3.1.16 \(\int (b \coth (c+d x))^n \, dx\) [16]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3557, 371} \begin {gather*} \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)} \end {gather*}

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 3557

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]

Rubi steps

\begin {align*} \int (b \coth (c+d x))^n \, dx &=-\frac {b \text {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*}

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Mathematica [A]
time = 0.03, size = 51, normalized size = 1.06 \begin {gather*} \frac {\coth (c+d x) (b \coth (c+d x))^n \, _2F_1\left (1,\frac {1+n}{2};1+\frac {1+n}{2};\coth ^2(c+d x)\right )}{d (1+n)} \end {gather*}

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 = 1.71, size = 0, normalized size = 0.00 \[\int \left (b \coth \left (d x +c \right )\right )^{n}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \coth {\left (c + d x \right )}\right )^{n}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*coth(c + d*x))^n,x)

[Out]

int((b*coth(c + d*x))^n, x)

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