3.15 \(\int \coth ^n(a+b x) \, dx\)

Optimal. Leaf size=43 \[ \frac{\coth ^{n+1}(a+b x) \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};\coth ^2(a+b x)\right )}{b (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0223006, antiderivative size = 43, 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 = {3476, 364} \[ \frac{\coth ^{n+1}(a+b x) \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};\coth ^2(a+b x)\right )}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[Coth[a + b*x]^n,x]

[Out]

(Coth[a + b*x]^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Coth[a + b*x]^2])/(b*(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 \coth ^n(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^n}{-1+x^2} \, dx,x,\coth (a+b x)\right )}{b}\\ &=\frac{\coth ^{1+n}(a+b x) \, _2F_1\left (1,\frac{1+n}{2};\frac{3+n}{2};\coth ^2(a+b x)\right )}{b (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0625768, size = 45, normalized size = 1.05 \[ \frac{\coth ^{n+1}(a+b x) \, _2F_1\left (1,\frac{n+1}{2};\frac{n+1}{2}+1;\coth ^2(a+b x)\right )}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[a + b*x]^n,x]

[Out]

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

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Maple [F]  time = 0.165, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm coth} \left (bx+a\right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(coth(b*x + a)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(coth(b*x + a)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(a + b*x)**n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(coth(b*x + a)^n, x)