\(\int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx\) [705]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx=\text {Int}\left (\frac {1}{(a+b \sec (c+d x))^{2/3}},x\right ) \]

[Out]

Unintegrable(1/(a+b*sec(d*x+c))^(2/3),x)

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 14, 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 = {} \[ \int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx=\int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx \]

[In]

Int[(a + b*Sec[c + d*x])^(-2/3),x]

[Out]

Defer[Int][(a + b*Sec[c + d*x])^(-2/3), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.73 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx=\int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.34 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86

\[\int \frac {1}{\left (a +b \sec \left (d x +c \right )\right )^{\frac {2}{3}}}d x\]

[In]

int(1/(a+b*sec(d*x+c))^(2/3),x)

[Out]

int(1/(a+b*sec(d*x+c))^(2/3),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx=\text {Timed out} \]

[In]

integrate(1/(a+b*sec(d*x+c))^(2/3),x, algorithm="fricas")

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx=\int \frac {1}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]

[In]

integrate(1/(a+b*sec(d*x+c))**(2/3),x)

[Out]

Integral((a + b*sec(c + d*x))**(-2/3), x)

Maxima [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx=\int { \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}}} \,d x } \]

[In]

integrate(1/(a+b*sec(d*x+c))^(2/3),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.91 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx=\int { \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}}} \,d x } \]

[In]

integrate(1/(a+b*sec(d*x+c))^(2/3),x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 14.44 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(a+b \sec (c+d x))^{2/3}} \, dx=\int \frac {1}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{2/3}} \,d x \]

[In]

int(1/(a + b/cos(c + d*x))^(2/3),x)

[Out]

int(1/(a + b/cos(c + d*x))^(2/3), x)