3.784 \(\int \frac{(d \sec (e+f x))^n}{\sqrt{a+b \sec (e+f x)}} \, dx\)

Optimal. Leaf size=27 \[ \text{Unintegrable}\left (\frac{(d \sec (e+f x))^n}{\sqrt{a+b \sec (e+f x)}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0701936, 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 = {} \[ \int \frac{(d \sec (e+f x))^n}{\sqrt{a+b \sec (e+f x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{(d \sec (e+f x))^n}{\sqrt{a+b \sec (e+f x)}} \, dx &=\int \frac{(d \sec (e+f x))^n}{\sqrt{a+b \sec (e+f x)}} \, dx\\ \end{align*}

Mathematica [A]  time = 2.45017, size = 0, normalized size = 0. \[ \int \frac{(d \sec (e+f x))^n}{\sqrt{a+b \sec (e+f x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.193, size = 0, normalized size = 0. \begin{align*} \int{ \left ( d\sec \left ( fx+e \right ) \right ) ^{n}{\frac{1}{\sqrt{a+b\sec \left ( fx+e \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*sec(f*x + e))^n/sqrt(b*sec(f*x + e) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((d*sec(f*x + e))^n/sqrt(b*sec(f*x + e) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(f*x+e))**n/(a+b*sec(f*x+e))**(1/2),x)

[Out]

Integral((d*sec(e + f*x))**n/sqrt(a + b*sec(e + f*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*sec(f*x + e))^n/sqrt(b*sec(f*x + e) + a), x)