3.37 \(\int \sec ^n(a+b x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0328775, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3772, 2643} \[ -\frac{\sin (a+b x) \sec ^{n-1}(a+b x) \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(a+b x)\right )}{b (1-n) \sqrt{\sin ^2(a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*x]^n,x]

[Out]

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

Rule 3772

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x])^(n - 1)*((Sin[c + d*x]/b)^(n - 1)
*Int[1/(Sin[c + d*x]/b)^n, x]), x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

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

Mathematica [A]  time = 0.0495195, size = 61, normalized size = 0.87 \[ \frac{\sqrt{-\tan ^2(a+b x)} \csc (a+b x) \sec ^{n-1}(a+b x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n}{2},\frac{n+2}{2},\sec ^2(a+b x)\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[a + b*x]^n,x]

[Out]

(Csc[a + b*x]*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Sec[a + b*x]^2]*Sec[a + b*x]^(-1 + n)*Sqrt[-Tan[a + b*x]^
2])/(b*n)

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Maple [F]  time = 0.291, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( bx+a \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x+a)^n,x)

[Out]

int(sec(b*x+a)^n,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^n,x, algorithm="maxima")

[Out]

integrate(sec(b*x + a)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^n,x, algorithm="fricas")

[Out]

integral(sec(b*x + a)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)**n,x)

[Out]

Integral(sec(a + b*x)**n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^n,x, algorithm="giac")

[Out]

integrate(sec(b*x + a)^n, x)