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

Optimal. Leaf size=47 \[ \frac {2 a \tan (e+f x) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};\sec (e+f x)+1\right )}{f \sqrt {a-a \sec (e+f x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3806, 65} \[ \frac {2 a \tan (e+f x) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};\sec (e+f x)+1\right )}{f \sqrt {a-a \sec (e+f x)}} \]

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]]*Tan[e + f*x])/(f*Sqrt[a - a*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 (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 {2 a \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};1+\sec (e+f x)\right ) \tan (e+f x)}{f \sqrt {a-a \sec (e+f x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 72.78, size = 213, normalized size = 4.53 \[ \frac {2^{n-\frac {1}{2}} e^{\frac {1}{2} i (e+f (1-2 n) x)} \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^{n-\frac {1}{2}} \csc \left (\frac {e}{2}+\frac {f x}{2}\right ) \sqrt {a-a \sec (e+f x)} \left ((n+1) e^{i f n x} \, _2F_1\left (1,\frac {1-n}{2};\frac {n+2}{2};-e^{2 i (e+f x)}\right )-n e^{i (e+f (n+1) x)} \, _2F_1\left (1,1-\frac {n}{2};\frac {n+3}{2};-e^{2 i (e+f x)}\right )\right ) (-\sec (e+f x))^n \sec ^{-n-\frac {1}{2}}(e+f x)}{f n (n+1)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-a \sec \left (f x + e\right ) + a} \left (-\sec \left (f x + e\right )\right )^{n}, x\right ) \]

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)

________________________________________________________________________________________

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

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]

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

________________________________________________________________________________________

maple [F]  time = 1.13, size = 0, normalized size = 0.00 \[ \int \left (-\sec \left (f x +e \right )\right )^{n} \sqrt {a -a \sec \left (f x +e \right )}\, dx \]

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________