\(\int \frac {(a+b \sec (c+d x))^n}{\sqrt {\tan (c+d x)}} \, dx\) [364]

Optimal result

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

[Out]

Unintegrable((a+b*sec(d*x+c))^n/tan(d*x+c)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.06 (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 = {} \[ \int \frac {(a+b \sec (c+d x))^n}{\sqrt {\tan (c+d x)}} \, dx=\int \frac {(a+b \sec (c+d x))^n}{\sqrt {\tan (c+d x)}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 23.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(a+b \sec (c+d x))^n}{\sqrt {\tan (c+d x)}} \, dx=\int \frac {(a+b \sec (c+d x))^n}{\sqrt {\tan (c+d x)}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.33 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

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

[In]

int((a+b*sec(d*x+c))^n/tan(d*x+c)^(1/2),x)

[Out]

int((a+b*sec(d*x+c))^n/tan(d*x+c)^(1/2),x)

Fricas [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \sec (c+d x))^n}{\sqrt {\tan (c+d x)}} \, dx=\int { \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\tan \left (d x + c\right )}} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))^n/tan(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.49 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b \sec (c+d x))^n}{\sqrt {\tan (c+d x)}} \, dx=\int \frac {\left (a + b \sec {\left (c + d x \right )}\right )^{n}}{\sqrt {\tan {\left (c + d x \right )}}}\, dx \]

[In]

integrate((a+b*sec(d*x+c))**n/tan(d*x+c)**(1/2),x)

[Out]

Integral((a + b*sec(c + d*x))**n/sqrt(tan(c + d*x)), x)

Maxima [N/A]

Not integrable

Time = 1.93 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \sec (c+d x))^n}{\sqrt {\tan (c+d x)}} \, dx=\int { \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\tan \left (d x + c\right )}} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))^n/tan(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

integrate((b*sec(d*x + c) + a)^n/sqrt(tan(d*x + c)), x)

Giac [N/A]

Not integrable

Time = 0.75 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \sec (c+d x))^n}{\sqrt {\tan (c+d x)}} \, dx=\int { \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\tan \left (d x + c\right )}} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))^n/tan(d*x+c)^(1/2),x, algorithm="giac")

[Out]

integrate((b*sec(d*x + c) + a)^n/sqrt(tan(d*x + c)), x)

Mupad [N/A]

Not integrable

Time = 18.95 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(a+b \sec (c+d x))^n}{\sqrt {\tan (c+d x)}} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}} \,d x \]

[In]

int((a + b/cos(c + d*x))^n/tan(c + d*x)^(1/2),x)

[Out]

int((a + b/cos(c + d*x))^n/tan(c + d*x)^(1/2), x)