\(\int (e x)^m (a+b \sec (c+d x^n))^p \, dx\) [71]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int (e x)^m \left (a+b \sec \left (c+d x^n\right )\right )^p \, dx=x^{-m} (e x)^m \text {Int}\left (x^m \left (a+b \sec \left (c+d x^n\right )\right )^p,x\right ) \]

[Out]

(e*x)^m*Unintegrable(x^m*(a+b*sec(c+d*x^n))^p,x)/(x^m)

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 20, 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 = {} \[ \int (e x)^m \left (a+b \sec \left (c+d x^n\right )\right )^p \, dx=\int (e x)^m \left (a+b \sec \left (c+d x^n\right )\right )^p \, dx \]

[In]

Int[(e*x)^m*(a + b*Sec[c + d*x^n])^p,x]

[Out]

((e*x)^m*Defer[Int][x^m*(a + b*Sec[c + d*x^n])^p, x])/x^m

Rubi steps \begin{align*} \text {integral}& = \left (x^{-m} (e x)^m\right ) \int x^m \left (a+b \sec \left (c+d x^n\right )\right )^p \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (e x)^m \left (a+b \sec \left (c+d x^n\right )\right )^p \, dx=\int (e x)^m \left (a+b \sec \left (c+d x^n\right )\right )^p \, dx \]

[In]

Integrate[(e*x)^m*(a + b*Sec[c + d*x^n])^p,x]

[Out]

Integrate[(e*x)^m*(a + b*Sec[c + d*x^n])^p, x]

Maple [N/A] (verified)

Not integrable

Time = 0.93 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \left (e x \right )^{m} {\left (a +b \sec \left (c +d \,x^{n}\right )\right )}^{p}d x\]

[In]

int((e*x)^m*(a+b*sec(c+d*x^n))^p,x)

[Out]

int((e*x)^m*(a+b*sec(c+d*x^n))^p,x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (e x)^m \left (a+b \sec \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{m} {\left (b \sec \left (d x^{n} + c\right ) + a\right )}^{p} \,d x } \]

[In]

integrate((e*x)^m*(a+b*sec(c+d*x^n))^p,x, algorithm="fricas")

[Out]

integral((e*x)^m*(b*sec(d*x^n + c) + a)^p, x)

Sympy [N/A]

Not integrable

Time = 47.83 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int (e x)^m \left (a+b \sec \left (c+d x^n\right )\right )^p \, dx=\int \left (e x\right )^{m} \left (a + b \sec {\left (c + d x^{n} \right )}\right )^{p}\, dx \]

[In]

integrate((e*x)**m*(a+b*sec(c+d*x**n))**p,x)

[Out]

Integral((e*x)**m*(a + b*sec(c + d*x**n))**p, x)

Maxima [N/A]

Not integrable

Time = 2.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (e x)^m \left (a+b \sec \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{m} {\left (b \sec \left (d x^{n} + c\right ) + a\right )}^{p} \,d x } \]

[In]

integrate((e*x)^m*(a+b*sec(c+d*x^n))^p,x, algorithm="maxima")

[Out]

integrate((e*x)^m*(b*sec(d*x^n + c) + a)^p, x)

Giac [N/A]

Not integrable

Time = 0.76 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (e x)^m \left (a+b \sec \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{m} {\left (b \sec \left (d x^{n} + c\right ) + a\right )}^{p} \,d x } \]

[In]

integrate((e*x)^m*(a+b*sec(c+d*x^n))^p,x, algorithm="giac")

[Out]

integrate((e*x)^m*(b*sec(d*x^n + c) + a)^p, x)

Mupad [N/A]

Not integrable

Time = 12.91 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int (e x)^m \left (a+b \sec \left (c+d x^n\right )\right )^p \, dx=\int {\left (a+\frac {b}{\cos \left (c+d\,x^n\right )}\right )}^p\,{\left (e\,x\right )}^m \,d x \]

[In]

int((a + b/cos(c + d*x^n))^p*(e*x)^m,x)

[Out]

int((a + b/cos(c + d*x^n))^p*(e*x)^m, x)