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

Optimal. Leaf size=49 \[ -\frac{(b \sec (e+f x))^{n+1} \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};\sec ^2(e+f x)\right )}{b f (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0382708, antiderivative size = 49, normalized size of antiderivative = 1., 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 = {2622, 364} \[ -\frac{(b \sec (e+f x))^{n+1} \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};\sec ^2(e+f x)\right )}{b f (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2622

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

Rubi steps

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

Mathematica [A]  time = 0.33198, size = 92, normalized size = 1.88 \[ \frac{(b \sec (e+f x))^n \left (\, _2F_1(1,-n;1-n;\cos (e+f x))-2^n \sec ^2\left (\frac{1}{2} (e+f x)\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac{1}{2} \cos (e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )}{2 f n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.403, size = 0, normalized size = 0. \begin{align*} \int \csc \left ( fx+e \right ) \left ( b\sec \left ( fx+e \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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