\(\int \frac {(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{10/3}} \, dx\) [229]

Optimal result

Integrand size = 29, antiderivative size = 29 \[ \int \frac {(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{10/3}} \, dx=\text {Int}\left (\frac {(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{10/3}},x\right ) \]

[Out]

Unintegrable((a+b*sec(f*x+e))^(4/3)/(c+d*sec(f*x+e))^(10/3),x)

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{10/3}} \, dx=\int \frac {(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{10/3}} \, dx \]

[In]

Int[(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(10/3),x]

[Out]

Defer[Int][(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(10/3), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{10/3}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 130.90 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{10/3}} \, dx=\int \frac {(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{10/3}} \, dx \]

[In]

Integrate[(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(10/3),x]

[Out]

Integrate[(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(10/3), x]

Maple [N/A] (verified)

Not integrable

Time = 0.63 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86

\[\int \frac {\left (a +b \sec \left (f x +e \right )\right )^{\frac {4}{3}}}{\left (c +d \sec \left (f x +e \right )\right )^{\frac {10}{3}}}d x\]

[In]

int((a+b*sec(f*x+e))^(4/3)/(c+d*sec(f*x+e))^(10/3),x)

[Out]

int((a+b*sec(f*x+e))^(4/3)/(c+d*sec(f*x+e))^(10/3),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{10/3}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*sec(f*x+e))^(4/3)/(c+d*sec(f*x+e))^(10/3),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{10/3}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*sec(f*x+e))**(4/3)/(c+d*sec(f*x+e))**(10/3),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 2.38 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{10/3}} \, dx=\int { \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {4}{3}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {10}{3}}} \,d x } \]

[In]

integrate((a+b*sec(f*x+e))^(4/3)/(c+d*sec(f*x+e))^(10/3),x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e) + a)^(4/3)/(d*sec(f*x + e) + c)^(10/3), x)

Giac [N/A]

Not integrable

Time = 4.90 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{10/3}} \, dx=\int { \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {4}{3}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {10}{3}}} \,d x } \]

[In]

integrate((a+b*sec(f*x+e))^(4/3)/(c+d*sec(f*x+e))^(10/3),x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e) + a)^(4/3)/(d*sec(f*x + e) + c)^(10/3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (e+f x))^{4/3}}{(c+d \sec (e+f x))^{10/3}} \, dx=\text {Hanged} \]

[In]

int((a + b/cos(e + f*x))^(4/3)/(c + d/cos(e + f*x))^(10/3),x)

[Out]

\text{Hanged}