\(\int \frac {(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx\) [351]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx=\text {Int}\left (\frac {(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 25, 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 = {} \[ \int \frac {(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx \]

[In]

Int[(e*Tan[c + d*x])^m/(a + b*Sec[c + d*x])^(3/2),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 19.97 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.13 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \frac {(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx=\int { \frac {\left (e \tan \left (d x + c\right )\right )^{m}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integral(sqrt(b*sec(d*x + c) + a)*(e*tan(d*x + c))^m/(b^2*sec(d*x + c)^2 + 2*a*b*sec(d*x + c) + a^2), x)

Sympy [N/A]

Not integrable

Time = 4.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {\left (e \tan {\left (c + d x \right )}\right )^{m}}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((e*tan(d*x+c))**m/(a+b*sec(d*x+c))**(3/2),x)

[Out]

Integral((e*tan(c + d*x))**m/(a + b*sec(c + d*x))**(3/2), x)

Maxima [N/A]

Not integrable

Time = 1.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx=\int { \frac {\left (e \tan \left (d x + c\right )\right )^{m}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.42 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx=\int { \frac {\left (e \tan \left (d x + c\right )\right )^{m}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 28.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {{\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^m}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

[In]

int((e*tan(c + d*x))^m/(a + b/cos(c + d*x))^(3/2),x)

[Out]

int((e*tan(c + d*x))^m/(a + b/cos(c + d*x))^(3/2), x)