\(\int (a+b \sec (c+d x))^n (e \tan (c+d x))^m \, dx\) [352]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int (a+b \sec (c+d x))^n (e \tan (c+d x))^m \, dx=\text {Int}\left ((a+b \sec (c+d x))^n (e \tan (c+d x))^m,x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 23, 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 = {} \[ \int (a+b \sec (c+d x))^n (e \tan (c+d x))^m \, dx=\int (a+b \sec (c+d x))^n (e \tan (c+d x))^m \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 7.76 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (a+b \sec (c+d x))^n (e \tan (c+d x))^m \, dx=\int (a+b \sec (c+d x))^n (e \tan (c+d x))^m \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.87 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (a+b \sec (c+d x))^n (e \tan (c+d x))^m \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \left (e \tan \left (d x + c\right )\right )^{m} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 33.98 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int (a+b \sec (c+d x))^n (e \tan (c+d x))^m \, dx=\int \left (e \tan {\left (c + d x \right )}\right )^{m} \left (a + b \sec {\left (c + d x \right )}\right )^{n}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.89 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (a+b \sec (c+d x))^n (e \tan (c+d x))^m \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \left (e \tan \left (d x + c\right )\right )^{m} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.83 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (a+b \sec (c+d x))^n (e \tan (c+d x))^m \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \left (e \tan \left (d x + c\right )\right )^{m} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 15.76 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int (a+b \sec (c+d x))^n (e \tan (c+d x))^m \, dx=\int {\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^m\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]

[In]

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

[Out]

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