3.16 \(\int (b \coth (c+d x))^n \, dx\)

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

[Out]

((b*Coth[c + d*x])^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Coth[c + d*x]^2])/(b*d*(1 + n))

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Rubi [A]  time = 0.0270079, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3476, 364} \[ \frac{(b \coth (c+d x))^{n+1} \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};\coth ^2(c+d x)\right )}{b d (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(b*Coth[c + d*x])^n,x]

[Out]

((b*Coth[c + d*x])^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Coth[c + d*x]^2])/(b*d*(1 + n))

Rule 3476

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[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 (b \coth (c+d x))^n \, dx &=-\frac{b \operatorname{Subst}\left (\int \frac{x^n}{-b^2+x^2} \, dx,x,b \coth (c+d x)\right )}{d}\\ &=\frac{(b \coth (c+d x))^{1+n} \, _2F_1\left (1,\frac{1+n}{2};\frac{3+n}{2};\coth ^2(c+d x)\right )}{b d (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.041099, size = 51, normalized size = 1.06 \[ \frac{\coth (c+d x) (b \coth (c+d x))^n \, _2F_1\left (1,\frac{n+1}{2};\frac{n+1}{2}+1;\coth ^2(c+d x)\right )}{d (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Coth[c + d*x])^n,x]

[Out]

(Coth[c + d*x]*(b*Coth[c + d*x])^n*Hypergeometric2F1[1, (1 + n)/2, 1 + (1 + n)/2, Coth[c + d*x]^2])/(d*(1 + n)
)

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Maple [F]  time = 0.191, size = 0, normalized size = 0. \begin{align*} \int \left ( b{\rm coth} \left (dx+c\right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*coth(d*x+c))^n,x)

[Out]

int((b*coth(d*x+c))^n,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*coth(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((b*coth(d*x + c))^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*coth(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((b*coth(d*x + c))^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*coth(d*x+c))**n,x)

[Out]

Integral((b*coth(c + d*x))**n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*coth(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((b*coth(d*x + c))^n, x)