∫ tanh−1 (tanh(a+bx))3 _______________________________

Optimal. Leaf size=60 \[ 3 b^3 x-\frac {3 b \tanh ^{-1}(\tanh (a+b x))^2}{2 x}-\frac {\tanh ^{-1}(\tanh (a+b x))^3}{2 x^2}-3 b^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \log (x) \]

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

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Rubi [A]
time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2199, 2189, 29} \begin {gather*} -3 b^2 \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )-\frac {\tanh ^{-1}(\tanh (a+b x))^3}{2 x^2}-\frac {3 b \tanh ^{-1}(\tanh (a+b x))^2}{2 x}+3 b^3 x \end {gather*}

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

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

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]

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

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Mathematica [A]
time = 0.03, size = 66, normalized size = 1.10 \begin {gather*} b^3 x-\frac {3 b \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2}{x}-\frac {\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^3}{2 x^2}+3 b^2 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right ) \log (x) \end {gather*}

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

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method	result	size

default	\(-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{2 x^{2}}+\frac {3 b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{x}+2 b \left (\ln \left (x \right ) \arctanh \left (\tanh \left (b x +a \right )\right )-b \left (x \ln \left (x \right )-x \right )\right )\right )}{2}\)	\(59\)
risch	\(\text {Expression too large to display}\)	\(4445\)

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

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Maxima [A]
time = 0.35, size = 72, normalized size = 1.20 \begin {gather*} 3 \, {\left (b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right ) \log \left (x\right ) - {\left (b {\left (x + \frac {a}{b}\right )} \log \left (x\right ) - b {\left (x + \frac {a \log \left (x\right )}{b}\right )}\right )} b\right )} b - \frac {3 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{2 \, x} - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{2 \, x^{2}} \end {gather*}

3*(b*arctanh(tanh(b*x + a))*log(x) - (b*(x + a/b)*log(x) - b*(x + a*log(x)/b))*b)*b - 3/2*b*arctanh(tanh(b*x +

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Fricas [A]
time = 0.32, size = 37, normalized size = 0.62 \begin {gather*} \frac {2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} \log \left (x\right ) - 6 \, a^{2} b x - a^{3}}{2 \, x^{2}} \end {gather*}

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{3}}\, dx \end {gather*}

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Giac [A]
time = 0.40, size = 31, normalized size = 0.52 \begin {gather*} b^{3} x + 3 \, a b^{2} \log \left ({\left | x \right |}\right ) - \frac {6 \, a^{2} b x + a^{3}}{2 \, x^{2}} \end {gather*}

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Mupad [B]
time = 0.20, size = 365, normalized size = 6.08 \begin {gather*} \frac {9\,b^2\,\ln \left (\frac {{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{4}-\frac {{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^3}{16\,x^2}-\frac {9\,b^2\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{4}-\frac {3\,b^3\,x}{2}+\frac {{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^3}{16\,x^2}-\frac {3\,b\,{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2}{8\,x}-\frac {3\,b^2\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \left (x\right )}{2}+\frac {3\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2}{16\,x^2}-\frac {3\,{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{16\,x^2}-\frac {3\,b\,{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2}{8\,x}+\frac {3\,b^2\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \left (x\right )}{2}-3\,b^3\,x\,\ln \left (x\right )+\frac {3\,b\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{4\,x} \end {gather*}

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

16*x^2) - (9*b^2*log(1/(exp(2*a)*exp(2*b*x) + 1)))/4 - (3*b^3*x)/2 + log(1/(exp(2*a)*exp(2*b*x) + 1))^3/(16*x^

2) - (3*b*log(1/(exp(2*a)*exp(2*b*x) + 1))^2)/(8*x) - (3*b^2*log(1/(exp(2*a)*exp(2*b*x) + 1))*log(x))/2 + (3*l

og(1/(exp(2*a)*exp(2*b*x) + 1))*log((exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1))^2)/(16*x^2) - (3*log(1/(e

xp(2*a)*exp(2*b*x) + 1))^2*log((exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)))/(16*x^2) - (3*b*log((exp(2*a)

*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1))^2)/(8*x) + (3*b^2*log((exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1))

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

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3.1.61 \(\int \frac {\tanh ^{-1}(\tanh (a+b x))^3}{x^3} \, dx\) [61]