3.3.100 \(\int (-\sec (e+f x))^n \sqrt {1+\sec (e+f x)} \, dx\) [300]

Optimal. Leaf size=64 \[ -\frac {\, _2F_1\left (\frac {1}{2},n;1+n;\sec (e+f x)\right ) (-\sec (e+f x))^n \tan (e+f x)}{f n \sqrt {1-\sec (e+f x)} \sqrt {1+\sec (e+f x)}} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, 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 = {3891, 66} \begin {gather*} -\frac {\tan (e+f x) (-\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], 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])))

Rule 3891

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 {1+\sec (e+f x)} \, dx &=\frac {\tan (e+f x) \text {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 {\, _2F_1\left (\frac {1}{2},n;1+n;\sec (e+f x)\right ) (-\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*}

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Mathematica [A]
time = 0.06, size = 67, normalized size = 1.05 \begin {gather*} \frac {2 \, _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 {1+\sec (e+f x)}} \end {gather*}

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]]*(-Sec[e + f*x])^n*Sec[e + f*x]^(1 - n)*Sin[e + f*x])/(
f*Sqrt[1 + Sec[e + f*x]])

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \left (-\sec \left (f x +e \right )\right )^{n} \sqrt {1+\sec \left (f x +e \right )}\, dx\]

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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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