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

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\sqrt{\tan (c+d x)} (a+b \sec (c+d x))^n,x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0428503, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (a+b \sec (c+d x))^n \sqrt{\tan (c+d x)} \, dx \]

Verification is Not applicable to the result.

[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*} \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\\ \end{align*}

Mathematica [A]  time = 5.28421, size = 0, normalized size = 0. \[ \int (a+b \sec (c+d x))^n \sqrt{\tan (c+d x)} \, dx \]

Verification is Not applicable to the result.

[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 [A]  time = 0.318, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\sec \left ( dx+c \right ) \right ) ^{n}\sqrt{\tan \left ( dx+c \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sqrt{\tan \left (d x + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sqrt{\tan \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sec{\left (c + d x \right )}\right )^{n} \sqrt{\tan{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sqrt{\tan \left (d x + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)