3.93 \(\int \frac {x^m}{\tanh ^{-1}(\tanh (a+b x))^2} \, dx\)

Optimal. Leaf size=65 \[ -\frac {x^m \, _2F_1\left (1,m;m+1;\frac {b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac {x^m}{b \tanh ^{-1}(\tanh (a+b x))} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2168, 2164} \[ -\frac {x^m \, _2F_1\left (1,m;m+1;\frac {b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac {x^m}{b \tanh ^{-1}(\tanh (a+b x))} \]

Antiderivative was successfully verified.

[In]

Int[x^m/ArcTanh[Tanh[a + b*x]]^2,x]

[Out]

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

Rule 2168

Int[(u_)^(m_)*(v_)^(n_.), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(u^(m + 1)*v^
n)/(a*(m + 1)), x] - Dist[(b*n)/(a*(m + 1)), Int[u^(m + 1)*v^(n - 1), x], x] /; NeQ[b*u - a*v, 0]] /; FreeQ[{m
, n}, x] && PiecewiseLinearQ[u, v, x] && NeQ[m, -1] && ((LtQ[m, -1] && GtQ[n, 0] &&  !(ILtQ[m + n, -2] && (Fra
ctionQ[m] || GeQ[2*n + m + 1, 0]))) || (IGtQ[n, 0] && IGtQ[m, 0] && LeQ[n, m]) || (IGtQ[n, 0] &&  !IntegerQ[m]
) || (ILtQ[m, 0] &&  !IntegerQ[n]))

Rubi steps

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

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Mathematica [A]  time = 0.55, size = 51, normalized size = 0.78 \[ \frac {x^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac {b x}{\tanh ^{-1}(\tanh (a+b x))-b x}\right )}{(m+1) \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m/ArcTanh[Tanh[a + b*x]]^2,x]

[Out]

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

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fricas [F]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/arctanh(tanh(b*x+a))^2,x, algorithm="fricas")

[Out]

integral(x^m/arctanh(tanh(b*x + a))^2, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/arctanh(tanh(b*x+a))^2,x, algorithm="giac")

[Out]

integrate(x^m/arctanh(tanh(b*x + a))^2, x)

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maple [F]  time = 2.78, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/arctanh(tanh(b*x+a))^2,x)

[Out]

int(x^m/arctanh(tanh(b*x+a))^2,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/arctanh(tanh(b*x+a))^2,x, algorithm="maxima")

[Out]

integrate(x^m/arctanh(tanh(b*x + a))^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^m}{{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/atanh(tanh(a + b*x))^2,x)

[Out]

int(x^m/atanh(tanh(a + b*x))^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m/atanh(tanh(b*x+a))**2,x)

[Out]

Integral(x**m/atanh(tanh(a + b*x))**2, x)

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