Integrand size = 15, antiderivative size = 66 \[ \int \frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{x^2} \, dx=\frac {b \arctan \left (\frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{\sqrt {b x-\text {arctanh}(\tanh (a+b x))}}\right )}{\sqrt {b x-\text {arctanh}(\tanh (a+b x))}}-\frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{x} \] Output:
b*arctan(arctanh(tanh(b*x+a))^(1/2)/(b*x-arctanh(tanh(b*x+a)))^(1/2))/(b*x -arctanh(tanh(b*x+a)))^(1/2)-arctanh(tanh(b*x+a))^(1/2)/x
Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{x^2} \, dx=-\frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{x}-\frac {b \text {arctanh}\left (\frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{\sqrt {-b x+\text {arctanh}(\tanh (a+b x))}}\right )}{\sqrt {-b x+\text {arctanh}(\tanh (a+b x))}} \] Input:
Integrate[Sqrt[ArcTanh[Tanh[a + b*x]]]/x^2,x]
Output:
-(Sqrt[ArcTanh[Tanh[a + b*x]]]/x) - (b*ArcTanh[Sqrt[ArcTanh[Tanh[a + b*x]] ]/Sqrt[-(b*x) + ArcTanh[Tanh[a + b*x]]]])/Sqrt[-(b*x) + ArcTanh[Tanh[a + b *x]]]
Time = 0.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2599, 2592}
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.
\(\displaystyle \int \frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{x^2} \, dx\) |
\(\Big \downarrow \) 2599 |
\(\displaystyle \frac {1}{2} b \int \frac {1}{x \sqrt {\text {arctanh}(\tanh (a+b x))}}dx-\frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{x}\) |
\(\Big \downarrow \) 2592 |
\(\displaystyle \frac {b \arctan \left (\frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{\sqrt {b x-\text {arctanh}(\tanh (a+b x))}}\right )}{\sqrt {b x-\text {arctanh}(\tanh (a+b x))}}-\frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{x}\) |
Input:
Int[Sqrt[ArcTanh[Tanh[a + b*x]]]/x^2,x]
Output:
(b*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]] )/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]] - Sqrt[ArcTanh[Tanh[a + b*x]]]/x
Int[1/((u_)*Sqrt[v_]), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simpli fy[D[v, x]]}, Simp[2*(ArcTan[Sqrt[v]/Rt[(b*u - a*v)/a, 2]]/(a*Rt[(b*u - a*v )/a, 2])), x] /; NeQ[b*u - a*v, 0] && PosQ[(b*u - a*v)/a]] /; PiecewiseLine arQ[u, v, x]
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 = 0.21 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.95
method | result | size |
default | \(2 b \left (-\frac {\sqrt {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )}}{2 b x}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )}}{\sqrt {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x}}\right )}{2 \sqrt {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x}}\right )\) | \(63\) |
Input:
int(arctanh(tanh(b*x+a))^(1/2)/x^2,x,method=_RETURNVERBOSE)
Output:
2*b*(-1/2*arctanh(tanh(b*x+a))^(1/2)/b/x-1/2/(arctanh(tanh(b*x+a))-b*x)^(1 /2)*arctanh(arctanh(tanh(b*x+a))^(1/2)/(arctanh(tanh(b*x+a))-b*x)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{x^2} \, dx=\left [\frac {\sqrt {a} b x \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, \sqrt {b x + a} a}{2 \, a x}, \frac {\sqrt {-a} b x \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x + a}}\right ) - \sqrt {b x + a} a}{a x}\right ] \] Input:
integrate(arctanh(tanh(b*x+a))^(1/2)/x^2,x, algorithm="fricas")
Output:
[1/2*(sqrt(a)*b*x*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) - 2*sqrt(b* x + a)*a)/(a*x), (sqrt(-a)*b*x*arctan(sqrt(-a)/sqrt(b*x + a)) - sqrt(b*x + a)*a)/(a*x)]
\[ \int \frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{x^2} \, dx=\int \frac {\sqrt {\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}}{x^{2}}\, dx \] Input:
integrate(atanh(tanh(b*x+a))**(1/2)/x**2,x)
Output:
Integral(sqrt(atanh(tanh(a + b*x)))/x**2, x)
\[ \int \frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{x^2} \, dx=\int { \frac {\sqrt {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}}{x^{2}} \,d x } \] Input:
integrate(arctanh(tanh(b*x+a))^(1/2)/x^2,x, algorithm="maxima")
Output:
integrate(sqrt(arctanh(tanh(b*x + a)))/x^2, x)
Time = 0.11 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{x^2} \, dx=\frac {1}{2} \, \sqrt {2} b {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {\sqrt {2} \sqrt {b x + a}}{b x}\right )} \] Input:
integrate(arctanh(tanh(b*x+a))^(1/2)/x^2,x, algorithm="giac")
Output:
1/2*sqrt(2)*b*(sqrt(2)*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) - sqrt(2)*s qrt(b*x + a)/(b*x))
Time = 9.22 (sec) , antiderivative size = 341, normalized size of antiderivative = 5.17 \[ \int \frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{x^2} \, dx=-\frac {\sqrt {\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}-\frac {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}}}{x}+\frac {\sqrt {2}\,b\,\ln \left (\frac {\sqrt {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x}\,\left (\sqrt {2}\,b\,x-\sqrt {2}\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )+\sqrt {\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}-\frac {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}}\,\sqrt {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x}\,2{}\mathrm {i}\right )\,1{}\mathrm {i}}{x}\right )\,1{}\mathrm {i}}{2\,\sqrt {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x}} \] Input:
int(atanh(tanh(a + b*x))^(1/2)/x^2,x)
Output:
(2^(1/2)*b*log(((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)^(1/2)*((log((2*exp(2*a)*exp(2*b* x))/(exp(2*a)*exp(2*b*x) + 1))/2 - log(2/(exp(2*a)*exp(2*b*x) + 1))/2)^(1/ 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)^(1/2)*2i - 2^(1/2)*(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) + 2^(1/2)*b*x)*1i)/x)*1i)/(2*(log(2/(exp(2*a)*exp(2*b*x) + 1)) - log((2*ex p(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*x)^(1/2)) - (log((2*ex p(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1))/2 - log(2/(exp(2*a)*exp(2*b* x) + 1))/2)^(1/2)/x
\[ \int \frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{x^2} \, dx=\frac {-2 \sqrt {\mathit {atanh} \left (\tanh \left (b x +a \right )\right )}+\left (\int \frac {\sqrt {\mathit {atanh} \left (\tanh \left (b x +a \right )\right )}}{\mathit {atanh} \left (\tanh \left (b x +a \right )\right ) x}d x \right ) b x}{2 x} \] Input:
int(atanh(tanh(b*x+a))^(1/2)/x^2,x)
Output:
( - 2*sqrt(atanh(tanh(a + b*x))) + int(sqrt(atanh(tanh(a + b*x)))/(atanh(t anh(a + b*x))*x),x)*b*x)/(2*x)