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

Optimal. Leaf size=56 \[ \frac {\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac {x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {x^2}{2 b} \]

[Out]

1/2*x^2/b+x*(b*x-arctanh(tanh(b*x+a)))/b^2+(b*x-arctanh(tanh(b*x+a)))^2*ln(arctanh(tanh(b*x+a)))/b^3

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Rubi [A]  time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2159, 2158, 2157, 29} \[ \frac {x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac {x^2}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2/ArcTanh[Tanh[a + b*x]],x]

[Out]

x^2/(2*b) + (x*(b*x - ArcTanh[Tanh[a + b*x]]))/b^2 + ((b*x - ArcTanh[Tanh[a + b*x]])^2*Log[ArcTanh[Tanh[a + b*
x]]])/b^3

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2157

Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Dist[1/c, Subst[Int[x^m, x], x, u], x]] /; FreeQ[m,
x] && PiecewiseLinearQ[u, x]

Rule 2158

Int[(v_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(b*x)/a, x] - Dist[(b*u
- a*v)/a, Int[1/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x]

Rule 2159

Int[(v_)^(n_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[v^n/(a*n), x] - Dis
t[(b*u - a*v)/a, Int[v^(n - 1)/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && GtQ[n, 0] && Ne
Q[n, 1]

Rubi steps

\begin {align*} \int \frac {x^2}{\tanh ^{-1}(\tanh (a+b x))} \, dx &=\frac {x^2}{2 b}-\frac {\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right ) \int \frac {x}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{b}\\ &=\frac {x^2}{2 b}+\frac {x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2 \int \frac {1}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{b^2}\\ &=\frac {x^2}{2 b}+\frac {x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{b^3}\\ &=\frac {x^2}{2 b}+\frac {x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^3}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 55, normalized size = 0.98 \[ \frac {\left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^3}-\frac {x \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )}{b^2}+\frac {x^2}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^2/(2*b) - (x*(-(b*x) + ArcTanh[Tanh[a + b*x]]))/b^2 + ((-(b*x) + ArcTanh[Tanh[a + b*x]])^2*Log[ArcTanh[Tanh[
a + b*x]]])/b^3

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fricas [A]  time = 0.38, size = 29, normalized size = 0.52 \[ \frac {b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} \log \left (b x + a\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^2*x^2 - 2*a*b*x + 2*a^2*log(b*x + a))/b^3

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giac [A]  time = 0.18, size = 30, normalized size = 0.54 \[ \frac {a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3}} + \frac {b x^{2} - 2 \, a x}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*log(abs(b*x + a))/b^3 + 1/2*(b*x^2 - 2*a*x)/b^2

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maple [B]  time = 0.14, size = 111, normalized size = 1.98 \[ \frac {x^{2}}{2 b}-\frac {a x}{b^{2}}-\frac {x \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{b^{2}}+\frac {\ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right ) a^{2}}{b^{3}}+\frac {2 \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right ) a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{b^{3}}+\frac {\ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2/b-1/b^2*a*x-1/b^2*x*(arctanh(tanh(b*x+a))-b*x-a)+1/b^3*ln(arctanh(tanh(b*x+a)))*a^2+2/b^3*ln(arctanh(t
anh(b*x+a)))*a*(arctanh(tanh(b*x+a))-b*x-a)+1/b^3*ln(arctanh(tanh(b*x+a)))*(arctanh(tanh(b*x+a))-b*x-a)^2

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maxima [A]  time = 0.52, size = 29, normalized size = 0.52 \[ \frac {a^{2} \log \left (b x + a\right )}{b^{3}} + \frac {b x^{2} - 2 \, a x}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*log(b*x + a)/b^3 + 1/2*(b*x^2 - 2*a*x)/b^2

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mupad [B]  time = 0.27, size = 234, normalized size = 4.18 \[ \frac {x^2}{2\,b}+\frac {x\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}{2\,b^2}+\frac {\ln \left (\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\right )\,\left ({\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2-4\,a\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )+4\,a^2\right )}{4\,b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/atanh(tanh(a + b*x)),x)

[Out]

x^2/(2*b) + (x*(log(2/(exp(2*a)*exp(2*b*x) + 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + 2*
b*x))/(2*b^2) + (log(log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) - log(2/(exp(2*a)*exp(2*b*x) + 1))
)*((2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + log(2/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*x)^2
- 4*a*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + log(2/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*
x) + 4*a^2))/(4*b^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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