\(\int (b \coth ^2(c+d x))^n \, dx\) [17]

Optimal result

Integrand size = 12, antiderivative size = 57 \[ \int \left (b \coth ^2(c+d x)\right )^n \, dx=\frac {\coth (c+d x) \left (b \coth ^2(c+d x)\right )^n \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+2 n),\frac {1}{2} (3+2 n),\coth ^2(c+d x)\right )}{d (1+2 n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3739, 3557, 371} \[ \int \left (b \coth ^2(c+d x)\right )^n \, dx=\frac {\coth (c+d x) \left (b \coth ^2(c+d x)\right )^n \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (2 n+1),\frac {1}{2} (2 n+3),\coth ^2(c+d x)\right )}{d (2 n+1)} \]

[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 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]

Rule 3739

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
]])

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 \operatorname {Hypergeometric2F1}\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] (verified)

Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int \left (b \coth ^2(c+d x)\right )^n \, dx=\frac {\coth (c+d x) \left (b \coth ^2(c+d x)\right )^n \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+n,\frac {3}{2}+n,\coth ^2(c+d x)\right )}{d (1+2 n)} \]

[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*(1 + 2*n))

Maple [F]

\[\int \left (\coth \left (d x +c \right )^{2} b \right )^{n}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \left (b \coth ^2(c+d x)\right )^n \, dx=\int { \left (b \coth \left (d x + c\right )^{2}\right )^{n} \,d x } \]

[In]

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

[Out]

integral((b*coth(d*x + c)^2)^n, x)

Sympy [F]

\[ \int \left (b \coth ^2(c+d x)\right )^n \, dx=\int \left (b \coth ^{2}{\left (c + d x \right )}\right )^{n}\, dx \]

[In]

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

[Out]

Integral((b*coth(c + d*x)**2)**n, x)

Maxima [F]

\[ \int \left (b \coth ^2(c+d x)\right )^n \, dx=\int { \left (b \coth \left (d x + c\right )^{2}\right )^{n} \,d x } \]

[In]

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

[Out]

integrate((b*coth(d*x + c)^2)^n, x)

Giac [F]

\[ \int \left (b \coth ^2(c+d x)\right )^n \, dx=\int { \left (b \coth \left (d x + c\right )^{2}\right )^{n} \,d x } \]

[In]

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

[Out]

integrate((b*coth(d*x + c)^2)^n, x)

Mupad [F(-1)]

Timed out. \[ \int \left (b \coth ^2(c+d x)\right )^n \, dx=\int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^2\right )}^n \,d x \]

[In]

int((b*coth(c + d*x)^2)^n,x)

[Out]

int((b*coth(c + d*x)^2)^n, x)