\(\int \sec ^2(c+d x) \sqrt {a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx\) [120]
Optimal result
Integrand size = 33, antiderivative size = 101 \[
\int \sec ^2(c+d x) \sqrt {a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\frac {2 a (5 A+7 B) \tan (c+d x)}{15 d \sqrt {a+a \sec (c+d x)}}+\frac {2 (5 A-2 B) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{15 d}+\frac {2 B (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{5 a d}
\]
[Out]
2/5*B*(a+a*sec(d*x+c))^(3/2)*tan(d*x+c)/a/d+2/15*a*(5*A+7*B)*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+2/15*(5*A-2*B
)*(a+a*sec(d*x+c))^(1/2)*tan(d*x+c)/d
Rubi [A] (verified)
Time = 0.27 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of
steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4095, 4086, 3877}
\[
\int \sec ^2(c+d x) \sqrt {a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\frac {2 (5 A-2 B) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{15 d}+\frac {2 a (5 A+7 B) \tan (c+d x)}{15 d \sqrt {a \sec (c+d x)+a}}+\frac {2 B \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 a d}
\]
[In]
Int[Sec[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]
[Out]
(2*a*(5*A + 7*B)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(5*A - 2*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c
+ d*x])/(15*d) + (2*B*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*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 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 4095
Int[csc[(e_.) + (f_.)*(x_)]^2*(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 + 1)/(b*f*(m + 2))), x] + Dist[1/(b*(m + 2)),
Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[b*B*(m + 1) + (A*b*(m + 2) - a*B)*Csc[e + f*x], x], x], x] /; Fr
eeQ[{a, b, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && !LtQ[m, -1]
Rubi steps \begin{align*}
\text {integral}& = \frac {2 B (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{5 a d}+\frac {2 \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {3 a B}{2}+\frac {1}{2} a (5 A-2 B) \sec (c+d x)\right ) \, dx}{5 a} \\ & = \frac {2 (5 A-2 B) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{15 d}+\frac {2 B (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{5 a d}+\frac {1}{15} (5 A+7 B) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {2 a (5 A+7 B) \tan (c+d x)}{15 d \sqrt {a+a \sec (c+d x)}}+\frac {2 (5 A-2 B) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{15 d}+\frac {2 B (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{5 a d} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.21 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.79
\[
\int \sec ^2(c+d x) \sqrt {a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\frac {2 (5 A+7 B+(5 A+4 B) \cos (c+d x)+(5 A+4 B) \cos (2 (c+d x))) \sec (c+d x) \sqrt {a (1+\sec (c+d x))} \tan (c+d x)}{15 d (1+\cos (c+d x))}
\]
[In]
Integrate[Sec[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]
[Out]
(2*(5*A + 7*B + (5*A + 4*B)*Cos[c + d*x] + (5*A + 4*B)*Cos[2*(c + d*x)])*Sec[c + d*x]*Sqrt[a*(1 + Sec[c + d*x]
)]*Tan[c + d*x])/(15*d*(1 + Cos[c + d*x]))
Maple [A] (verified)
Time = 3.54 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.79
| | |
method | result | size |
| | |
default |
\(\frac {2 \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (10 A \sin \left (d x +c \right )+8 B \sin \left (d x +c \right )+5 A \tan \left (d x +c \right )+4 B \tan \left (d x +c \right )+3 B \tan \left (d x +c \right ) \sec \left (d x +c \right )\right )}{15 d \left (\cos \left (d x +c \right )+1\right )}\) |
\(80\) |
parts |
\(\frac {2 A \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (2 \sin \left (d x +c \right )+\tan \left (d x +c \right )\right )}{3 d \left (\cos \left (d x +c \right )+1\right )}+\frac {2 B \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (8 \sin \left (d x +c \right )+4 \tan \left (d x +c \right )+3 \sec \left (d x +c \right ) \tan \left (d x +c \right )\right )}{15 d \left (\cos \left (d x +c \right )+1\right )}\) |
\(104\) |
| | |
|
|
|
[In]
int(sec(d*x+c)^2*(a+a*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)),x,method=_RETURNVERBOSE)
[Out]
2/15/d*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)+1)*(10*A*sin(d*x+c)+8*B*sin(d*x+c)+5*A*tan(d*x+c)+4*B*tan(d*x+c)+3
*B*tan(d*x+c)*sec(d*x+c))
Fricas [A] (verification not implemented)
none
Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.86
\[
\int \sec ^2(c+d x) \sqrt {a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\frac {2 \, {\left (2 \, {\left (5 \, A + 4 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (5 \, A + 4 \, B\right )} \cos \left (d x + c\right ) + 3 \, B\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{15 \, {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )}}
\]
[In]
integrate(sec(d*x+c)^2*(a+a*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)),x, algorithm="fricas")
[Out]
2/15*(2*(5*A + 4*B)*cos(d*x + c)^2 + (5*A + 4*B)*cos(d*x + c) + 3*B)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*s
in(d*x + c)/(d*cos(d*x + c)^3 + d*cos(d*x + c)^2)
Sympy [F]
\[
\int \sec ^2(c+d x) \sqrt {a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\int \sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \left (A + B \sec {\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx
\]
[In]
integrate(sec(d*x+c)**2*(a+a*sec(d*x+c))**(1/2)*(A+B*sec(d*x+c)),x)
[Out]
Integral(sqrt(a*(sec(c + d*x) + 1))*(A + B*sec(c + d*x))*sec(c + d*x)**2, x)
Maxima [F]
\[
\int \sec ^2(c+d x) \sqrt {a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} \sqrt {a \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{2} \,d x }
\]
[In]
integrate(sec(d*x+c)^2*(a+a*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)),x, algorithm="maxima")
[Out]
4/15*(15*(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*d*cos(2*d*x + 2*c)^2 + A
*d*sin(2*d*x + 2*c)^2 + 2*A*d*cos(2*d*x + 2*c) + A*d)*integrate((((cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 3*cos(6
*d*x + 6*c)*cos(2*d*x + 2*c) + 3*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(8*d*x + 8*c)*sin
(2*d*x + 2*c) + 3*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 3*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(8*d*x + 8*c) + 3*cos(2*d*x + 2*
c)*sin(6*d*x + 6*c) + 3*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 3*cos(6*d*x +
6*c)*sin(2*d*x + 2*c) - 3*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(8*d*x + 8*c) + 3*cos(2*
d*x + 2*c)*sin(6*d*x + 6*c) + 3*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 3*cos(
6*d*x + 6*c)*sin(2*d*x + 2*c) - 3*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(8*d*x + 8*c)*cos(2*d*x + 2*c) + 3*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 3*cos(4*d*x + 4*c)*co
s(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 3*sin(6*d*x + 6*c)*sin(2*d*x + 2*c)
+ 3*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*(3*cos(6*d*x + 6*c) + 3*cos(4*d*x + 4*c) +
cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + cos(8*d*x + 8*c)^2 + 6*(3*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x
+ 6*c) + 9*cos(6*d*x + 6*c)^2 + 9*cos(4*d*x + 4*c)^2 + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^
2 + 2*(3*sin(6*d*x + 6*c) + 3*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + sin(8*d*x + 8*c)^2 + 6*(
3*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 9*sin(6*d*x + 6*c)^2 + 9*sin(4*d*x + 4*c)^2 + 6*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*(3*cos(6*d*x + 6*c) + 3*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + cos(8*d*x + 8*c)^2 + 6*(
3*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 9*cos(6*d*x + 6*c)^2 + 9*cos(4*d*x + 4*c)^2 + 6*cos(
4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(3*sin(6*d*x + 6*c) + 3*sin(4*d*x + 4*c) + sin(2*d*x +
2*c))*sin(8*d*x + 8*c) + sin(8*d*x + 8*c)^2 + 6*(3*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 9*s
in(6*d*x + 6*c)^2 + 9*sin(4*d*x + 4*c)^2 + 6*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*sin(1/2*a
rctan2(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 + 2*B)*d*cos(2*d*x + 2*c)^2 + (A + 2*B)*d*sin(2*d*x + 2*c)^2 + 2*(A + 2*B)*d*cos(2*d*
x + 2*c) + (A + 2*B)*d)*integrate((((cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 3*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) +
3*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 3*sin(6*d*x +
6*c)*sin(2*d*x + 2*c) + 3*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(8*d*x + 8*c) + 3*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 3*cos(2*
d*x + 2*c)*sin(4*d*x + 4*c) - cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 3*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 3*cos(
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(8*d*x + 8*c) + 3*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) +
3*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 3*cos(6*d*x + 6*c)*sin(2*d*x + 2*c)
- 3*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*cos(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - (cos(8*d*x + 8*c
)*cos(2*d*x + 2*c) + 3*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 3*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2
*c)^2 + sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 3*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 3*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*(3*cos(6*d*x + 6*c) + 3*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*d*x +
8*c) + cos(8*d*x + 8*c)^2 + 6*(3*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 9*cos(6*d*x + 6*c)^2
+ 9*cos(4*d*x + 4*c)^2 + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + 2*(3*sin(6*d*x + 6*c) + 3*
sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + sin(8*d*x + 8*c)^2 + 6*(3*sin(4*d*x + 4*c) + sin(2*d*x
+ 2*c))*sin(6*d*x + 6*c) + 9*sin(6*d*x + 6*c)^2 + 9*sin(4*d*x + 4*c)^2 + 6*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*(3*cos(6*d*x + 6*c) + 3*
cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + cos(8*d*x + 8*c)^2 + 6*(3*cos(4*d*x + 4*c) + cos(2*d*x
+ 2*c))*cos(6*d*x + 6*c) + 9*cos(6*d*x + 6*c)^2 + 9*cos(4*d*x + 4*c)^2 + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c)
+ cos(2*d*x + 2*c)^2 + 2*(3*sin(6*d*x + 6*c) + 3*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + sin(8
*d*x + 8*c)^2 + 6*(3*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 9*sin(6*d*x + 6*c)^2 + 9*sin(4*d*
x + 4*c)^2 + 6*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) -
(5*(A + 2*B)*cos(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1))*sin(2*d*x + 2*c) - (5*(A + 2*B)*cos(2*d*
x + 2*c) + 5*A + 4*B)*sin(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)))*sqrt(a))/((d*cos(2*d*x + 2*c)^
2 + d*sin(2*d*x + 2*c)^2 + 2*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]
\[
\int \sec ^2(c+d x) \sqrt {a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} \sqrt {a \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{2} \,d x }
\]
[In]
integrate(sec(d*x+c)^2*(a+a*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)),x, algorithm="giac")
[Out]
sage0*x
Mupad [B] (verification not implemented)
Time = 18.78 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.10
\[
\int \sec ^2(c+d x) \sqrt {a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx=-\frac {4\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}-1\right )\,\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left (A\,5{}\mathrm {i}+B\,4{}\mathrm {i}+A\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,5{}\mathrm {i}+A\,{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,10{}\mathrm {i}+A\,{\mathrm {e}}^{c\,3{}\mathrm {i}+d\,x\,3{}\mathrm {i}}\,5{}\mathrm {i}+A\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,5{}\mathrm {i}+B\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,4{}\mathrm {i}+B\,{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,14{}\mathrm {i}+B\,{\mathrm {e}}^{c\,3{}\mathrm {i}+d\,x\,3{}\mathrm {i}}\,4{}\mathrm {i}+B\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,4{}\mathrm {i}\right )}{15\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}
\]
[In]
int(((A + B/cos(c + d*x))*(a + a/cos(c + d*x))^(1/2))/cos(c + d*x)^2,x)
[Out]
-(4*(exp(c*1i + d*x*1i) - 1)*(a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(A*5i + B*4i + A*ex
p(c*1i + d*x*1i)*5i + A*exp(c*2i + d*x*2i)*10i + A*exp(c*3i + d*x*3i)*5i + A*exp(c*4i + d*x*4i)*5i + B*exp(c*1
i + d*x*1i)*4i + B*exp(c*2i + d*x*2i)*14i + B*exp(c*3i + d*x*3i)*4i + B*exp(c*4i + d*x*4i)*4i))/(15*d*(exp(c*1
i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)^2)