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

Optimal. Leaf size=28 \[ \text {Int}\left (\frac {(d \sec (e+f x))^n}{\sqrt {a+b \sec (e+f x)}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 3.01, size = 0, normalized size = 0.00 \[ \int \frac {(d \sec (e+f x))^n}{\sqrt {a+b \sec (e+f x)}} \, dx \]

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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fricas [A]  time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (d \sec \left (f x + e\right )\right )^{n}}{\sqrt {b \sec \left (f x + e\right ) + a}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \sec \left (f x + e\right )\right )^{n}}{\sqrt {b \sec \left (f x + e\right ) + a}}\,{d x} \]

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)

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maple [A]  time = 1.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \sec \left (f x +e \right )\right )^{n}}{\sqrt {a +b \sec \left (f x +e \right )}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \sec \left (f x + e\right )\right )^{n}}{\sqrt {b \sec \left (f x + e\right ) + a}}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^n}{\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \sec {\left (e + f x \right )}\right )^{n}}{\sqrt {a + b \sec {\left (e + f x \right )}}}\, dx \]

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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