\(\int \frac {x^m}{\coth ^{-1}(\tanh (a+b x))} \, dx\) [158]

Optimal result

Integrand size = 13, antiderivative size = 53 \[ \int \frac {x^m}{\coth ^{-1}(\tanh (a+b x))} \, dx=-\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(1+m) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]

[Out]

-x^(1+m)*hypergeom([1, 1+m],[2+m],b*x/(b*x-arccoth(tanh(b*x+a))))/(1+m)/(b*x-arccoth(tanh(b*x+a)))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2195} \[ \int \frac {x^m}{\coth ^{-1}(\tanh (a+b x))} \, dx=-\frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(m+1) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]

[In]

Int[x^m/ArcCoth[Tanh[a + b*x]],x]

[Out]

-((x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (b*x)/(b*x - ArcCoth[Tanh[a + b*x]])])/((1 + m)*(b*x - ArcCoth
[Tanh[a + b*x]])))

Rule 2195

Int[(v_)^(n_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(v^(n + 1)/((n + 1)
*(b*u - a*v)))*Hypergeometric2F1[1, n + 1, n + 2, (-a)*(v/(b*u - a*v))], x] /; NeQ[b*u - a*v, 0]] /; Piecewise
LinearQ[u, v, x] &&  !IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(1+m) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.96 \[ \int \frac {x^m}{\coth ^{-1}(\tanh (a+b x))} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{-b x+\coth ^{-1}(\tanh (a+b x))}\right )}{(1+m) \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )} \]

[In]

Integrate[x^m/ArcCoth[Tanh[a + b*x]],x]

[Out]

(x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*x)/(-(b*x) + ArcCoth[Tanh[a + b*x]]))])/((1 + m)*(-(b*x) +
ArcCoth[Tanh[a + b*x]]))

Maple [F]

\[\int \frac {x^{m}}{\operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )}d x\]

[In]

int(x^m/arccoth(tanh(b*x+a)),x)

[Out]

int(x^m/arccoth(tanh(b*x+a)),x)

Fricas [F]

\[ \int \frac {x^m}{\coth ^{-1}(\tanh (a+b x))} \, dx=\int { \frac {x^{m}}{\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )} \,d x } \]

[In]

integrate(x^m/arccoth(tanh(b*x+a)),x, algorithm="fricas")

[Out]

integral(x^m/arccoth(tanh(b*x + a)), x)

Sympy [F]

\[ \int \frac {x^m}{\coth ^{-1}(\tanh (a+b x))} \, dx=\int \frac {x^{m}}{\operatorname {acoth}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]

[In]

integrate(x**m/acoth(tanh(b*x+a)),x)

[Out]

Integral(x**m/acoth(tanh(a + b*x)), x)

Maxima [F]

\[ \int \frac {x^m}{\coth ^{-1}(\tanh (a+b x))} \, dx=\int { \frac {x^{m}}{\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )} \,d x } \]

[In]

integrate(x^m/arccoth(tanh(b*x+a)),x, algorithm="maxima")

[Out]

integrate(x^m/arccoth(tanh(b*x + a)), x)

Giac [F]

\[ \int \frac {x^m}{\coth ^{-1}(\tanh (a+b x))} \, dx=\int { \frac {x^{m}}{\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )} \,d x } \]

[In]

integrate(x^m/arccoth(tanh(b*x+a)),x, algorithm="giac")

[Out]

integrate(x^m/arccoth(tanh(b*x + a)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^m}{\coth ^{-1}(\tanh (a+b x))} \, dx=\int \frac {x^m}{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )} \,d x \]

[In]

int(x^m/acoth(tanh(a + b*x)),x)

[Out]

int(x^m/acoth(tanh(a + b*x)), x)