3.304 \(\int (d \sec (e+f x))^n \sqrt{1+\sec (e+f x)} \, dx\)

Optimal. Leaf size=64 \[ -\frac{\tan (e+f x) (d \sec (e+f x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},n,n+1,\sec (e+f x)\right )}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}} \]

[Out]

-((Hypergeometric2F1[1/2, n, 1 + n, Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]
*Sqrt[1 + Sec[e + f*x]]))

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Rubi [A]  time = 0.0566329, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3806, 64} \[ -\frac{\tan (e+f x) (d \sec (e+f x))^n \, _2F_1\left (\frac{1}{2},n;n+1;\sec (e+f x)\right )}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}} \]

Antiderivative was successfully verified.

[In]

Int[(d*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]],x]

[Out]

-((Hypergeometric2F1[1/2, n, 1 + n, Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]
*Sqrt[1 + Sec[e + f*x]]))

Rule 3806

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(a^2*d*
Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]]), Subst[Int[(d*x)^(n - 1)/Sqrt[a - b*x], x]
, x, Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0]

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-(d/(b*c)), 0])))

Rubi steps

\begin{align*} \int (d \sec (e+f x))^n \sqrt{1+\sec (e+f x)} \, dx &=-\frac{(d \tan (e+f x)) \operatorname{Subst}\left (\int \frac{(d x)^{-1+n}}{\sqrt{1-x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt{1-\sec (e+f x)} \sqrt{1+\sec (e+f x)}}\\ &=-\frac{\, _2F_1\left (\frac{1}{2},n;1+n;\sec (e+f x)\right ) (d \sec (e+f x))^n \tan (e+f x)}{f n \sqrt{1-\sec (e+f x)} \sqrt{1+\sec (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.0398213, size = 67, normalized size = 1.05 \[ \frac{2 \sin (e+f x) \sec ^{1-n}(e+f x) (d \sec (e+f x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},1-n,\frac{3}{2},1-\sec (e+f x)\right )}{f \sqrt{\sec (e+f x)+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]],x]

[Out]

(2*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Sec[e + f*x]^(1 - n)*(d*Sec[e + f*x])^n*Sin[e + f*x])/
(f*Sqrt[1 + Sec[e + f*x]])

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Maple [F]  time = 0.173, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sec \left ( fx+e \right ) \right ) ^{n}\sqrt{1+\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*(1+sec(f*x+e))^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \sec \left (f x + e\right )\right )^{n} \sqrt{\sec \left (f x + e\right ) + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*sec(f*x + e))^n*sqrt(sec(f*x + e) + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (d \sec \left (f x + e\right )\right )^{n} \sqrt{\sec \left (f x + e\right ) + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((d*sec(f*x + e))^n*sqrt(sec(f*x + e) + 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \sec{\left (e + f x \right )}\right )^{n} \sqrt{\sec{\left (e + f x \right )} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d*sec(e + f*x))**n*sqrt(sec(e + f*x) + 1), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \sec \left (f x + e\right )\right )^{n} \sqrt{\sec \left (f x + e\right ) + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*sec(f*x + e))^n*sqrt(sec(f*x + e) + 1), x)