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

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

[Out]

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

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Rubi [A]  time = 0.0440214, 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 = {} \[ \int \frac{(a+b \sec (c+d x))^n}{\sqrt{\tan (c+d x)}} \, dx \]

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 \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\\ \end{align*}

Mathematica [A]  time = 4.8888, size = 0, normalized size = 0. \[ \int \frac{(a+b \sec (c+d x))^n}{\sqrt{\tan (c+d x)}} \, dx \]

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]

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Maple [A]  time = 0.297, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\sec \left ( dx+c \right ) \right ) ^{n}{\frac{1}{\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)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\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)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\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)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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)