3.2.25 \(\int \frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x} \, dx\) [125]

3.2.25.1 Optimal result

Integrand size = 15, antiderivative size = 91 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x} \, dx=2 \arctan \left (\frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{\sqrt {b x-\text {arctanh}(\tanh (a+b x))}}\right ) (b x-\text {arctanh}(\tanh (a+b x)))^{3/2}-2 (b x-\text {arctanh}(\tanh (a+b x))) \sqrt {\text {arctanh}(\tanh (a+b x))}+\frac {2}{3} \text {arctanh}(\tanh (a+b x))^{3/2} \]

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

3.2.25.2 Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.88 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x} \, dx=-\frac {2}{3} \left (3 b x \sqrt {\text {arctanh}(\tanh (a+b x))}-4 \text {arctanh}(\tanh (a+b x))^{3/2}+3 \text {arctanh}\left (\frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{\sqrt {-b x+\text {arctanh}(\tanh (a+b x))}}\right ) (-b x+\text {arctanh}(\tanh (a+b x)))^{3/2}\right ) \]

input

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

output

(-2*(3*b*x*Sqrt[ArcTanh[Tanh[a + b*x]]] - 4*ArcTanh[Tanh[a + b*x]]^(3/2) + 
 3*ArcTanh[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[-(b*x) + ArcTanh[Tanh[a + b*x 
]]]]*(-(b*x) + ArcTanh[Tanh[a + b*x]])^(3/2)))/3
 

3.2.25.3 Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2590, 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.

\(\displaystyle \int \frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x} \, dx\)

\(\Big \downarrow \) 2590

\(\displaystyle \frac {2}{3} \text {arctanh}(\tanh (a+b x))^{3/2}-(b x-\text {arctanh}(\tanh (a+b x))) \int \frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{x}dx\)

\(\Big \downarrow \) 2590

\(\displaystyle \frac {2}{3} \text {arctanh}(\tanh (a+b x))^{3/2}-(b x-\text {arctanh}(\tanh (a+b x))) \left (2 \sqrt {\text {arctanh}(\tanh (a+b x))}-(b x-\text {arctanh}(\tanh (a+b x))) \int \frac {1}{x \sqrt {\text {arctanh}(\tanh (a+b x))}}dx\right )\)

\(\Big \downarrow \) 2592

\(\displaystyle \frac {2}{3} \text {arctanh}(\tanh (a+b x))^{3/2}-(b x-\text {arctanh}(\tanh (a+b x))) \left (2 \sqrt {\text {arctanh}(\tanh (a+b x))}-2 \sqrt {b x-\text {arctanh}(\tanh (a+b x))} \arctan \left (\frac {\sqrt {\text {arctanh}(\tanh (a+b x))}}{\sqrt {b x-\text {arctanh}(\tanh (a+b x))}}\right )\right )\)

input

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

output

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

3.2.25.3.1 Defintions of rubi rules used

rule 2590

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]
 

rule 2592

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]
 

3.2.25.4 Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.44

method	result	size
default	\(\frac {2 \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}}{3}+2 \sqrt {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )}\, a +2 \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right ) \sqrt {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )}-\frac {2 \left (a^{2}+2 a \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )+\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )}}{\sqrt {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x}}\right )}{\sqrt {\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x}}\)	\(131\)

input

int(arctanh(tanh(b*x+a))^(3/2)/x,x,method=_RETURNVERBOSE)
 

output

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

3.2.25.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.97 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x} \, dx=\left [a^{\frac {3}{2}} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + \frac {2}{3} \, {\left (b x + 4 \, a\right )} \sqrt {b x + a}, 2 \, \sqrt {-a} a \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + \frac {2}{3} \, {\left (b x + 4 \, a\right )} \sqrt {b x + a}\right ] \]

input

integrate(arctanh(tanh(b*x+a))^(3/2)/x,x, algorithm="fricas")
 

output

[a^(3/2)*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2/3*(b*x + 4*a)*sq 
rt(b*x + a), 2*sqrt(-a)*a*arctan(sqrt(b*x + a)*sqrt(-a)/a) + 2/3*(b*x + 4* 
a)*sqrt(b*x + a)]
 

3.2.25.6 Sympy [F]

\[ \int \frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x} \, dx=\int \frac {\operatorname {atanh}^{\frac {3}{2}}{\left (\tanh {\left (a + b x \right )} \right )}}{x}\, dx \]

input

integrate(atanh(tanh(b*x+a))**(3/2)/x,x)
 

output

Integral(atanh(tanh(a + b*x))**(3/2)/x, x)
 

3.2.25.7 Maxima [F]

\[ \int \frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x} \, dx=\int { \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{\frac {3}{2}}}{x} \,d x } \]

input

integrate(arctanh(tanh(b*x+a))^(3/2)/x,x, algorithm="maxima")
 

output

integrate(arctanh(tanh(b*x + a))^(3/2)/x, x)
 

3.2.25.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.63 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x} \, dx=\frac {1}{3} \, \sqrt {2} {\left (\frac {3 \, \sqrt {2} a^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \sqrt {2} {\left (b x + a\right )}^{\frac {3}{2}} + 3 \, \sqrt {2} \sqrt {b x + a} a\right )} \]

input

integrate(arctanh(tanh(b*x+a))^(3/2)/x,x, algorithm="giac")
 

output

1/3*sqrt(2)*(3*sqrt(2)*a^2*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + sqrt( 
2)*(b*x + a)^(3/2) + 3*sqrt(2)*sqrt(b*x + a)*a)
 

3.2.25.9 Mupad [B] (verification not implemented)

Time = 8.14 (sec) , antiderivative size = 501, normalized size of antiderivative = 5.51 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^{3/2}}{x} \, 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}}\,\left (\frac {4\,b\,\left (\frac {\ln \left (\frac {2}{{\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}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}+b\,x\right )}{3}-2\,b\,\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 )\right )}{b}+\frac {2\,b\,x\,\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}}}{3}+\frac {\sqrt {2}\,\ln \left (\frac {\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 )\,4{}\mathrm {i}}{x\,\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}}\right )\,{\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 )}^{3/2}\,1{}\mathrm {i}}{4} \]

input

int(atanh(tanh(a + b*x))^(3/2)/x,x)
 

output

((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)*((4*b*(log(2/(exp(2*a)*exp(2*b*x) + 1))/2 - l 
og((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1))/2 + b*x))/3 - 2*b*(l 
og(2/(exp(2*a)*exp(2*b*x) + 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*ex 
p(2*b*x) + 1)) + 2*b*x)))/b + (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)*e 
xp(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)*4i)/(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)))*(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)*1i)/4 + (2*b*x*(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))/3