frac {2 \left (1408 B \cos \left (d x +c \right )^{5}+1280 C \cos \left (d x +c \right )^{5}+704 B \cos \left (d x +c \right )^{4}+640 C \cos \left (d x +c \right )^{4}+528 B \cos \left (d x +c \right )^{3}+480 C \cos \left (d x +c \right )^{3}+440 B \cos \left (d x +c \right )^{2}+400 C \cos \left (d x +c \right )^{2}+385 B \cos \left (d x +c \right )+350 C \cos \left (d x +c \right )+315 C \right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \tan \left (d x +c \right ) \sec \left (d x +c \right )^{4}}{3465 d \left (\cos \left (d x +c \right )+1\right )}\)	\(152\)
parts	\(\frac {2 B \left (128 \cos \left (d x +c \right )^{4}+64 \cos \left (d x +c \right )^{3}+48 \cos \left (d x +c \right )^{2}+40 \cos \left (d x +c \right )+35\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \tan \left (d x +c \right ) \sec \left (d x +c \right )^{3}}{315 d \left (\cos \left (d x +c \right )+1\right )}+\frac {2 C \left (256 \cos \left (d x +c \right )^{5}+128 \cos \left (d x +c \right )^{4}+96 \cos \left (d x +c \right )^{3}+80 \cos \left (d x +c \right )^{2}+70 \cos \left (d x +c \right )+63\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \tan \left (d x +c \right ) \sec \left (d x +c \right )^{4}}{693 d \left (\cos \left (d x +c \right )+1\right )}\)	\(176\)

[In]

int(sec(d*x+c)^4*(B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+a*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/3465/d*(1408*B*cos(d*x+c)^5+1280*C*cos(d*x+c)^5+704*B*cos(d*x+c)^4+640*C*cos(d*x+c)^4+528*B*cos(d*x+c)^3+480
*C*cos(d*x+c)^3+440*B*cos(d*x+c)^2+400*C*cos(d*x+c)^2+385*B*cos(d*x+c)+350*C*cos(d*x+c)+315*C)*(a*(1+sec(d*x+c
)))^(1/2)/(cos(d*x+c)+1)*tan(d*x+c)*sec(d*x+c)^4

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.60 \[ \int \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (128 \, {\left (11 \, B + 10 \, C\right )} \cos \left (d x + c\right )^{5} + 64 \, {\left (11 \, B + 10 \, C\right )} \cos \left (d x + c\right )^{4} + 48 \, {\left (11 \, B + 10 \, C\right )} \cos \left (d x + c\right )^{3} + 40 \, {\left (11 \, B + 10 \, C\right )} \cos \left (d x + c\right )^{2} + 35 \, {\left (11 \, B + 10 \, C\right )} \cos \left (d x + c\right ) + 315 \, C\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}} \]

[In]

integrate(sec(d*x+c)^4*(B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+a*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

2/3465*(128*(11*B + 10*C)*cos(d*x + c)^5 + 64*(11*B + 10*C)*cos(d*x + c)^4 + 48*(11*B + 10*C)*cos(d*x + c)^3 +
40*(11*B + 10*C)*cos(d*x + c)^2 + 35*(11*B + 10*C)*cos(d*x + c) + 315*C)*sqrt((a*cos(d*x + c) + a)/cos(d*x +
c))*sin(d*x + c)/(d*cos(d*x + c)^6 + d*cos(d*x + c)^5)

Sympy [F]

\[ \int \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \left (B + C \sec {\left (c + d x \right )}\right ) \sec ^{5}{\left (c + d x \right )}\, dx \]

[In]

integrate(sec(d*x+c)**4*(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))*(B + C*sec(c + d*x))*sec(c + d*x)**5, x)

Maxima [F]

\[ \int \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} \sqrt {a \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{4} \,d x } \]

[In]

integrate(sec(d*x+c)^4*(B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+a*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

-32/3465*((cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*(11*(63*B*sin(6*d*x + 6*c)
+ 9*(11*B + 10*C)*sin(4*d*x + 4*c) + 4*(11*B + 10*C)*sin(2*d*x + 2*c))*cos(11/2*arctan2(sin(2*d*x + 2*c), cos(
2*d*x + 2*c) + 1)) - (693*B*cos(6*d*x + 6*c) + 99*(11*B + 10*C)*cos(4*d*x + 4*c) + 44*(11*B + 10*C)*cos(2*d*x
+ 2*c) + 88*B + 80*C)*sin(11/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))*sqrt(a) - 3465*(((B + 2*C)*d*
cos(2*d*x + 2*c)^6 + (B + 2*C)*d*sin(2*d*x + 2*c)^6 + 6*(B + 2*C)*d*cos(2*d*x + 2*c)^5 + 15*(B + 2*C)*d*cos(2*
d*x + 2*c)^4 + 20*(B + 2*C)*d*cos(2*d*x + 2*c)^3 + 3*((B + 2*C)*d*cos(2*d*x + 2*c)^2 + 2*(B + 2*C)*d*cos(2*d*x
+ 2*c) + (B + 2*C)*d)*sin(2*d*x + 2*c)^4 + 15*(B + 2*C)*d*cos(2*d*x + 2*c)^2 + 6*(B + 2*C)*d*cos(2*d*x + 2*c)
+ 3*((B + 2*C)*d*cos(2*d*x + 2*c)^4 + 4*(B + 2*C)*d*cos(2*d*x + 2*c)^3 + 6*(B + 2*C)*d*cos(2*d*x + 2*c)^2 + 4
*(B + 2*C)*d*cos(2*d*x + 2*c) + (B + 2*C)*d)*sin(2*d*x + 2*c)^2 + (B + 2*C)*d)*integrate((((cos(14*d*x + 14*c)
*cos(2*d*x + 2*c) + 6*cos(12*d*x + 12*c)*cos(2*d*x + 2*c) + 15*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 20*cos(8*
d*x + 8*c)*cos(2*d*x + 2*c) + 15*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos
(2*d*x + 2*c)^2 + sin(14*d*x + 14*c)*sin(2*d*x + 2*c) + 6*sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 15*sin(10*d*x
+ 10*c)*sin(2*d*x + 2*c) + 20*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 15*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 6*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))) + (c
os(2*d*x + 2*c)*sin(14*d*x + 14*c) + 6*cos(2*d*x + 2*c)*sin(12*d*x + 12*c) + 15*cos(2*d*x + 2*c)*sin(10*d*x +
10*c) + 20*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 15*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 6*cos(2*d*x + 2*c)*sin(4
*d*x + 4*c) - cos(14*d*x + 14*c)*sin(2*d*x + 2*c) - 6*cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 15*cos(10*d*x + 10
*c)*sin(2*d*x + 2*c) - 20*cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 15*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 6*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*arctan2(sin(2*d*x +
2*c), cos(2*d*x + 2*c) + 1)) - ((cos(2*d*x + 2*c)*sin(14*d*x + 14*c) + 6*cos(2*d*x + 2*c)*sin(12*d*x + 12*c) +
15*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 20*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 15*cos(2*d*x + 2*c)*sin(6*d*x
+ 6*c) + 6*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(14*d*x + 14*c)*sin(2*d*x + 2*c) - 6*cos(12*d*x + 12*c)*sin
(2*d*x + 2*c) - 15*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 20*cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 15*cos(6*d*x +
6*c)*sin(2*d*x + 2*c) - 6*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*cos(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*
c))) - (cos(14*d*x + 14*c)*cos(2*d*x + 2*c) + 6*cos(12*d*x + 12*c)*cos(2*d*x + 2*c) + 15*cos(10*d*x + 10*c)*co
s(2*d*x + 2*c) + 20*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 15*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 6*cos(4*d*x + 4
*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(14*d*x + 14*c)*sin(2*d*x + 2*c) + 6*sin(12*d*x + 12*c)*sin(2*d
*x + 2*c) + 15*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 20*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 15*sin(6*d*x + 6*c
)*sin(2*d*x + 2*c) + 6*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*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))/(((2*(6*cos(12*d*x + 12*c) +
15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos
(14*d*x + 14*c) + cos(14*d*x + 14*c)^2 + 12*(15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c)
+ 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 36*cos(12*d*x + 12*c)^2 + 30*(20*cos(8*d*x + 8*
c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 225*cos(10*d*x + 10*c)^
2 + 40*(15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 400*cos(8*d*x + 8*c)^2
+ 30*(6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 225*cos(6*d*x + 6*c)^2 + 36*cos(4*d*x + 4*c)^
2 + 12*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(6*sin(12*d*x + 12*c) + 15*sin(10*d*x + 10*c
) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + si
n(14*d*x + 14*c)^2 + 12*(15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c
) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 36*sin(12*d*x + 12*c)^2 + 30*(20*sin(8*d*x + 8*c) + 15*sin(6*d*x +
6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 225*sin(10*d*x + 10*c)^2 + 40*(15*sin(6*d*x
+ 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 400*sin(8*d*x + 8*c)^2 + 30*(6*sin(4*d*x +
4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 225*sin(6*d*x + 6*c)^2 + 36*sin(4*d*x + 4*c)^2 + 12*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*(6*
cos(12*d*x + 12*c) + 15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) +
cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + cos(14*d*x + 14*c)^2 + 12*(15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c)
+ 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 36*cos(12*d*x + 12*c)^2 +
30*(20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 22
5*cos(10*d*x + 10*c)^2 + 40*(15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 4
00*cos(8*d*x + 8*c)^2 + 30*(6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 225*cos(6*d*x + 6*c)^2 +
36*cos(4*d*x + 4*c)^2 + 12*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(6*sin(12*d*x + 12*c) +
15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*si
n(14*d*x + 14*c) + sin(14*d*x + 14*c)^2 + 12*(15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c
) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 36*sin(12*d*x + 12*c)^2 + 30*(20*sin(8*d*x + 8
*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 225*sin(10*d*x + 10*c)
^2 + 40*(15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 400*sin(8*d*x + 8*c)^
2 + 30*(6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 225*sin(6*d*x + 6*c)^2 + 36*sin(4*d*x + 4*c)
^2 + 12*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) + (B*d*cos(2*d*x +
2*c)^6 + B*d*sin(2*d*x + 2*c)^6 + 6*B*d*cos(2*d*x + 2*c)^5 + 15*B*d*cos(2*d*x + 2*c)^4 + 20*B*d*cos(2*d*x + 2
*c)^3 + 3*(B*d*cos(2*d*x + 2*c)^2 + 2*B*d*cos(2*d*x + 2*c) + B*d)*sin(2*d*x + 2*c)^4 + 15*B*d*cos(2*d*x + 2*c)
^2 + 6*B*d*cos(2*d*x + 2*c) + 3*(B*d*cos(2*d*x + 2*c)^4 + 4*B*d*cos(2*d*x + 2*c)^3 + 6*B*d*cos(2*d*x + 2*c)^2
+ 4*B*d*cos(2*d*x + 2*c) + B*d)*sin(2*d*x + 2*c)^2 + B*d)*integrate((((cos(14*d*x + 14*c)*cos(2*d*x + 2*c) + 6
*cos(12*d*x + 12*c)*cos(2*d*x + 2*c) + 15*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 20*cos(8*d*x + 8*c)*cos(2*d*x
+ 2*c) + 15*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin
(14*d*x + 14*c)*sin(2*d*x + 2*c) + 6*sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 15*sin(10*d*x + 10*c)*sin(2*d*x + 2
*c) + 20*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 15*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 6*sin(4*d*x + 4*c)*sin(2*d
*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + (cos(2*d*x + 2*c)*sin(1
4*d*x + 14*c) + 6*cos(2*d*x + 2*c)*sin(12*d*x + 12*c) + 15*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 20*cos(2*d*x
+ 2*c)*sin(8*d*x + 8*c) + 15*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 6*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(14*
d*x + 14*c)*sin(2*d*x + 2*c) - 6*cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 15*cos(10*d*x + 10*c)*sin(2*d*x + 2*c)
- 20*cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 15*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 6*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(14*d*x + 14*c) + 6*cos(2*d*x + 2*c)*sin(12*d*x + 12*c) + 15*cos(2*d*x + 2*c)*
sin(10*d*x + 10*c) + 20*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 15*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 6*cos(2*d*x
+ 2*c)*sin(4*d*x + 4*c) - cos(14*d*x + 14*c)*sin(2*d*x + 2*c) - 6*cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 15*co
s(10*d*x + 10*c)*sin(2*d*x + 2*c) - 20*cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 15*cos(6*d*x + 6*c)*sin(2*d*x + 2*c
) - 6*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(14*d*x +
14*c)*cos(2*d*x + 2*c) + 6*cos(12*d*x + 12*c)*cos(2*d*x + 2*c) + 15*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 20*c
os(8*d*x + 8*c)*cos(2*d*x + 2*c) + 15*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c)
+ cos(2*d*x + 2*c)^2 + sin(14*d*x + 14*c)*sin(2*d*x + 2*c) + 6*sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 15*sin(10
*d*x + 10*c)*sin(2*d*x + 2*c) + 20*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 15*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) +
6*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*sin(5/2*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*(6*cos(12*d*x + 12*c) + 15*cos(10*d*x + 10*c)
+ 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + cos
(14*d*x + 14*c)^2 + 12*(15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c)
+ cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 36*cos(12*d*x + 12*c)^2 + 30*(20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6
*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 225*cos(10*d*x + 10*c)^2 + 40*(15*cos(6*d*x
+ 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 400*cos(8*d*x + 8*c)^2 + 30*(6*cos(4*d*x +
4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 225*cos(6*d*x + 6*c)^2 + 36*cos(4*d*x + 4*c)^2 + 12*cos(4*d*x + 4*
c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(6*sin(12*d*x + 12*c) + 15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*
c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + sin(14*d*x + 14*c)^2 +
12*(15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))
*sin(12*d*x + 12*c) + 36*sin(12*d*x + 12*c)^2 + 30*(20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x +
4*c) + sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 225*sin(10*d*x + 10*c)^2 + 40*(15*sin(6*d*x + 6*c) + 6*sin(4*d*x
+ 4*c) + sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 400*sin(8*d*x + 8*c)^2 + 30*(6*sin(4*d*x + 4*c) + sin(2*d*x + 2
*c))*sin(6*d*x + 6*c) + 225*sin(6*d*x + 6*c)^2 + 36*sin(4*d*x + 4*c)^2 + 12*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*(6*cos(12*d*x + 12*c) +
15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos
(14*d*x + 14*c) + cos(14*d*x + 14*c)^2 + 12*(15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c)
+ 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*d*x + 12*c) + 36*cos(12*d*x + 12*c)^2 + 30*(20*cos(8*d*x + 8*
c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(10*d*x + 10*c) + 225*cos(10*d*x + 10*c)^
2 + 40*(15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 400*cos(8*d*x + 8*c)^2
+ 30*(6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 225*cos(6*d*x + 6*c)^2 + 36*cos(4*d*x + 4*c)^
2 + 12*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(6*sin(12*d*x + 12*c) + 15*sin(10*d*x + 10*c
) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + si
n(14*d*x + 14*c)^2 + 12*(15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c
) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 36*sin(12*d*x + 12*c)^2 + 30*(20*sin(8*d*x + 8*c) + 15*sin(6*d*x +
6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 225*sin(10*d*x + 10*c)^2 + 40*(15*sin(6*d*x
+ 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 400*sin(8*d*x + 8*c)^2 + 30*(6*sin(4*d*x +
4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 225*sin(6*d*x + 6*c)^2 + 36*sin(4*d*x + 4*c)^2 + 12*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))*sqrt(a))/(d*cos(2*d*x + 2*c)^6 + d*sin
(2*d*x + 2*c)^6 + 6*d*cos(2*d*x + 2*c)^5 + 15*d*cos(2*d*x + 2*c)^4 + 3*(d*cos(2*d*x + 2*c)^2 + 2*d*cos(2*d*x +
2*c) + d)*sin(2*d*x + 2*c)^4 + 20*d*cos(2*d*x + 2*c)^3 + 15*d*cos(2*d*x + 2*c)^2 + 3*(d*cos(2*d*x + 2*c)^4 +
4*d*cos(2*d*x + 2*c)^3 + 6*d*cos(2*d*x + 2*c)^2 + 4*d*cos(2*d*x + 2*c) + d)*sin(2*d*x + 2*c)^2 + 6*d*cos(2*d*x
+ 2*c) + d)

Giac [F]

\[ \int \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} \sqrt {a \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{4} \,d x } \]

[In]

integrate(sec(d*x+c)^4*(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 = 27.28 (sec) , antiderivative size = 626, normalized size of antiderivative = 2.72 \[ \int \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \left (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 {B\,32{}\mathrm {i}}{9\,d}+\frac {C\,128{}\mathrm {i}}{9\,d}+{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {C\,256{}\mathrm {i}}{33\,d}+\frac {\left (352\,B+704\,C\right )\,1{}\mathrm {i}}{99\,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 )}^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 {B\,32{}\mathrm {i}}{11\,d}+{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {B\,32{}\mathrm {i}}{11\,d}-\frac {\left (32\,B+64\,C\right )\,1{}\mathrm {i}}{11\,d}\right )-\frac {\left (32\,B+64\,C\right )\,1{}\mathrm {i}}{11\,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 )}^5}+\frac {\left (\frac {B\,32{}\mathrm {i}}{5\,d}-\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (352\,B+320\,C\right )\,1{}\mathrm {i}}{1155\,d}\right )\,\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}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}-\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 (352\,B+896\,C\right )\,1{}\mathrm {i}}{693\,d}+\frac {\left (3168\,B+6336\,C\right )\,1{}\mathrm {i}}{693\,d}\right )+\frac {B\,32{}\mathrm {i}}{7\,d}+\frac {\left (3168\,B-6336\,C\right )\,1{}\mathrm {i}}{693\,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 )}^3}-\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 (2816\,B+2560\,C\right )\,1{}\mathrm {i}}{3465\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )}-\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 (1408\,B+1280\,C\right )\,1{}\mathrm {i}}{3465\,d\,\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 )} \]

[In]

int(((B/cos(c + d*x) + C/cos(c + d*x)^2)*(a + a/cos(c + d*x))^(1/2))/cos(c + d*x)^4,x)

[Out]

(((B*32i)/(5*d) - (exp(c*1i + d*x*1i)*(352*B + 320*C)*1i)/(1155*d))*(a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i
+ d*x*1i)/2))^(1/2))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)^2) - ((a + a/(exp(- c*1i - d*x*1i)/2 +
exp(c*1i + d*x*1i)/2))^(1/2)*((B*32i)/(11*d) + exp(c*1i + d*x*1i)*((B*32i)/(11*d) - ((32*B + 64*C)*1i)/(11*d)
) - ((32*B + 64*C)*1i)/(11*d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)^5) - ((a + a/(exp(- c*1i -
d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*((C*128i)/(9*d) - (B*32i)/(9*d) + exp(c*1i + d*x*1i)*((C*256i)/(33*d)
+ ((352*B + 704*C)*1i)/(99*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)*((B*32i)/(7*d) - exp(c*1i + d*x*1i)*(((352*B + 896*C)*1i)/(693*d) +
((3168*B + 6336*C)*1i)/(693*d)) + ((3168*B - 6336*C)*1i)/(693*d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*
2i) + 1)^3) - (exp(c*1i + d*x*1i)*(a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(2816*B + 2560
*C)*1i)/(3465*d*(exp(c*1i + d*x*1i) + 1)) - (exp(c*1i + d*x*1i)*(a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*
x*1i)/2))^(1/2)*(1408*B + 1280*C)*1i)/(3465*d*(exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1))