Integrand size = 15, antiderivative size = 81 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x^2} \, dx=-3 b \arctan \left (\frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{\sqrt {b x-\text {arctanh}(\tanh (a+b x))}}\right ) \sqrt {b x-\text {arctanh}(\tanh (a+b x))}+3 b \sqrt {\text {arctanh}(\tanh (a+b x))}-\frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x} \]
-arctanh(tanh(b*x+a))^(3/2)/x-3*b*arctan(arctanh(tanh(b*x+a))^(1/2)/(b*x-a rctanh(tanh(b*x+a)))^(1/2))*(b*x-arctanh(tanh(b*x+a)))^(1/2)+3*b*arctanh(t anh(b*x+a))^(1/2)
Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.98 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x^2} \, dx=3 b \sqrt {\text {arctanh}(\tanh (a+b x))}-\frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x}-3 b \text {arctanh}\left (\frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{\sqrt {-b x+\text {arctanh}(\tanh (a+b x))}}\right ) \sqrt {-b x+\text {arctanh}(\tanh (a+b x))} \]
3*b*Sqrt[ArcTanh[Tanh[a + b*x]]] - ArcTanh[Tanh[a + b*x]]^(3/2)/x - 3*b*Ar cTanh[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[-(b*x) + ArcTanh[Tanh[a + b*x]]]]* Sqrt[-(b*x) + ArcTanh[Tanh[a + b*x]]]
Time = 0.26 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2599, 2590, 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.
\(\displaystyle \int \frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x^2} \, dx\) |
\(\Big \downarrow \) 2599 |
\(\displaystyle \frac {3}{2} b \int \frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{x}dx-\frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x}\) |
\(\Big \downarrow \) 2590 |
\(\displaystyle \frac {3}{2} b \left (2 \sqrt {\text {arctanh}(\tanh (a+b x))}-(b x-\text {arctanh}(\tanh (a+b x))) \int \frac {1}{x \sqrt {\text {arctanh}(\tanh (a+b x))}}dx\right )-\frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x}\) |
\(\Big \downarrow \) 2592 |
\(\displaystyle \frac {3}{2} b \left (2 \sqrt {\text {arctanh}(\tanh (a+b x))}-2 \sqrt {b x-\text {arctanh}(\tanh (a+b x))} \arctan \left (\frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{\sqrt {b x-\text {arctanh}(\tanh (a+b x))}}\right )\right )-\frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x}\) |
(3*b*(-2*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b *x]]]]*Sqrt[b*x - ArcTanh[Tanh[a + b*x]]] + 2*Sqrt[ArcTanh[Tanh[a + b*x]]] ))/2 - ArcTanh[Tanh[a + b*x]]^(3/2)/x
3.2.26.3.1 Defintions of rubi rules used
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[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 = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.05
method | result | size |
default | \(2 b \left (\sqrt {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )}+\frac {\left (-\frac {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )}{2}+\frac {b x}{2}\right ) \sqrt {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )}}{x b}-\frac {3 \sqrt {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x}\, \operatorname {arctanh}\left (\frac {\sqrt {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )}}{\sqrt {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x}}\right )}{2}\right )\) | \(85\) |
2*b*(arctanh(tanh(b*x+a))^(1/2)+(-1/2*arctanh(tanh(b*x+a))+1/2*b*x)*arctan h(tanh(b*x+a))^(1/2)/x/b-3/2*(arctanh(tanh(b*x+a))-b*x)^(1/2)*arctanh(arct anh(tanh(b*x+a))^(1/2)/(arctanh(tanh(b*x+a))-b*x)^(1/2)))
Time = 0.25 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.26 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x^2} \, dx=\left [\frac {3 \, \sqrt {a} b x \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (2 \, b x - a\right )} \sqrt {b x + a}}{2 \, x}, \frac {3 \, \sqrt {-a} b x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (2 \, b x - a\right )} \sqrt {b x + a}}{x}\right ] \]
[1/2*(3*sqrt(a)*b*x*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(2*b* x - a)*sqrt(b*x + a))/x, (3*sqrt(-a)*b*x*arctan(sqrt(b*x + a)*sqrt(-a)/a) + (2*b*x - a)*sqrt(b*x + a))/x]
\[ \int \frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x^2} \, dx=\int \frac {\operatorname {atanh}^{\frac {3}{2}}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{2}}\, dx \]
\[ \int \frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x^2} \, dx=\int { \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{\frac {3}{2}}}{x^{2}} \,d x } \]
Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.85 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x^2} \, dx=\frac {\sqrt {2} {\left (\frac {3 \, \sqrt {2} a b^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 2 \, \sqrt {2} \sqrt {b x + a} b^{2} - \frac {\sqrt {2} \sqrt {b x + a} a b}{x}\right )}}{2 \, b} \]
1/2*sqrt(2)*(3*sqrt(2)*a*b^2*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + 2*s qrt(2)*sqrt(b*x + a)*b^2 - sqrt(2)*sqrt(b*x + a)*a*b/x)/b
Time = 5.34 (sec) , antiderivative size = 459, normalized size of antiderivative = 5.67 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x^2} \, dx=3\,b\,\sqrt {\frac {\ln \left (\frac {{\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 {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}}+\frac {\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\sqrt {\frac {\ln \left (\frac {{\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 {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}}}{2\,x}-\frac {\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\sqrt {\frac {\ln \left (\frac {{\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 {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}}}{2\,x}+b\,\ln \left (-\frac {4\,\sqrt {2}\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-4\,\sqrt {2}\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+8\,\sqrt {\frac {\ln \left (\frac {{\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 {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}}\,\sqrt {\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 (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-2\,b\,x}+4\,\sqrt {2}\,b\,x}{x\,\sqrt {\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 (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-2\,b\,x}}\right )\,\sqrt {\frac {9\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{8}-\frac {9\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{8}-\frac {9\,b\,x}{4}} \]
3*b*(log((exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1))/2 - log(1/(exp(2 *a)*exp(2*b*x) + 1))/2)^(1/2) + (log(1/(exp(2*a)*exp(2*b*x) + 1))*(log((ex p(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1))/2 - log(1/(exp(2*a)*exp(2*b* x) + 1))/2)^(1/2))/(2*x) - (log((exp(2*a)*exp(2*b*x))/(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 - log(1/(ex p(2*a)*exp(2*b*x) + 1))/2)^(1/2))/(2*x) + b*log(-(4*2^(1/2)*log(1/(exp(2*a )*exp(2*b*x) + 1)) - 4*2^(1/2)*log((exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b *x) + 1)) + 8*(log((exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1))/2 - lo g(1/(exp(2*a)*exp(2*b*x) + 1))/2)^(1/2)*(log((exp(2*a)*exp(2*b*x))/(exp(2* a)*exp(2*b*x) + 1)) - log(1/(exp(2*a)*exp(2*b*x) + 1)) - 2*b*x)^(1/2) + 4* 2^(1/2)*b*x)/(x*(log((exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) - lo g(1/(exp(2*a)*exp(2*b*x) + 1)) - 2*b*x)^(1/2)))*((9*log((exp(2*a)*exp(2*b* x))/(exp(2*a)*exp(2*b*x) + 1)))/8 - (9*log(1/(exp(2*a)*exp(2*b*x) + 1)))/8 - (9*b*x)/4)^(1/2)