\(\int \sec ^4(2 (a+b x)) \sqrt {c \tan (a+b x) \tan (2 (a+b x))} \, dx\) [603]
Optimal result
Integrand size = 31, antiderivative size = 157 \[
\int \sec ^4(2 (a+b x)) \sqrt {c \tan (a+b x) \tan (2 (a+b x))} \, dx=-\frac {2 c \tan (2 a+2 b x)}{5 b \sqrt {-c+c \sec (2 a+2 b x)}}+\frac {c \sec ^3(2 a+2 b x) \tan (2 a+2 b x)}{7 b \sqrt {-c+c \sec (2 a+2 b x)}}-\frac {4 \sqrt {-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{35 b}-\frac {6 (-c+c \sec (2 a+2 b x))^{3/2} \tan (2 a+2 b x)}{35 b c}
\]
[Out]
-6/35*(-c+c*sec(2*b*x+2*a))^(3/2)*tan(2*b*x+2*a)/b/c-2/5*c*tan(2*b*x+2*a)/b/(-c+c*sec(2*b*x+2*a))^(1/2)+1/7*c*
sec(2*b*x+2*a)^3*tan(2*b*x+2*a)/b/(-c+c*sec(2*b*x+2*a))^(1/2)-4/35*(-c+c*sec(2*b*x+2*a))^(1/2)*tan(2*b*x+2*a)/
b
Rubi [A] (verified)
Time = 0.56 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of
steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {4482, 3888, 3885, 4086, 3877}
\[
\int \sec ^4(2 (a+b x)) \sqrt {c \tan (a+b x) \tan (2 (a+b x))} \, dx=\frac {c \tan (2 a+2 b x) \sec ^3(2 a+2 b x)}{7 b \sqrt {c \sec (2 a+2 b x)-c}}-\frac {6 \tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}}{35 b c}-\frac {4 \tan (2 a+2 b x) \sqrt {c \sec (2 a+2 b x)-c}}{35 b}-\frac {2 c \tan (2 a+2 b x)}{5 b \sqrt {c \sec (2 a+2 b x)-c}}
\]
[In]
Int[Sec[2*(a + b*x)]^4*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]
[Out]
(-2*c*Tan[2*a + 2*b*x])/(5*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (c*Sec[2*a + 2*b*x]^3*Tan[2*a + 2*b*x])/(7*b*Sqr
t[-c + c*Sec[2*a + 2*b*x]]) - (4*Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(35*b) - (6*(-c + c*Sec[2*a +
2*b*x])^(3/2)*Tan[2*a + 2*b*x])/(35*b*c)
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 4482
Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]
Rubi steps \begin{align*}
\text {integral}& = \int \sec ^4(2 a+2 b x) \sqrt {-c+c \sec (2 a+2 b x)} \, dx \\ & = \frac {c \sec ^3(2 a+2 b x) \tan (2 a+2 b x)}{7 b \sqrt {-c+c \sec (2 a+2 b x)}}-\frac {6}{7} \int \sec ^3(2 a+2 b x) \sqrt {-c+c \sec (2 a+2 b x)} \, dx \\ & = \frac {c \sec ^3(2 a+2 b x) \tan (2 a+2 b x)}{7 b \sqrt {-c+c \sec (2 a+2 b x)}}-\frac {6 (-c+c \sec (2 a+2 b x))^{3/2} \tan (2 a+2 b x)}{35 b c}-\frac {12 \int \sec (2 a+2 b x) \sqrt {-c+c \sec (2 a+2 b x)} \left (\frac {3 c}{2}+c \sec (2 a+2 b x)\right ) \, dx}{35 c} \\ & = \frac {c \sec ^3(2 a+2 b x) \tan (2 a+2 b x)}{7 b \sqrt {-c+c \sec (2 a+2 b x)}}-\frac {4 \sqrt {-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{35 b}-\frac {6 (-c+c \sec (2 a+2 b x))^{3/2} \tan (2 a+2 b x)}{35 b c}-\frac {2}{5} \int \sec (2 a+2 b x) \sqrt {-c+c \sec (2 a+2 b x)} \, dx \\ & = -\frac {2 c \tan (2 a+2 b x)}{5 b \sqrt {-c+c \sec (2 a+2 b x)}}+\frac {c \sec ^3(2 a+2 b x) \tan (2 a+2 b x)}{7 b \sqrt {-c+c \sec (2 a+2 b x)}}-\frac {4 \sqrt {-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{35 b}-\frac {6 (-c+c \sec (2 a+2 b x))^{3/2} \tan (2 a+2 b x)}{35 b c} \\
\end{align*}
Mathematica [A] (verified)
Time = 1.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.41
\[
\int \sec ^4(2 (a+b x)) \sqrt {c \tan (a+b x) \tan (2 (a+b x))} \, dx=-\frac {(7 \cos (3 (a+b x))+2 \cos (7 (a+b x))) \csc (a+b x) \sec ^3(2 (a+b x)) \sqrt {c \tan (a+b x) \tan (2 (a+b x))}}{35 b}
\]
[In]
Integrate[Sec[2*(a + b*x)]^4*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]
[Out]
-1/35*((7*Cos[3*(a + b*x)] + 2*Cos[7*(a + b*x)])*Csc[a + b*x]*Sec[2*(a + b*x)]^3*Sqrt[c*Tan[a + b*x]*Tan[2*(a
+ b*x)]])/b
Maple [A] (verified)
Time = 7.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.57
| | |
| method | result | size |
| | |
| default |
\(-\frac {\sqrt {2}\, \cot \left (x b +a \right ) \sqrt {\frac {c \sin \left (x b +a \right )^{2}}{2 \cos \left (x b +a \right )^{2}-1}}\, \left (128 \cos \left (x b +a \right )^{6}-224 \cos \left (x b +a \right )^{4}+140 \cos \left (x b +a \right )^{2}-35\right ) \sqrt {4}}{70 b \left (2 \cos \left (x b +a \right )^{2}-1\right )^{3}}\) |
\(90\) |
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[In]
int(sec(2*b*x+2*a)^4*(c*tan(b*x+a)*tan(2*b*x+2*a))^(1/2),x,method=_RETURNVERBOSE)
[Out]
-1/70*2^(1/2)/b*cot(b*x+a)*(c*sin(b*x+a)^2/(2*cos(b*x+a)^2-1))^(1/2)*(128*cos(b*x+a)^6-224*cos(b*x+a)^4+140*co
s(b*x+a)^2-35)/(2*cos(b*x+a)^2-1)^3*4^(1/2)
Fricas [A] (verification not implemented)
none
Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.68
\[
\int \sec ^4(2 (a+b x)) \sqrt {c \tan (a+b x) \tan (2 (a+b x))} \, dx=-\frac {\sqrt {2} {\left (35 \, \tan \left (b x + a\right )^{6} - 35 \, \tan \left (b x + a\right )^{4} + 49 \, \tan \left (b x + a\right )^{2} - 9\right )} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{35 \, {\left (b \tan \left (b x + a\right )^{7} - 3 \, b \tan \left (b x + a\right )^{5} + 3 \, b \tan \left (b x + a\right )^{3} - b \tan \left (b x + a\right )\right )}}
\]
[In]
integrate(sec(2*b*x+2*a)^4*(c*tan(b*x+a)*tan(2*b*x+2*a))^(1/2),x, algorithm="fricas")
[Out]
-1/35*sqrt(2)*(35*tan(b*x + a)^6 - 35*tan(b*x + a)^4 + 49*tan(b*x + a)^2 - 9)*sqrt(-c*tan(b*x + a)^2/(tan(b*x
+ a)^2 - 1))/(b*tan(b*x + a)^7 - 3*b*tan(b*x + a)^5 + 3*b*tan(b*x + a)^3 - b*tan(b*x + a))
Sympy [F(-1)]
Timed out. \[
\int \sec ^4(2 (a+b x)) \sqrt {c \tan (a+b x) \tan (2 (a+b x))} \, dx=\text {Timed out}
\]
[In]
integrate(sec(2*b*x+2*a)**4*(c*tan(b*x+a)*tan(2*b*x+2*a))**(1/2),x)
[Out]
Timed out
Maxima [F]
\[
\int \sec ^4(2 (a+b x)) \sqrt {c \tan (a+b x) \tan (2 (a+b x))} \, dx=\int { \sqrt {c \tan \left (2 \, b x + 2 \, a\right ) \tan \left (b x + a\right )} \sec \left (2 \, b x + 2 \, a\right )^{4} \,d x }
\]
[In]
integrate(sec(2*b*x+2*a)^4*(c*tan(b*x+a)*tan(2*b*x+2*a))^(1/2),x, algorithm="maxima")
[Out]
-8/35*(70*(b*cos(4*b*x + 4*a)^2 + b*sin(4*b*x + 4*a)^2 + 2*b*cos(4*b*x + 4*a) + b)*(cos(4*b*x + 4*a)^2 + sin(4
*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)^(3/4)*sqrt(c)*integrate(-(((cos(20*b*x + 20*a)*cos(4*b*x + 4*a) + 4*co
s(16*b*x + 16*a)*cos(4*b*x + 4*a) + 6*cos(12*b*x + 12*a)*cos(4*b*x + 4*a) + 4*cos(8*b*x + 8*a)*cos(4*b*x + 4*a
) + cos(4*b*x + 4*a)^2 + sin(20*b*x + 20*a)*sin(4*b*x + 4*a) + 4*sin(16*b*x + 16*a)*sin(4*b*x + 4*a) + 6*sin(1
2*b*x + 12*a)*sin(4*b*x + 4*a) + 4*sin(8*b*x + 8*a)*sin(4*b*x + 4*a) + sin(4*b*x + 4*a)^2)*cos(1/2*arctan2(sin
(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)) + (cos(4*b*x + 4*a)*sin(20*b*x + 20*a) + 4*cos(4*b*x + 4*a)*sin(16*b*x
+ 16*a) + 6*cos(4*b*x + 4*a)*sin(12*b*x + 12*a) + 4*cos(4*b*x + 4*a)*sin(8*b*x + 8*a) - cos(20*b*x + 20*a)*sin
(4*b*x + 4*a) - 4*cos(16*b*x + 16*a)*sin(4*b*x + 4*a) - 6*cos(12*b*x + 12*a)*sin(4*b*x + 4*a) - 4*cos(8*b*x +
8*a)*sin(4*b*x + 4*a))*sin(1/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)))*cos(5/2*arctan2(sin(4*b*x +
4*a), cos(4*b*x + 4*a))) + ((cos(4*b*x + 4*a)*sin(20*b*x + 20*a) + 4*cos(4*b*x + 4*a)*sin(16*b*x + 16*a) + 6*c
os(4*b*x + 4*a)*sin(12*b*x + 12*a) + 4*cos(4*b*x + 4*a)*sin(8*b*x + 8*a) - cos(20*b*x + 20*a)*sin(4*b*x + 4*a)
- 4*cos(16*b*x + 16*a)*sin(4*b*x + 4*a) - 6*cos(12*b*x + 12*a)*sin(4*b*x + 4*a) - 4*cos(8*b*x + 8*a)*sin(4*b*
x + 4*a))*cos(1/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)) - (cos(20*b*x + 20*a)*cos(4*b*x + 4*a) + 4
*cos(16*b*x + 16*a)*cos(4*b*x + 4*a) + 6*cos(12*b*x + 12*a)*cos(4*b*x + 4*a) + 4*cos(8*b*x + 8*a)*cos(4*b*x +
4*a) + cos(4*b*x + 4*a)^2 + sin(20*b*x + 20*a)*sin(4*b*x + 4*a) + 4*sin(16*b*x + 16*a)*sin(4*b*x + 4*a) + 6*si
n(12*b*x + 12*a)*sin(4*b*x + 4*a) + 4*sin(8*b*x + 8*a)*sin(4*b*x + 4*a) + sin(4*b*x + 4*a)^2)*sin(1/2*arctan2(
sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)))*sin(5/2*arctan2(sin(4*b*x + 4*a), cos(4*b*x + 4*a))))/(((2*(4*cos(1
6*b*x + 16*a) + 6*cos(12*b*x + 12*a) + 4*cos(8*b*x + 8*a) + cos(4*b*x + 4*a))*cos(20*b*x + 20*a) + cos(20*b*x
+ 20*a)^2 + 8*(6*cos(12*b*x + 12*a) + 4*cos(8*b*x + 8*a) + cos(4*b*x + 4*a))*cos(16*b*x + 16*a) + 16*cos(16*b*
x + 16*a)^2 + 12*(4*cos(8*b*x + 8*a) + cos(4*b*x + 4*a))*cos(12*b*x + 12*a) + 36*cos(12*b*x + 12*a)^2 + 16*cos
(8*b*x + 8*a)^2 + 8*cos(8*b*x + 8*a)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + 2*(4*sin(16*b*x + 16*a) + 6*sin(1
2*b*x + 12*a) + 4*sin(8*b*x + 8*a) + sin(4*b*x + 4*a))*sin(20*b*x + 20*a) + sin(20*b*x + 20*a)^2 + 8*(6*sin(12
*b*x + 12*a) + 4*sin(8*b*x + 8*a) + sin(4*b*x + 4*a))*sin(16*b*x + 16*a) + 16*sin(16*b*x + 16*a)^2 + 12*(4*sin
(8*b*x + 8*a) + sin(4*b*x + 4*a))*sin(12*b*x + 12*a) + 36*sin(12*b*x + 12*a)^2 + 16*sin(8*b*x + 8*a)^2 + 8*sin
(8*b*x + 8*a)*sin(4*b*x + 4*a) + sin(4*b*x + 4*a)^2)*cos(1/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1))
^2 + (2*(4*cos(16*b*x + 16*a) + 6*cos(12*b*x + 12*a) + 4*cos(8*b*x + 8*a) + cos(4*b*x + 4*a))*cos(20*b*x + 20*
a) + cos(20*b*x + 20*a)^2 + 8*(6*cos(12*b*x + 12*a) + 4*cos(8*b*x + 8*a) + cos(4*b*x + 4*a))*cos(16*b*x + 16*a
) + 16*cos(16*b*x + 16*a)^2 + 12*(4*cos(8*b*x + 8*a) + cos(4*b*x + 4*a))*cos(12*b*x + 12*a) + 36*cos(12*b*x +
12*a)^2 + 16*cos(8*b*x + 8*a)^2 + 8*cos(8*b*x + 8*a)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + 2*(4*sin(16*b*x +
16*a) + 6*sin(12*b*x + 12*a) + 4*sin(8*b*x + 8*a) + sin(4*b*x + 4*a))*sin(20*b*x + 20*a) + sin(20*b*x + 20*a)
^2 + 8*(6*sin(12*b*x + 12*a) + 4*sin(8*b*x + 8*a) + sin(4*b*x + 4*a))*sin(16*b*x + 16*a) + 16*sin(16*b*x + 16*
a)^2 + 12*(4*sin(8*b*x + 8*a) + sin(4*b*x + 4*a))*sin(12*b*x + 12*a) + 36*sin(12*b*x + 12*a)^2 + 16*sin(8*b*x
+ 8*a)^2 + 8*sin(8*b*x + 8*a)*sin(4*b*x + 4*a) + sin(4*b*x + 4*a)^2)*sin(1/2*arctan2(sin(4*b*x + 4*a), -cos(4*
b*x + 4*a) - 1))^2)*(cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)^(1/4)), x) + (7*cos(7/2
*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1))*sin(4*b*x + 4*a) + (7*cos(4*b*x + 4*a) + 2)*sin(7/2*arctan2
(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)))*sqrt(c))/((b*cos(4*b*x + 4*a)^2 + b*sin(4*b*x + 4*a)^2 + 2*b*cos(4
*b*x + 4*a) + b)*(cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)^(3/4))
Giac [F(-1)]
Timed out. \[
\int \sec ^4(2 (a+b x)) \sqrt {c \tan (a+b x) \tan (2 (a+b x))} \, dx=\text {Timed out}
\]
[In]
integrate(sec(2*b*x+2*a)^4*(c*tan(b*x+a)*tan(2*b*x+2*a))^(1/2),x, algorithm="giac")
[Out]
Timed out
Mupad [B] (verification not implemented)
Time = 34.13 (sec) , antiderivative size = 463, normalized size of antiderivative = 2.95
\[
\int \sec ^4(2 (a+b x)) \sqrt {c \tan (a+b x) \tan (2 (a+b x))} \, dx=-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,\sqrt {\frac {c\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+1\right )}}\,16{}\mathrm {i}}{35\,b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}+\frac {\left (\frac {8{}\mathrm {i}}{7\,b}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,8{}\mathrm {i}}{7\,b}\right )\,\sqrt {\frac {c\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+1\right )}}}{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+1\right )}^3}-\frac {\left (\frac {8{}\mathrm {i}}{5\,b}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,64{}\mathrm {i}}{35\,b}\right )\,\sqrt {\frac {c\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+1\right )}}}{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+1\right )}^2}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,\sqrt {\frac {c\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+1\right )}}\,8{}\mathrm {i}}{35\,b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+1\right )}
\]
[In]
int((c*tan(a + b*x)*tan(2*a + 2*b*x))^(1/2)/cos(2*a + 2*b*x)^4,x)
[Out]
((8i/(7*b) - (exp(a*2i + b*x*2i)*8i)/(7*b))*((c*(exp(a*2i + b*x*2i)*1i - 1i)*(exp(a*4i + b*x*4i)*1i - 1i))/((e
xp(a*2i + b*x*2i) + 1)*(exp(a*4i + b*x*4i) + 1)))^(1/2))/((exp(a*2i + b*x*2i) - 1)*(exp(a*4i + b*x*4i) + 1)^3)
- (exp(a*2i + b*x*2i)*((c*(exp(a*2i + b*x*2i)*1i - 1i)*(exp(a*4i + b*x*4i)*1i - 1i))/((exp(a*2i + b*x*2i) + 1
)*(exp(a*4i + b*x*4i) + 1)))^(1/2)*16i)/(35*b*(exp(a*2i + b*x*2i) - 1)) - ((8i/(5*b) - (exp(a*2i + b*x*2i)*64i
)/(35*b))*((c*(exp(a*2i + b*x*2i)*1i - 1i)*(exp(a*4i + b*x*4i)*1i - 1i))/((exp(a*2i + b*x*2i) + 1)*(exp(a*4i +
b*x*4i) + 1)))^(1/2))/((exp(a*2i + b*x*2i) - 1)*(exp(a*4i + b*x*4i) + 1)^2) - (exp(a*2i + b*x*2i)*((c*(exp(a*
2i + b*x*2i)*1i - 1i)*(exp(a*4i + b*x*4i)*1i - 1i))/((exp(a*2i + b*x*2i) + 1)*(exp(a*4i + b*x*4i) + 1)))^(1/2)
*8i)/(35*b*(exp(a*2i + b*x*2i) - 1)*(exp(a*4i + b*x*4i) + 1))