Integrand size = 15, antiderivative size = 78 \[ \int \frac {1}{x \text {arctanh}(\tanh (a+b x))^{3/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{\sqrt {b x-\text {arctanh}(\tanh (a+b x))}}\right )}{(b x-\text {arctanh}(\tanh (a+b x)))^{3/2}}-\frac {2}{(b x-\text {arctanh}(\tanh (a+b x))) \sqrt {\text {arctanh}(\tanh (a+b x))}} \] Output:
-2*arctan(arctanh(tanh(b*x+a))^(1/2)/(b*x-arctanh(tanh(b*x+a)))^(1/2))/(b* x-arctanh(tanh(b*x+a)))^(3/2)-2/(b*x-arctanh(tanh(b*x+a)))/arctanh(tanh(b* x+a))^(1/2)
Time = 0.12 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x \text {arctanh}(\tanh (a+b x))^{3/2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{\sqrt {-b x+\text {arctanh}(\tanh (a+b x))}}\right )}{(-b x+\text {arctanh}(\tanh (a+b x)))^{3/2}}+\frac {2}{\sqrt {\text {arctanh}(\tanh (a+b x))} (-b x+\text {arctanh}(\tanh (a+b x)))} \] Input:
Integrate[1/(x*ArcTanh[Tanh[a + b*x]]^(3/2)),x]
Output:
(-2*ArcTanh[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[-(b*x) + ArcTanh[Tanh[a + b* x]]]])/(-(b*x) + ArcTanh[Tanh[a + b*x]])^(3/2) + 2/(Sqrt[ArcTanh[Tanh[a + b*x]]]*(-(b*x) + ArcTanh[Tanh[a + b*x]]))
Time = 0.27 (sec) , antiderivative size = 78, 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 = {2594, 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.
\(\displaystyle \int \frac {1}{x \text {arctanh}(\tanh (a+b x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 2594 |
\(\displaystyle -\frac {\int \frac {1}{x \sqrt {\text {arctanh}(\tanh (a+b x))}}dx}{b x-\text {arctanh}(\tanh (a+b x))}-\frac {2}{(b x-\text {arctanh}(\tanh (a+b x))) \sqrt {\text {arctanh}(\tanh (a+b x))}}\) |
\(\Big \downarrow \) 2592 |
\(\displaystyle -\frac {2 \arctan \left (\frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{\sqrt {b x-\text {arctanh}(\tanh (a+b x))}}\right )}{(b x-\text {arctanh}(\tanh (a+b x)))^{3/2}}-\frac {2}{(b x-\text {arctanh}(\tanh (a+b x))) \sqrt {\text {arctanh}(\tanh (a+b x))}}\) |
Input:
Int[1/(x*ArcTanh[Tanh[a + b*x]]^(3/2)),x]
Output:
(-2*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]] ])/(b*x - ArcTanh[Tanh[a + b*x]])^(3/2) - 2/((b*x - ArcTanh[Tanh[a + b*x]] )*Sqrt[ArcTanh[Tanh[a + b*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[(v_)^(n_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[ D[v, x]]}, Simp[v^(n + 1)/((n + 1)*(b*u - a*v)), x] - Simp[a*((n + 1)/((n + 1)*(b*u - a*v))) Int[v^(n + 1)/u, x], x] /; NeQ[b*u - a*v, 0]] /; Piecew iseLinearQ[u, v, x] && LtQ[n, -1]
Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {2}{\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) \sqrt {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )}}{\sqrt {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x}}\right )}{\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right )^{\frac {3}{2}}}\) | \(68\) |
Input:
int(1/x/arctanh(tanh(b*x+a))^(3/2),x,method=_RETURNVERBOSE)
Output:
2/(arctanh(tanh(b*x+a))-b*x)/arctanh(tanh(b*x+a))^(1/2)-2/(arctanh(tanh(b* x+a))-b*x)^(3/2)*arctanh(arctanh(tanh(b*x+a))^(1/2)/(arctanh(tanh(b*x+a))- b*x)^(1/2))
Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.37 \[ \int \frac {1}{x \text {arctanh}(\tanh (a+b x))^{3/2}} \, dx=\left [\frac {{\left (b x + a\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, \sqrt {b x + a} a}{a^{2} b x + a^{3}}, \frac {2 \, {\left ({\left (b x + a\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x + a}}\right ) + \sqrt {b x + a} a\right )}}{a^{2} b x + a^{3}}\right ] \] Input:
integrate(1/x/arctanh(tanh(b*x+a))^(3/2),x, algorithm="fricas")
Output:
[((b*x + a)*sqrt(a)*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*sqrt( b*x + a)*a)/(a^2*b*x + a^3), 2*((b*x + a)*sqrt(-a)*arctan(sqrt(-a)/sqrt(b* x + a)) + sqrt(b*x + a)*a)/(a^2*b*x + a^3)]
\[ \int \frac {1}{x \text {arctanh}(\tanh (a+b x))^{3/2}} \, dx=\int \frac {1}{x \operatorname {atanh}^{\frac {3}{2}}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \] Input:
integrate(1/x/atanh(tanh(b*x+a))**(3/2),x)
Output:
Integral(1/(x*atanh(tanh(a + b*x))**(3/2)), x)
\[ \int \frac {1}{x \text {arctanh}(\tanh (a+b x))^{3/2}} \, dx=\int { \frac {1}{x \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/x/arctanh(tanh(b*x+a))^(3/2),x, algorithm="maxima")
Output:
integrate(1/(x*arctanh(tanh(b*x + a))^(3/2)), x)
Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.47 \[ \int \frac {1}{x \text {arctanh}(\tanh (a+b x))^{3/2}} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {2}{\sqrt {b x + a} a} \] Input:
integrate(1/x/arctanh(tanh(b*x+a))^(3/2),x, algorithm="giac")
Output:
2*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a) + 2/(sqrt(b*x + a)*a)
Time = 8.32 (sec) , antiderivative size = 614, normalized size of antiderivative = 7.87 \[ \int \frac {1}{x \text {arctanh}(\tanh (a+b x))^{3/2}} \, dx =\text {Too large to display} \] Input:
int(1/(x*atanh(tanh(a + b*x))^(3/2)),x)
Output:
(2^(1/2)*log((((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)*((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 - 8*a^3 - 6*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)^2 + 12*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))*1i)/(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)^(1/2))) *2i)/(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)^(3/2) - (8*(log((2*exp(2*a)*exp(2*b*x))/(ex p(2*a)*exp(2*b*x) + 1))/2 - log(2/(exp(2*a)*exp(2*b*x) + 1))/2)^(1/2))/((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)))*(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))
\[ \int \frac {1}{x \text {arctanh}(\tanh (a+b x))^{3/2}} \, dx=\int \frac {\sqrt {\mathit {atanh} \left (\tanh \left (b x +a \right )\right )}}{\mathit {atanh} \left (\tanh \left (b x +a \right )\right )^{2} x}d x \] Input:
int(1/x/atanh(tanh(b*x+a))^(3/2),x)
Output:
int(sqrt(atanh(tanh(a + b*x)))/(atanh(tanh(a + b*x))**2*x),x)