3.17 \(\int (b \coth ^2(c+d x))^n \, dx\)

Optimal. Leaf size=57 \[ \frac{\coth (c+d x) \left (b \coth ^2(c+d x)\right )^n \, _2F_1\left (1,\frac{1}{2} (2 n+1);\frac{1}{2} (2 n+3);\coth ^2(c+d x)\right )}{d (2 n+1)} \]

[Out]

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

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Rubi [A]  time = 0.040331, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3658, 3476, 364} \[ \frac{\coth (c+d x) \left (b \coth ^2(c+d x)\right )^n \, _2F_1\left (1,\frac{1}{2} (2 n+1);\frac{1}{2} (2 n+3);\coth ^2(c+d x)\right )}{d (2 n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(b*Coth[c + d*x]^2)^n,x]

[Out]

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

Rule 3658

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Tan[e + f*x]^n)^FracPart[p])/(Tan[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

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 \left (b \coth ^2(c+d x)\right )^n \, dx &=\left (\coth ^{-2 n}(c+d x) \left (b \coth ^2(c+d x)\right )^n\right ) \int \coth ^{2 n}(c+d x) \, dx\\ &=-\frac{\left (\coth ^{-2 n}(c+d x) \left (b \coth ^2(c+d x)\right )^n\right ) \operatorname{Subst}\left (\int \frac{x^{2 n}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac{\coth (c+d x) \left (b \coth ^2(c+d x)\right )^n \, _2F_1\left (1,\frac{1}{2} (1+2 n);\frac{1}{2} (3+2 n);\coth ^2(c+d x)\right )}{d (1+2 n)}\\ \end{align*}

Mathematica [A]  time = 0.0478515, size = 47, normalized size = 0.82 \[ \frac{\coth (c+d x) \left (b \coth ^2(c+d x)\right )^n \, _2F_1\left (1,n+\frac{1}{2};n+\frac{3}{2};\coth ^2(c+d x)\right )}{2 d n+d} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Coth[c + d*x]^2)^n,x]

[Out]

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

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Maple [F]  time = 0.247, size = 0, normalized size = 0. \begin{align*} \int \left ( b \left ({\rm coth} \left (dx+c\right ) \right ) ^{2} \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*coth(d*x+c)^2)^n,x)

[Out]

int((b*coth(d*x+c)^2)^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 )^{2}\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*coth(d*x+c)^2)^n,x, algorithm="maxima")

[Out]

integrate((b*coth(d*x + c)^2)^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 )^{2}\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*coth(d*x+c)^2)^n,x, algorithm="fricas")

[Out]

integral((b*coth(d*x + c)^2)^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \coth ^{2}{\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)**2)**n,x)

[Out]

Integral((b*coth(c + d*x)**2)**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 )^{2}\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*coth(d*x+c)^2)^n,x, algorithm="giac")

[Out]

integrate((b*coth(d*x + c)^2)^n, x)