\(\int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx\) [613]
Optimal result
Integrand size = 31, antiderivative size = 110 \[
\int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\frac {4 c^2 \tan (2 a+2 b x)}{5 b \sqrt {-c+c \sec (2 a+2 b x)}}-\frac {c \sqrt {-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{5 b}+\frac {(-c+c \sec (2 a+2 b x))^{3/2} \tan (2 a+2 b x)}{5 b}
\]
[Out]
1/5*(-c+c*sec(2*b*x+2*a))^(3/2)*tan(2*b*x+2*a)/b+4/5*c^2*tan(2*b*x+2*a)/b/(-c+c*sec(2*b*x+2*a))^(1/2)-1/5*c*(-
c+c*sec(2*b*x+2*a))^(1/2)*tan(2*b*x+2*a)/b
Rubi [A] (verified)
Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of
steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {4482, 3883, 3878, 3877}
\[
\int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\frac {4 c^2 \tan (2 a+2 b x)}{5 b \sqrt {c \sec (2 a+2 b x)-c}}-\frac {c \tan (2 a+2 b x) \sqrt {c \sec (2 a+2 b x)-c}}{5 b}+\frac {\tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}}{5 b}
\]
[In]
Int[Sec[2*(a + b*x)]^2*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]
[Out]
(4*c^2*Tan[2*a + 2*b*x])/(5*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) - (c*Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*
x])/(5*b) + ((-c + c*Sec[2*a + 2*b*x])^(3/2)*Tan[2*a + 2*b*x])/(5*b)
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 3878
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-b)*Cot[e + f*x]*(
(a + b*Csc[e + f*x])^(m - 1)/(f*m)), x] + Dist[a*((2*m - 1)/m), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1),
x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && IntegerQ[2*m]
Rule 3883
Int[csc[(e_.) + (f_.)*(x_)]^2*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-Cot[e + f*x])*(
(a + b*Csc[e + f*x])^m/(f*(m + 1))), x] + Dist[a*(m/(b*(m + 1))), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x],
x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 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 ^2(2 a+2 b x) (-c+c \sec (2 a+2 b x))^{3/2} \, dx \\ & = \frac {(-c+c \sec (2 a+2 b x))^{3/2} \tan (2 a+2 b x)}{5 b}-\frac {3}{5} \int \sec (2 a+2 b x) (-c+c \sec (2 a+2 b x))^{3/2} \, dx \\ & = -\frac {c \sqrt {-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{5 b}+\frac {(-c+c \sec (2 a+2 b x))^{3/2} \tan (2 a+2 b x)}{5 b}+\frac {1}{5} (4 c) \int \sec (2 a+2 b x) \sqrt {-c+c \sec (2 a+2 b x)} \, dx \\ & = \frac {4 c^2 \tan (2 a+2 b x)}{5 b \sqrt {-c+c \sec (2 a+2 b x)}}-\frac {c \sqrt {-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{5 b}+\frac {(-c+c \sec (2 a+2 b x))^{3/2} \tan (2 a+2 b x)}{5 b} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.63 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.54
\[
\int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\frac {\cot (a+b x) (-2+4 \cot (a+b x) \cot (2 (a+b x))+\sec (2 (a+b x))) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2}}{5 b}
\]
[In]
Integrate[Sec[2*(a + b*x)]^2*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]
[Out]
(Cot[a + b*x]*(-2 + 4*Cot[a + b*x]*Cot[2*(a + b*x)] + Sec[2*(a + b*x)])*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2
))/(5*b)
Maple [A] (verified)
Time = 5.67 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.74
| | |
method | result | size |
| | |
default |
\(\frac {\sqrt {2}\, \cot \left (x b +a \right ) \sqrt {\frac {c \sin \left (x b +a \right )^{2}}{2 \cos \left (x b +a \right )^{2}-1}}\, \left (12 \cos \left (x b +a \right )^{4}-15 \cos \left (x b +a \right )^{2}+5\right ) c \sqrt {4}}{5 b \left (2 \cos \left (x b +a \right )^{2}-1\right )^{2}}\) |
\(81\) |
| | |
|
|
|
[In]
int(sec(2*b*x+2*a)^2*(c*tan(b*x+a)*tan(2*b*x+2*a))^(3/2),x,method=_RETURNVERBOSE)
[Out]
1/5*2^(1/2)/b*cot(b*x+a)*(c*sin(b*x+a)^2/(2*cos(b*x+a)^2-1))^(1/2)*(12*cos(b*x+a)^4-15*cos(b*x+a)^2+5)*c/(2*co
s(b*x+a)^2-1)^2*4^(1/2)
Fricas [A] (verification not implemented)
none
Time = 0.24 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.80
\[
\int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\frac {2 \, \sqrt {2} {\left (5 \, c \tan \left (b x + a\right )^{4} - 5 \, c \tan \left (b x + a\right )^{2} + 2 \, c\right )} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{5 \, {\left (b \tan \left (b x + a\right )^{5} - 2 \, b \tan \left (b x + a\right )^{3} + b \tan \left (b x + a\right )\right )}}
\]
[In]
integrate(sec(2*b*x+2*a)^2*(c*tan(b*x+a)*tan(2*b*x+2*a))^(3/2),x, algorithm="fricas")
[Out]
2/5*sqrt(2)*(5*c*tan(b*x + a)^4 - 5*c*tan(b*x + a)^2 + 2*c)*sqrt(-c*tan(b*x + a)^2/(tan(b*x + a)^2 - 1))/(b*ta
n(b*x + a)^5 - 2*b*tan(b*x + a)^3 + b*tan(b*x + a))
Sympy [F(-1)]
Timed out. \[
\int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\text {Timed out}
\]
[In]
integrate(sec(2*b*x+2*a)**2*(c*tan(b*x+a)*tan(2*b*x+2*a))**(3/2),x)
[Out]
Timed out
Maxima [F]
\[
\int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\int { \left (c \tan \left (2 \, b x + 2 \, a\right ) \tan \left (b x + a\right )\right )^{\frac {3}{2}} \sec \left (2 \, b x + 2 \, a\right )^{2} \,d x }
\]
[In]
integrate(sec(2*b*x+2*a)^2*(c*tan(b*x+a)*tan(2*b*x+2*a))^(3/2),x, algorithm="maxima")
[Out]
-2/5*(10*(cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)^(1/4)*((b*c*cos(4*b*x + 4*a)^2 + b
*c*sin(4*b*x + 4*a)^2 + 2*b*c*cos(4*b*x + 4*a) + b*c)*integrate(-(cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*
cos(4*b*x + 4*a) + 1)^(1/4)*(((cos(12*b*x + 12*a)*cos(4*b*x + 4*a) + 2*cos(8*b*x + 8*a)*cos(4*b*x + 4*a) + cos
(4*b*x + 4*a)^2 + sin(12*b*x + 12*a)*sin(4*b*x + 4*a) + 2*sin(8*b*x + 8*a)*sin(4*b*x + 4*a) + sin(4*b*x + 4*a)
^2)*cos(3/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)) + (cos(4*b*x + 4*a)*sin(12*b*x + 12*a) + 2*cos(4
*b*x + 4*a)*sin(8*b*x + 8*a) - cos(12*b*x + 12*a)*sin(4*b*x + 4*a) - 2*cos(8*b*x + 8*a)*sin(4*b*x + 4*a))*sin(
3/2*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(12*b*x + 12*a) + 2*cos(4*b*x + 4*a)*sin(8*b*x + 8*a) - cos(12*b*x + 12*a)*sin(4*b*x + 4
*a) - 2*cos(8*b*x + 8*a)*sin(4*b*x + 4*a))*cos(3/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)) - (cos(12
*b*x + 12*a)*cos(4*b*x + 4*a) + 2*cos(8*b*x + 8*a)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + sin(12*b*x + 12*a)*
sin(4*b*x + 4*a) + 2*sin(8*b*x + 8*a)*sin(4*b*x + 4*a) + sin(4*b*x + 4*a)^2)*sin(3/2*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))))/((cos(4*b*x + 4*a)^4 + sin(4*b
*x + 4*a)^4 + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*cos(12*b*x + 12*a)^2 + 4*(cos
(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*cos(8*b*x + 8*a)^2 + 2*cos(4*b*x + 4*a)^3 + (co
s(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(12*b*x + 12*a)^2 + 4*(cos(4*b*x + 4*a)^2 +
sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(8*b*x + 8*a)^2 + (2*cos(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a)
+ 1)*sin(4*b*x + 4*a)^2 + 2*(cos(4*b*x + 4*a)^3 + cos(4*b*x + 4*a)*sin(4*b*x + 4*a)^2 + 2*(cos(4*b*x + 4*a)^2
+ sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*cos(8*b*x + 8*a) + 2*cos(4*b*x + 4*a)^2 + cos(4*b*x + 4*a))*cos
(12*b*x + 12*a) + 4*(cos(4*b*x + 4*a)^3 + cos(4*b*x + 4*a)*sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a)^2 + cos(4*b
*x + 4*a))*cos(8*b*x + 8*a) + cos(4*b*x + 4*a)^2 + 2*(sin(4*b*x + 4*a)^3 + 2*(cos(4*b*x + 4*a)^2 + sin(4*b*x +
4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(8*b*x + 8*a) + (cos(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(4*b*x +
4*a))*sin(12*b*x + 12*a) + 4*(sin(4*b*x + 4*a)^3 + (cos(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(4*b*x +
4*a))*sin(8*b*x + 8*a))*cos(3/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1))^2 + (cos(4*b*x + 4*a)^4 + si
n(4*b*x + 4*a)^4 + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*cos(12*b*x + 12*a)^2 + 4
*(cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*cos(8*b*x + 8*a)^2 + 2*cos(4*b*x + 4*a)^3
+ (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(12*b*x + 12*a)^2 + 4*(cos(4*b*x + 4*a
)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(8*b*x + 8*a)^2 + (2*cos(4*b*x + 4*a)^2 + 2*cos(4*b*x +
4*a) + 1)*sin(4*b*x + 4*a)^2 + 2*(cos(4*b*x + 4*a)^3 + cos(4*b*x + 4*a)*sin(4*b*x + 4*a)^2 + 2*(cos(4*b*x + 4*
a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*cos(8*b*x + 8*a) + 2*cos(4*b*x + 4*a)^2 + cos(4*b*x + 4*a)
)*cos(12*b*x + 12*a) + 4*(cos(4*b*x + 4*a)^3 + cos(4*b*x + 4*a)*sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a)^2 + co
s(4*b*x + 4*a))*cos(8*b*x + 8*a) + cos(4*b*x + 4*a)^2 + 2*(sin(4*b*x + 4*a)^3 + 2*(cos(4*b*x + 4*a)^2 + sin(4*
b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(8*b*x + 8*a) + (cos(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(4*
b*x + 4*a))*sin(12*b*x + 12*a) + 4*(sin(4*b*x + 4*a)^3 + (cos(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(4*b
*x + 4*a))*sin(8*b*x + 8*a))*sin(3/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1))^2), x) + 3*(b*c*cos(4*b
*x + 4*a)^2 + b*c*sin(4*b*x + 4*a)^2 + 2*b*c*cos(4*b*x + 4*a) + b*c)*integrate(-(cos(4*b*x + 4*a)^2 + sin(4*b*
x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)^(1/4)*(((cos(12*b*x + 12*a)*cos(4*b*x + 4*a) + 2*cos(8*b*x + 8*a)*cos(4*b
*x + 4*a) + cos(4*b*x + 4*a)^2 + sin(12*b*x + 12*a)*sin(4*b*x + 4*a) + 2*sin(8*b*x + 8*a)*sin(4*b*x + 4*a) + s
in(4*b*x + 4*a)^2)*cos(3/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)) + (cos(4*b*x + 4*a)*sin(12*b*x +
12*a) + 2*cos(4*b*x + 4*a)*sin(8*b*x + 8*a) - cos(12*b*x + 12*a)*sin(4*b*x + 4*a) - 2*cos(8*b*x + 8*a)*sin(4*b
*x + 4*a))*sin(3/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)))*cos(3/2*arctan2(sin(4*b*x + 4*a), cos(4*
b*x + 4*a))) + ((cos(4*b*x + 4*a)*sin(12*b*x + 12*a) + 2*cos(4*b*x + 4*a)*sin(8*b*x + 8*a) - cos(12*b*x + 12*a
)*sin(4*b*x + 4*a) - 2*cos(8*b*x + 8*a)*sin(4*b*x + 4*a))*cos(3/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a)
- 1)) - (cos(12*b*x + 12*a)*cos(4*b*x + 4*a) + 2*cos(8*b*x + 8*a)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + sin(
12*b*x + 12*a)*sin(4*b*x + 4*a) + 2*sin(8*b*x + 8*a)*sin(4*b*x + 4*a) + sin(4*b*x + 4*a)^2)*sin(3/2*arctan2(si
n(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)))*sin(3/2*arctan2(sin(4*b*x + 4*a), cos(4*b*x + 4*a))))/((cos(4*b*x + 4
*a)^4 + sin(4*b*x + 4*a)^4 + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*cos(12*b*x + 1
2*a)^2 + 4*(cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*cos(8*b*x + 8*a)^2 + 2*cos(4*b*x
+ 4*a)^3 + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(12*b*x + 12*a)^2 + 4*(cos(4
*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(8*b*x + 8*a)^2 + (2*cos(4*b*x + 4*a)^2 + 2*co
s(4*b*x + 4*a) + 1)*sin(4*b*x + 4*a)^2 + 2*(cos(4*b*x + 4*a)^3 + cos(4*b*x + 4*a)*sin(4*b*x + 4*a)^2 + 2*(cos(
4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*cos(8*b*x + 8*a) + 2*cos(4*b*x + 4*a)^2 + cos(4*
b*x + 4*a))*cos(12*b*x + 12*a) + 4*(cos(4*b*x + 4*a)^3 + cos(4*b*x + 4*a)*sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4
*a)^2 + cos(4*b*x + 4*a))*cos(8*b*x + 8*a) + cos(4*b*x + 4*a)^2 + 2*(sin(4*b*x + 4*a)^3 + 2*(cos(4*b*x + 4*a)^
2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(8*b*x + 8*a) + (cos(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) +
1)*sin(4*b*x + 4*a))*sin(12*b*x + 12*a) + 4*(sin(4*b*x + 4*a)^3 + (cos(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) +
1)*sin(4*b*x + 4*a))*sin(8*b*x + 8*a))*cos(3/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1))^2 + (cos(4*b*
x + 4*a)^4 + sin(4*b*x + 4*a)^4 + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*cos(12*b*
x + 12*a)^2 + 4*(cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*cos(8*b*x + 8*a)^2 + 2*cos(
4*b*x + 4*a)^3 + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(12*b*x + 12*a)^2 + 4*(
cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(8*b*x + 8*a)^2 + (2*cos(4*b*x + 4*a)^2 +
2*cos(4*b*x + 4*a) + 1)*sin(4*b*x + 4*a)^2 + 2*(cos(4*b*x + 4*a)^3 + cos(4*b*x + 4*a)*sin(4*b*x + 4*a)^2 + 2*
(cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*cos(8*b*x + 8*a) + 2*cos(4*b*x + 4*a)^2 + c
os(4*b*x + 4*a))*cos(12*b*x + 12*a) + 4*(cos(4*b*x + 4*a)^3 + cos(4*b*x + 4*a)*sin(4*b*x + 4*a)^2 + 2*cos(4*b*
x + 4*a)^2 + cos(4*b*x + 4*a))*cos(8*b*x + 8*a) + cos(4*b*x + 4*a)^2 + 2*(sin(4*b*x + 4*a)^3 + 2*(cos(4*b*x +
4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(8*b*x + 8*a) + (cos(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4
*a) + 1)*sin(4*b*x + 4*a))*sin(12*b*x + 12*a) + 4*(sin(4*b*x + 4*a)^3 + (cos(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*
a) + 1)*sin(4*b*x + 4*a))*sin(8*b*x + 8*a))*sin(3/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1))^2), x))*
sqrt(c) + (5*c*cos(5/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1))*sin(4*b*x + 4*a) + (5*c*cos(4*b*x + 4
*a) + 3*c)*sin(5/2*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)^
(1/4))
Giac [F(-1)]
Timed out. \[
\int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\text {Timed out}
\]
[In]
integrate(sec(2*b*x+2*a)^2*(c*tan(b*x+a)*tan(2*b*x+2*a))^(3/2),x, algorithm="giac")
[Out]
Timed out
Mupad [B] (verification not implemented)
Time = 45.17 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.35
\[
\int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\frac {2\,c\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}\,5{}\mathrm {i}+{\mathrm {e}}^{a\,6{}\mathrm {i}+b\,x\,6{}\mathrm {i}}\,5{}\mathrm {i}+{\mathrm {e}}^{a\,10{}\mathrm {i}+b\,x\,10{}\mathrm {i}}\,3{}\mathrm {i}+3{}\mathrm {i}\right )\,\sqrt {\frac {c\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+1\right )}}}{5\,b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+1\right )}^2}
\]
[In]
int((c*tan(a + b*x)*tan(2*a + 2*b*x))^(3/2)/cos(2*a + 2*b*x)^2,x)
[Out]
(2*c*(exp(a*4i + b*x*4i)*5i + exp(a*6i + b*x*6i)*5i + exp(a*10i + b*x*10i)*3i + 3i)*((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))/(5*b*(exp(a*2
i + b*x*2i) - 1)*(exp(a*4i + b*x*4i) + 1)^2)