\(\int (c+d x)^m (a+a \sec (e+f x))^n \, dx\) [21]

Optimal result

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

[Out]

Unintegrable((d*x+c)^m*(a+a*sec(f*x+e))^n,x)

Rubi [N/A]

Not integrable

Time = 0.05 (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 = {} \[ \int (c+d x)^m (a+a \sec (e+f x))^n \, dx=\int (c+d x)^m (a+a \sec (e+f x))^n \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (c+d x)^m (a+a \sec (e+f x))^n \, dx=\int (c+d x)^m (a+a \sec (e+f x))^n \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.32 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (c+d x)^m (a+a \sec (e+f x))^n \, dx=\int { {\left (d x + c\right )}^{m} {\left (a \sec \left (f x + e\right ) + a\right )}^{n} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int (c+d x)^m (a+a \sec (e+f x))^n \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (c+d x)^m (a+a \sec (e+f x))^n \, dx=\int { {\left (d x + c\right )}^{m} {\left (a \sec \left (f x + e\right ) + a\right )}^{n} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (c+d x)^m (a+a \sec (e+f x))^n \, dx=\int { {\left (d x + c\right )}^{m} {\left (a \sec \left (f x + e\right ) + a\right )}^{n} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 13.49 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int (c+d x)^m (a+a \sec (e+f x))^n \, dx=\int {\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^n\,{\left (c+d\,x\right )}^m \,d x \]

[In]

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

[Out]

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