\(\int (a+b \sec (c+d x))^n \sqrt {\tan (c+d x)} \, dx\) [363]

Optimal result

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

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

[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 (a+b \sec (c+d x))^n \sqrt {\tan (c+d x)} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 10.14 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (a+b \sec (c+d x))^n \sqrt {\tan (c+d x)} \, dx=\int (a+b \sec (c+d x))^n \sqrt {\tan (c+d x)} \, dx \]

[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 = 1.52 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

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

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

[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 = 6.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int (a+b \sec (c+d x))^n \sqrt {\tan (c+d x)} \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{n} \sqrt {\tan {\left (c + d x \right )}}\, dx \]

[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 = 1.75 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int (a+b \sec (c+d x))^n \sqrt {\tan (c+d x)} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sqrt {\tan \left (d x + c\right )} \,d x } \]

[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 = 0.58 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int (a+b \sec (c+d x))^n \sqrt {\tan (c+d x)} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sqrt {\tan \left (d x + c\right )} \,d x } \]

[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 = 14.74 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (a+b \sec (c+d x))^n \sqrt {\tan (c+d x)} \, dx=\int \sqrt {\mathrm {tan}\left (c+d\,x\right )}\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]

[In]

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

[Out]

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