3.85 \(\int \frac {x^m}{\tanh ^{-1}(\tanh (a+b x))} \, dx\)

Optimal. Leaf size=53 \[ -\frac {x^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{(m+1) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2164} \[ -\frac {x^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{(m+1) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^m/ArcTanh[Tanh[a + b*x]],x]

[Out]

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

Rule 2164

Int[(v_)^(n_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(v^(n + 1)*Hypergeo
metric2F1[1, n + 1, n + 2, -((a*v)/(b*u - a*v))])/((n + 1)*(b*u - a*v)), x] /; NeQ[b*u - a*v, 0]] /; Piecewise
LinearQ[u, v, x] &&  !IntegerQ[n]

Rubi steps

\begin {align*} \int \frac {x^m}{\tanh ^{-1}(\tanh (a+b x))} \, dx &=-\frac {x^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{(1+m) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 51, normalized size = 0.96 \[ \frac {x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac {b x}{\tanh ^{-1}(\tanh (a+b x))-b x}\right )}{(m+1) \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 1.04, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\arctanh \left (\tanh \left (b x +a \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^m}{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/atanh(tanh(a + b*x)),x)

[Out]

int(x^m/atanh(tanh(a + b*x)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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