3.3.26 \(\int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx\) [226]

3.3.26.1 Optimal result

Integrand size = 29, antiderivative size = 29 \[ \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx=\text {Int}\left (\frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}},x\right ) \]

output

Unintegrable((a+b*sec(f*x+e))^(2/3)/(c+d*sec(f*x+e))^(8/3),x)
 

3.3.26.2 Mathematica [N/A]

Not integrable

Time = 87.72 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx=\int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx \]

input

Integrate[(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(8/3),x]
 

output

Integrate[(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(8/3), x]
 

3.3.26.3 Rubi [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {3042, 4433}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a+b \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{2/3}}{\left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{8/3}}dx\)

\(\Big \downarrow \) 4433

\(\displaystyle \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}}dx\)

input

Int[(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(8/3),x]
 

output

$Aborted
 

3.3.26.3.1 Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4433

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*( 
d_.) + (c_))^(n_.), x_Symbol] :> Unintegrable[(a + b*Csc[e + f*x])^m*(c + d 
*Csc[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

3.3.26.4 Maple [N/A] (verified)

Not integrable

Time = 0.59 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86

\[\int \frac {\left (a +b \sec \left (f x +e \right )\right )^{\frac {2}{3}}}{\left (c +d \sec \left (f x +e \right )\right )^{\frac {8}{3}}}d x\]

input

int((a+b*sec(f*x+e))^(2/3)/(c+d*sec(f*x+e))^(8/3),x)
 

output

int((a+b*sec(f*x+e))^(2/3)/(c+d*sec(f*x+e))^(8/3),x)
 

3.3.26.5 Fricas [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx=\text {Timed out} \]

input

integrate((a+b*sec(f*x+e))^(2/3)/(c+d*sec(f*x+e))^(8/3),x, algorithm="fric 
as")
 

output

Timed out
 

3.3.26.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx=\text {Timed out} \]

input

integrate((a+b*sec(f*x+e))**(2/3)/(c+d*sec(f*x+e))**(8/3),x)
 

output

Timed out
 

3.3.26.7 Maxima [N/A]

Not integrable

Time = 1.40 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx=\int { \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {2}{3}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {8}{3}}} \,d x } \]

input

integrate((a+b*sec(f*x+e))^(2/3)/(c+d*sec(f*x+e))^(8/3),x, algorithm="maxi 
ma")
 

output

integrate((b*sec(f*x + e) + a)^(2/3)/(d*sec(f*x + e) + c)^(8/3), x)
 

3.3.26.8 Giac [N/A]

Not integrable

Time = 5.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx=\int { \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {2}{3}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {8}{3}}} \,d x } \]

input

integrate((a+b*sec(f*x+e))^(2/3)/(c+d*sec(f*x+e))^(8/3),x, algorithm="giac 
")
 

output

integrate((b*sec(f*x + e) + a)^(2/3)/(d*sec(f*x + e) + c)^(8/3), x)
 

3.3.26.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (e+f x))^{2/3}}{(c+d \sec (e+f x))^{8/3}} \, dx=\text {Hanged} \]

input

int((a + b/cos(e + f*x))^(2/3)/(c + d/cos(e + f*x))^(8/3),x)
 

output

\text{Hanged}