3.313 \(\int (-\sec (e+f x))^n \sqrt{a+a \sec (e+f x)} \, dx\)

Optimal. Leaf size=70 \[ \frac{2 a \sin (e+f x) (-\sec (e+f x))^n \sec ^{1-n}(e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},1-n,\frac{3}{2},1-\sec (e+f x)\right )}{f \sqrt{a \sec (e+f x)+a}} \]

[Out]

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

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Rubi [A]  time = 0.0764576, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {3806, 67, 65} \[ \frac{2 a \sin (e+f x) (-\sec (e+f x))^n \sec ^{1-n}(e+f x) \, _2F_1\left (\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right )}{f \sqrt{a \sec (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*(-Sec[e + f*x])^n*Sec[e + f*x]^(1 - n)*Sin[e + f*x])
/(f*Sqrt[a + a*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 67

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

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])

Rubi steps

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

Mathematica [A]  time = 0.120182, size = 71, normalized size = 1.01 \[ \frac{2 \tan \left (\frac{1}{2} (e+f x)\right ) \sqrt{a (\sec (e+f x)+1)} (-\sec (e+f x))^n \sec ^{-n}(e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},1-n,\frac{3}{2},1-\sec (e+f x)\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.189, size = 0, normalized size = 0. \begin{align*} \int \left ( -\sec \left ( fx+e \right ) \right ) ^{n}\sqrt{a+a\sec \left ( fx+e \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out