Integrand size = 13, antiderivative size = 95 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^4}{x^2} \, dx=4 b^2 x (b x-\text {arctanh}(\tanh (a+b x)))^2-2 b (b x-\text {arctanh}(\tanh (a+b x))) \text {arctanh}(\tanh (a+b x))^2+\frac {4}{3} b \text {arctanh}(\tanh (a+b x))^3-\frac {\text {arctanh}(\tanh (a+b x))^4}{x}-4 b (b x-\text {arctanh}(\tanh (a+b x)))^3 \log (x) \]
4*b^2*x*(b*x-arctanh(tanh(b*x+a)))^2-2*b*(b*x-arctanh(tanh(b*x+a)))*arctan h(tanh(b*x+a))^2+4/3*b*arctanh(tanh(b*x+a))^3-arctanh(tanh(b*x+a))^4/x-4*b *(b*x-arctanh(tanh(b*x+a)))^3*ln(x)
Time = 0.06 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^4}{x^2} \, dx=-\frac {\text {arctanh}(\tanh (a+b x))^4}{x}+\frac {2}{3} b^4 x^3 (5-6 \log (x))-12 b^2 x \text {arctanh}(\tanh (a+b x))^2 \log (x)+4 b \text {arctanh}(\tanh (a+b x))^3 (1+\log (x))+6 b^3 x^2 \text {arctanh}(\tanh (a+b x)) (-1+2 \log (x)) \]
-(ArcTanh[Tanh[a + b*x]]^4/x) + (2*b^4*x^3*(5 - 6*Log[x]))/3 - 12*b^2*x*Ar cTanh[Tanh[a + b*x]]^2*Log[x] + 4*b*ArcTanh[Tanh[a + b*x]]^3*(1 + Log[x]) + 6*b^3*x^2*ArcTanh[Tanh[a + b*x]]*(-1 + 2*Log[x])
Time = 0.31 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2599, 2590, 2590, 2589, 14}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\text {arctanh}(\tanh (a+b x))^4}{x^2} \, dx\) |
| \(\Big \downarrow \) 2599 |
| \(\displaystyle 4 b \int \frac {\text {arctanh}(\tanh (a+b x))^3}{x}dx-\frac {\text {arctanh}(\tanh (a+b x))^4}{x}\) |
| \(\Big \downarrow \) 2590 |
| \(\displaystyle 4 b \left (\frac {1}{3} \text {arctanh}(\tanh (a+b x))^3-(b x-\text {arctanh}(\tanh (a+b x))) \int \frac {\text {arctanh}(\tanh (a+b x))^2}{x}dx\right )-\frac {\text {arctanh}(\tanh (a+b x))^4}{x}\) |
| \(\Big \downarrow \) 2590 |
| \(\displaystyle 4 b \left (\frac {1}{3} \text {arctanh}(\tanh (a+b x))^3-(b x-\text {arctanh}(\tanh (a+b x))) \left (\frac {1}{2} \text {arctanh}(\tanh (a+b x))^2-(b x-\text {arctanh}(\tanh (a+b x))) \int \frac {\text {arctanh}(\tanh (a+b x))}{x}dx\right )\right )-\frac {\text {arctanh}(\tanh (a+b x))^4}{x}\) |
| \(\Big \downarrow \) 2589 |
| \(\displaystyle 4 b \left (\frac {1}{3} \text {arctanh}(\tanh (a+b x))^3-(b x-\text {arctanh}(\tanh (a+b x))) \left (\frac {1}{2} \text {arctanh}(\tanh (a+b x))^2-(b x-\text {arctanh}(\tanh (a+b x))) \left (b x-(b x-\text {arctanh}(\tanh (a+b x))) \int \frac {1}{x}dx\right )\right )\right )-\frac {\text {arctanh}(\tanh (a+b x))^4}{x}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle 4 b \left (\frac {1}{3} \text {arctanh}(\tanh (a+b x))^3-(b x-\text {arctanh}(\tanh (a+b x))) \left (\frac {1}{2} \text {arctanh}(\tanh (a+b x))^2-(b x-\text {arctanh}(\tanh (a+b x))) (b x-\log (x) (b x-\text {arctanh}(\tanh (a+b x))))\right )\right )-\frac {\text {arctanh}(\tanh (a+b x))^4}{x}\) |
-(ArcTanh[Tanh[a + b*x]]^4/x) + 4*b*(ArcTanh[Tanh[a + b*x]]^3/3 - (b*x - A rcTanh[Tanh[a + b*x]])*(ArcTanh[Tanh[a + b*x]]^2/2 - (b*x - ArcTanh[Tanh[a + b*x]])*(b*x - (b*x - ArcTanh[Tanh[a + b*x]])*Log[x])))
3.1.74.3.1 Defintions of rubi rules used
Int[(v_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[b*(x/a), x] - Simp[(b*u - a*v)/a Int[1/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x]
Int[(v_)^(n_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[ D[v, x]]}, Simp[v^n/(a*n), x] - Simp[(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] && NeQ[n, 1]
Int[(u_)^(m_)*(v_)^(n_.), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Sim plify[D[v, x]]}, Simp[u^(m + 1)*(v^n/(a*(m + 1))), x] - Simp[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] && (FractionQ[m] || GeQ[2*n + m + 1, 0]))) || (IGtQ [n, 0] && IGtQ[m, 0] && LeQ[n, m]) || (IGtQ[n, 0] && !IntegerQ[m]) || (ILt Q[m, 0] && !IntegerQ[n]))
Time = 0.23 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.74
| method | result | size |
| default | \(-\frac {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{4}}{x}+4 b \left (\ln \left (x \right ) \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{3}-3 b \left (b^{2} \left (\frac {x^{3} \ln \left (x \right )}{3}-\frac {x^{3}}{9}\right )+2 a b \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+2 b \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right ) \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+a^{2} \left (x \ln \left (x \right )-x \right )+2 a \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right ) \left (x \ln \left (x \right )-x \right )+\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2} \left (x \ln \left (x \right )-x \right )\right )\right )\) | \(165\) |
| parts | \(-\frac {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{4}}{x}+4 b \left (\ln \left (x \right ) \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{3}-3 b \left (b^{2} \left (\frac {x^{3} \ln \left (x \right )}{3}-\frac {x^{3}}{9}\right )+2 a b \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+2 b \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right ) \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+a^{2} \left (x \ln \left (x \right )-x \right )+2 a \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right ) \left (x \ln \left (x \right )-x \right )+\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2} \left (x \ln \left (x \right )-x \right )\right )\right )\) | \(165\) |
| risch | \(\text {Expression too large to display}\) | \(2404\) |
-arctanh(tanh(b*x+a))^4/x+4*b*(ln(x)*arctanh(tanh(b*x+a))^3-3*b*(b^2*(1/3* x^3*ln(x)-1/9*x^3)+2*a*b*(1/2*x^2*ln(x)-1/4*x^2)+2*b*(arctanh(tanh(b*x+a)) -b*x-a)*(1/2*x^2*ln(x)-1/4*x^2)+a^2*(x*ln(x)-x)+2*a*(arctanh(tanh(b*x+a))- b*x-a)*(x*ln(x)-x)+(arctanh(tanh(b*x+a))-b*x-a)^2*(x*ln(x)-x)))
Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.49 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^4}{x^2} \, dx=\frac {b^{4} x^{4} + 6 \, a b^{3} x^{3} + 18 \, a^{2} b^{2} x^{2} + 12 \, a^{3} b x \log \left (x\right ) - 3 \, a^{4}}{3 \, x} \]
\[ \int \frac {\text {arctanh}(\tanh (a+b x))^4}{x^2} \, dx=\int \frac {\operatorname {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{2}}\, dx \]
Time = 0.56 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.81 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^4}{x^2} \, dx=4 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3} \log \left (x\right ) - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{x} + \frac {2}{3} \, {\left (2 \, b^{3} x^{3} + 9 \, a b^{2} x^{2} + 18 \, a^{2} b x + 6 \, a^{3} \log \left (x\right ) - 6 \, \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3} \log \left (x\right )\right )} b \]
4*b*arctanh(tanh(b*x + a))^3*log(x) - arctanh(tanh(b*x + a))^4/x + 2/3*(2* b^3*x^3 + 9*a*b^2*x^2 + 18*a^2*b*x + 6*a^3*log(x) - 6*arctanh(tanh(b*x + a ))^3*log(x))*b
Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.46 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^4}{x^2} \, dx=\frac {1}{3} \, b^{4} x^{3} + 2 \, a b^{3} x^{2} + 6 \, a^{2} b^{2} x + 4 \, a^{3} b \log \left ({\left | x \right |}\right ) - \frac {a^{4}}{x} \]
Time = 0.16 (sec) , antiderivative size = 553, normalized size of antiderivative = 5.82 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^4}{x^2} \, dx=\ln \left (x\right )\,\left (4\,a^3\,b-\frac {b\,{\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 )}^3}{2}+3\,a\,b\,{\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 )}^2-6\,a^2\,b\,\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 )\right )-\frac {{\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+24\,a^2\,{\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+16\,a^4-8\,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 )}^3-32\,a^3\,\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 )}{16\,x}+\frac {b^4\,x^3}{3}+\frac {3\,b^2\,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}{2}-b^3\,x^2\,\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 ) \]
log(x)*(4*a^3*b - (b*(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)^3)/2 + 3*a*b*(2*a - l og((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + log(2/(exp(2*a)*ex p(2*b*x) + 1)) + 2*b*x)^2 - 6*a^2*b*(2*a - log((2*exp(2*a)*exp(2*b*x))/(ex p(2*a)*exp(2*b*x) + 1)) + log(2/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*x)) - ((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 + 24*a^2*(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 + 16*a^4 - 8*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)^3 - 32*a^3*(2*a - log((2*e xp(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))/(16*x) + (b^4*x^3)/3 + (3*b^2*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)/2 - b^3*x^2*(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)