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

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

[Out]

-hypergeom([1/2, n],[1+n],sec(f*x+e))*(d*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 = 0.06, 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 = {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 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])))

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 (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*}

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Mathematica [A]  time = 0.05, size = 67, normalized size = 1.05 \[ \frac {2 \sin (e+f x) \sec ^{1-n}(e+f x) (d \sec (e+f x))^n \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};1-\sec (e+f x)\right )}{f \sqrt {\sec (e+f x)+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(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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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d \sec \left (f x + e\right )\right )^{n} \sqrt {\sec \left (f x + e\right ) + 1}, x\right ) \]

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

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)

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maple [F]  time = 0.89, size = 0, normalized size = 0.00 \[ \int \left (d \sec \left (f x +e \right )\right )^{n} \sqrt {1+\sec \left (f x +e \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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