\(\int (a+b \sec (c+d x))^{4/3} \, dx\) [693]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int (a+b \sec (c+d x))^{4/3} \, dx=\text {Int}\left ((a+b \sec (c+d x))^{4/3},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (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 = {} \[ \int (a+b \sec (c+d x))^{4/3} \, dx=\int (a+b \sec (c+d x))^{4/3} \, dx \]

[In]

Int[(a + b*Sec[c + d*x])^(4/3),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 100.65 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int (a+b \sec (c+d x))^{4/3} \, dx=\int (a+b \sec (c+d x))^{4/3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.31 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 13.31 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int (a+b \sec (c+d x))^{4/3} \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{\frac {4}{3}}\, dx \]

[In]

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

[Out]

Integral((a + b*sec(c + d*x))**(4/3), x)

Maxima [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int (a+b \sec (c+d x))^{4/3} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.14 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int (a+b \sec (c+d x))^{4/3} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 15.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int (a+b \sec (c+d x))^{4/3} \, dx=\int {\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{4/3} \,d x \]

[In]

int((a + b/cos(c + d*x))^(4/3),x)

[Out]

int((a + b/cos(c + d*x))^(4/3), x)