3.291 \(\int (b \csc (e+f x))^n \sec ^4(e+f x) \, dx\)

Optimal. Leaf size=72 \[ \frac{b \sqrt{\cos ^2(e+f x)} \sec (e+f x) (b \csc (e+f x))^{n-1} \text{Hypergeometric2F1}\left (\frac{5}{2},\frac{1-n}{2},\frac{3-n}{2},\sin ^2(e+f x)\right )}{f (1-n)} \]

[Out]

(b*Sqrt[Cos[e + f*x]^2]*(b*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[5/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2]
*Sec[e + f*x])/(f*(1 - n))

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Rubi [A]  time = 0.0794841, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2631, 2577} \[ \frac{b \sqrt{\cos ^2(e+f x)} \sec (e+f x) (b \csc (e+f x))^{n-1} \, _2F_1\left (\frac{5}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)} \]

Antiderivative was successfully verified.

[In]

Int[(b*Csc[e + f*x])^n*Sec[e + f*x]^4,x]

[Out]

(b*Sqrt[Cos[e + f*x]^2]*(b*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[5/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2]
*Sec[e + f*x])/(f*(1 - n))

Rule 2631

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(a^2*(a*Csc[e
+ f*x])^(m - 1)*(b*Sec[e + f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1)*(b*Cos[e + f*x])^(n + 1))/b^2, Int[1/((a*Si
n[e + f*x])^m*(b*Cos[e + f*x])^n), x], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !SimplerQ[-m, -n]

Rule 2577

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b^(2*IntPart
[(n - 1)/2] + 1)*(b*Cos[e + f*x])^(2*FracPart[(n - 1)/2])*(a*Sin[e + f*x])^(m + 1)*Hypergeometric2F1[(1 + m)/2
, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2])/(a*f*(m + 1)*(Cos[e + f*x]^2)^FracPart[(n - 1)/2]), x] /; FreeQ[{a, b
, e, f, m, n}, x]

Rubi steps

\begin{align*} \int (b \csc (e+f x))^n \sec ^4(e+f x) \, dx &=\left (b^2 (b \csc (e+f x))^{-1+n} (b \sin (e+f x))^{-1+n}\right ) \int \sec ^4(e+f x) (b \sin (e+f x))^{-n} \, dx\\ &=\frac{b \sqrt{\cos ^2(e+f x)} (b \csc (e+f x))^{-1+n} \, _2F_1\left (\frac{5}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right ) \sec (e+f x)}{f (1-n)}\\ \end{align*}

Mathematica [A]  time = 0.508711, size = 77, normalized size = 1.07 \[ \frac{\tan (e+f x) \sec ^2(e+f x)^{-n/2} (b \csc (e+f x))^n \text{Hypergeometric2F1}\left (-\frac{n}{2}-1,\frac{1}{2}-\frac{n}{2},\frac{3}{2}-\frac{n}{2},-\tan ^2(e+f x)\right )}{f (1-n)} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Csc[e + f*x])^n*Sec[e + f*x]^4,x]

[Out]

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

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Maple [F]  time = 0.351, size = 0, normalized size = 0. \begin{align*} \int \left ( b\csc \left ( fx+e \right ) \right ) ^{n} \left ( \sec \left ( fx+e \right ) \right ) ^{4}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*csc(f*x+e))^n*sec(f*x+e)^4,x)

[Out]

int((b*csc(f*x+e))^n*sec(f*x+e)^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \csc \left (f x + e\right )\right )^{n} \sec \left (f x + e\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*csc(f*x+e))^n*sec(f*x+e)^4,x, algorithm="maxima")

[Out]

integrate((b*csc(f*x + e))^n*sec(f*x + e)^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b \csc \left (f x + e\right )\right )^{n} \sec \left (f x + e\right )^{4}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*csc(f*x+e))^n*sec(f*x+e)^4,x, algorithm="fricas")

[Out]

integral((b*csc(f*x + e))^n*sec(f*x + e)^4, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*csc(f*x+e))**n*sec(f*x+e)**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \csc \left (f x + e\right )\right )^{n} \sec \left (f x + e\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*csc(f*x+e))^n*sec(f*x+e)^4,x, algorithm="giac")

[Out]

integrate((b*csc(f*x + e))^n*sec(f*x + e)^4, x)