3.286 \(\int (b \csc (e+f x))^n \sec (e+f x) \, dx\)

Optimal. Leaf size=48 \[ \frac {(b \csc (e+f x))^{n+1} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\csc ^2(e+f x)\right )}{b f (n+1)} \]

[Out]

(b*csc(f*x+e))^(1+n)*hypergeom([1, 1/2+1/2*n],[3/2+1/2*n],csc(f*x+e)^2)/b/f/(1+n)

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Rubi [A]  time = 0.04, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2621, 364} \[ \frac {(b \csc (e+f x))^{n+1} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\csc ^2(e+f x)\right )}{b f (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(b*Csc[e + f*x])^n*Sec[e + f*x],x]

[Out]

((b*Csc[e + f*x])^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Csc[e + f*x]^2])/(b*f*(1 + n))

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2621

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 51, normalized size = 1.06 \[ -\frac {b (b \csc (e+f x))^{n-1} \, _2F_1\left (1,\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right )}{f (n-1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Csc[e + f*x])^n*Sec[e + f*x],x]

[Out]

-((b*(b*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[1, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2])/(f*(-1 + n)))

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fricas [F]  time = 1.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \csc \left (f x + e\right )\right )^{n} \sec \left (f x + e\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*csc(f*x+e))^n*sec(f*x+e),x, algorithm="fricas")

[Out]

integral((b*csc(f*x + e))^n*sec(f*x + e), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*csc(f*x+e))^n*sec(f*x+e),x, algorithm="giac")

[Out]

integrate((b*csc(f*x + e))^n*sec(f*x + e), x)

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maple [F]  time = 1.16, size = 0, normalized size = 0.00 \[ \int \left (b \csc \left (f x +e \right )\right )^{n} \sec \left (f x +e \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*csc(f*x+e))^n*sec(f*x+e),x)

[Out]

int((b*csc(f*x+e))^n*sec(f*x+e),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*csc(f*x+e))^n*sec(f*x+e),x, algorithm="maxima")

[Out]

integrate((b*csc(f*x + e))^n*sec(f*x + e), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b/sin(e + f*x))^n/cos(e + f*x),x)

[Out]

int((b/sin(e + f*x))^n/cos(e + f*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*csc(f*x+e))**n*sec(f*x+e),x)

[Out]

Integral((b*csc(e + f*x))**n*sec(e + f*x), x)

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