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

Optimal. Leaf size=45 \[ \frac {2 \tan (e+f x) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};1-\sec (e+f x)\right )}{f \sqrt {\sec (e+f x)+1}} \]

[Out]

2*hypergeom([1/2, 1-n],[3/2],1-sec(f*x+e))*tan(f*x+e)/f/(1+sec(f*x+e))^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3806, 65} \[ \frac {2 \tan (e+f x) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};1-\sec (e+f x)\right )}{f \sqrt {\sec (e+f x)+1}} \]

Antiderivative was successfully verified.

[In]

Int[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]]*Tan[e + f*x])/(f*Sqrt[1 + Sec[e + f*x]])

Rule 65

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

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]

Rubi steps

\begin {align*} \int \sec ^n(e+f x) \sqrt {1+\sec (e+f x)} \, dx &=-\frac {\tan (e+f x) \operatorname {Subst}\left (\int \frac {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 {2 \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};1-\sec (e+f x)\right ) \tan (e+f x)}{f \sqrt {1+\sec (e+f x)}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 45, normalized size = 1.00 \[ \frac {2 \tan (e+f x) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};1-\sec (e+f x)\right )}{f \sqrt {\sec (e+f x)+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[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]]*Tan[e + f*x])/(f*Sqrt[1 + Sec[e + f*x]])

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fricas [F]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sec \left (f x + e\right )^{n} \sqrt {\sec \left (f x + e\right ) + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sec \left (f x + e\right )^{n} \sqrt {\sec \left (f x + e\right ) + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 1.08, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{n}\left (f x +e \right )\right ) \sqrt {1+\sec \left (f x +e \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sec \left (f x + e\right )^{n} \sqrt {\sec \left (f x + e\right ) + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {\frac {1}{\cos \left (e+f\,x\right )}+1}\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sec {\left (e + f x \right )} + 1} \sec ^{n}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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