\(\int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))^3} \, dx\) [211]

Optimal result

Integrand size = 15, antiderivative size = 152 \[ \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))^3} \, dx=-\frac {3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x-\text {arctanh}(\tanh (a+b x))}}\right )}{4 \sqrt {b} (b x-\text {arctanh}(\tanh (a+b x)))^{5/2}}+\frac {3}{4 b \sqrt {x} (b x-\text {arctanh}(\tanh (a+b x)))^2}+\frac {1}{4 b^2 x^{3/2} (b x-\text {arctanh}(\tanh (a+b x)))}-\frac {1}{2 b \sqrt {x} \text {arctanh}(\tanh (a+b x))^2}+\frac {1}{4 b^2 x^{3/2} \text {arctanh}(\tanh (a+b x))} \] Output:

-3/4*arctanh(b^(1/2)*x^(1/2)/(b*x-arctanh(tanh(b*x+a)))^(1/2))/b^(1/2)/(b* 
x-arctanh(tanh(b*x+a)))^(5/2)+3/4/b/x^(1/2)/(b*x-arctanh(tanh(b*x+a)))^2+1 
/4/b^2/x^(3/2)/(b*x-arctanh(tanh(b*x+a)))-1/2/b/x^(1/2)/arctanh(tanh(b*x+a 
))^2+1/4/b^2/x^(3/2)/arctanh(tanh(b*x+a))
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))^3} \, dx=\frac {3 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-b x+\text {arctanh}(\tanh (a+b x))}}\right )}{4 \sqrt {b} (-b x+\text {arctanh}(\tanh (a+b x)))^{5/2}}+\frac {3 \sqrt {x}}{4 \text {arctanh}(\tanh (a+b x)) (-b x+\text {arctanh}(\tanh (a+b x)))^2}+\frac {\sqrt {x}}{2 \text {arctanh}(\tanh (a+b x))^2 (-b x+\text {arctanh}(\tanh (a+b x)))} \] Input:

Integrate[1/(Sqrt[x]*ArcTanh[Tanh[a + b*x]]^3),x]
 

Output:

(3*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[-(b*x) + ArcTanh[Tanh[a + b*x]]]])/(4*Sqr 
t[b]*(-(b*x) + ArcTanh[Tanh[a + b*x]])^(5/2)) + (3*Sqrt[x])/(4*ArcTanh[Tan 
h[a + b*x]]*(-(b*x) + ArcTanh[Tanh[a + b*x]])^2) + Sqrt[x]/(2*ArcTanh[Tanh 
[a + b*x]]^2*(-(b*x) + ArcTanh[Tanh[a + b*x]]))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2599, 2599, 2594, 2594, 2593}

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.

\(\displaystyle \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))^3} \, dx\)

\(\Big \downarrow \) 2599

\(\displaystyle -\frac {\int \frac {1}{x^{3/2} \text {arctanh}(\tanh (a+b x))^2}dx}{4 b}-\frac {1}{2 b \sqrt {x} \text {arctanh}(\tanh (a+b x))^2}\)

\(\Big \downarrow \) 2599

\(\displaystyle -\frac {-\frac {3 \int \frac {1}{x^{5/2} \text {arctanh}(\tanh (a+b x))}dx}{2 b}-\frac {1}{b x^{3/2} \text {arctanh}(\tanh (a+b x))}}{4 b}-\frac {1}{2 b \sqrt {x} \text {arctanh}(\tanh (a+b x))^2}\)

\(\Big \downarrow \) 2594

\(\displaystyle -\frac {-\frac {3 \left (\frac {b \int \frac {1}{x^{3/2} \text {arctanh}(\tanh (a+b x))}dx}{b x-\text {arctanh}(\tanh (a+b x))}+\frac {2}{3 x^{3/2} (b x-\text {arctanh}(\tanh (a+b x)))}\right )}{2 b}-\frac {1}{b x^{3/2} \text {arctanh}(\tanh (a+b x))}}{4 b}-\frac {1}{2 b \sqrt {x} \text {arctanh}(\tanh (a+b x))^2}\)

\(\Big \downarrow \) 2594

\(\displaystyle -\frac {-\frac {3 \left (\frac {b \left (\frac {b \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))}dx}{b x-\text {arctanh}(\tanh (a+b x))}+\frac {2}{\sqrt {x} (b x-\text {arctanh}(\tanh (a+b x)))}\right )}{b x-\text {arctanh}(\tanh (a+b x))}+\frac {2}{3 x^{3/2} (b x-\text {arctanh}(\tanh (a+b x)))}\right )}{2 b}-\frac {1}{b x^{3/2} \text {arctanh}(\tanh (a+b x))}}{4 b}-\frac {1}{2 b \sqrt {x} \text {arctanh}(\tanh (a+b x))^2}\)

\(\Big \downarrow \) 2593

\(\displaystyle -\frac {-\frac {3 \left (\frac {2}{3 x^{3/2} (b x-\text {arctanh}(\tanh (a+b x)))}+\frac {b \left (\frac {2}{\sqrt {x} (b x-\text {arctanh}(\tanh (a+b x)))}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x-\text {arctanh}(\tanh (a+b x))}}\right )}{(b x-\text {arctanh}(\tanh (a+b x)))^{3/2}}\right )}{b x-\text {arctanh}(\tanh (a+b x))}\right )}{2 b}-\frac {1}{b x^{3/2} \text {arctanh}(\tanh (a+b x))}}{4 b}-\frac {1}{2 b \sqrt {x} \text {arctanh}(\tanh (a+b x))^2}\)

Input:

Int[1/(Sqrt[x]*ArcTanh[Tanh[a + b*x]]^3),x]
 

Output:

-1/4*((-3*(2/(3*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])) + (b*((-2*Sqrt[b]* 
ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(b*x - ArcT 
anh[Tanh[a + b*x]])^(3/2) + 2/(Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]]))))/( 
b*x - ArcTanh[Tanh[a + b*x]])))/(2*b) - 1/(b*x^(3/2)*ArcTanh[Tanh[a + b*x] 
]))/b - 1/(2*b*Sqrt[x]*ArcTanh[Tanh[a + b*x]]^2)
 

Defintions of rubi rules used

rule 2593

Int[1/((u_)*Sqrt[v_]), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simpli 
fy[D[v, x]]}, Simp[-2*(ArcTanh[Sqrt[v]/Rt[-(b*u - a*v)/a, 2]]/(a*Rt[-(b*u - 
 a*v)/a, 2])), x] /; NeQ[b*u - a*v, 0] && NegQ[(b*u - a*v)/a]] /; Piecewise 
LinearQ[u, v, x]
 

rule 2594

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]
 

rule 2599

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]))
 

Maple [A] (verified)

Time = 1.32 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(\frac {\sqrt {x}}{2 \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{2}}+\frac {3 \sqrt {x}}{4 \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right )^{2} \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )}+\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\right )}{4 \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right )^{2} \sqrt {\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\)	\(112\)
default	\(\frac {\sqrt {x}}{2 \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{2}}+\frac {3 \sqrt {x}}{4 \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right )^{2} \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )}+\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\right )}{4 \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right )^{2} \sqrt {\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\)	\(112\)
risch	\(\text {Expression too large to display}\)	\(1837\)

Input:

int(1/x^(1/2)/arctanh(tanh(b*x+a))^3,x,method=_RETURNVERBOSE)
 

Output:

1/2*x^(1/2)/(arctanh(tanh(b*x+a))-b*x)/arctanh(tanh(b*x+a))^2+3/4/(arctanh 
(tanh(b*x+a))-b*x)^2*x^(1/2)/arctanh(tanh(b*x+a))+3/4/(arctanh(tanh(b*x+a) 
)-b*x)^2/((arctanh(tanh(b*x+a))-b*x)*b)^(1/2)*arctan(b*x^(1/2)/((arctanh(t 
anh(b*x+a))-b*x)*b)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))^3} \, dx=\left [-\frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) - 2 \, {\left (3 \, a b^{2} x + 5 \, a^{2} b\right )} \sqrt {x}}{8 \, {\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}}, -\frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) - {\left (3 \, a b^{2} x + 5 \, a^{2} b\right )} \sqrt {x}}{4 \, {\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}}\right ] \] Input:

integrate(1/x^(1/2)/arctanh(tanh(b*x+a))^3,x, algorithm="fricas")
 

Output:

[-1/8*(3*(b^2*x^2 + 2*a*b*x + a^2)*sqrt(-a*b)*log((b*x - a - 2*sqrt(-a*b)* 
sqrt(x))/(b*x + a)) - 2*(3*a*b^2*x + 5*a^2*b)*sqrt(x))/(a^3*b^3*x^2 + 2*a^ 
4*b^2*x + a^5*b), -1/4*(3*(b^2*x^2 + 2*a*b*x + a^2)*sqrt(a*b)*arctan(sqrt( 
a*b)/(b*sqrt(x))) - (3*a*b^2*x + 5*a^2*b)*sqrt(x))/(a^3*b^3*x^2 + 2*a^4*b^ 
2*x + a^5*b)]
 

Sympy [F]

\[ \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))^3} \, dx=\int \frac {1}{\sqrt {x} \operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \] Input:

integrate(1/x**(1/2)/atanh(tanh(b*x+a))**3,x)
 

Output:

Integral(1/(sqrt(x)*atanh(tanh(a + b*x))**3), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))^3} \, dx=\frac {3 \, b x^{\frac {3}{2}} + 5 \, a \sqrt {x}}{4 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )}} + \frac {3 \, \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{2}} \] Input:

integrate(1/x^(1/2)/arctanh(tanh(b*x+a))^3,x, algorithm="maxima")
 

Output:

1/4*(3*b*x^(3/2) + 5*a*sqrt(x))/(a^2*b^2*x^2 + 2*a^3*b*x + a^4) + 3/4*arct 
an(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^2)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))^3} \, dx=\frac {3 \, \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{2}} + \frac {3 \, b x^{\frac {3}{2}} + 5 \, a \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{2}} \] Input:

integrate(1/x^(1/2)/arctanh(tanh(b*x+a))^3,x, algorithm="giac")
 

Output:

3/4*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^2) + 1/4*(3*b*x^(3/2) + 5*a*s 
qrt(x))/((b*x + a)^2*a^2)
 

Mupad [B] (verification not implemented)

Time = 4.09 (sec) , antiderivative size = 741, normalized size of antiderivative = 4.88 \[ \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))^3} \, dx =\text {Too large to display} \] Input:

int(1/(x^(1/2)*atanh(tanh(a + b*x))^3),x)
 

Output:

(6*x^(1/2))/((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)))*(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) - (4*x^(1/2))/ 
((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*(log(2/(exp(2*a)*exp(2*b*x) + 1)) - log((2*exp(2*a)*e 
xp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*x)) + (3*2^(1/2)*log((b^(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)*(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) - 4*b^( 
1/2)*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)^(1/2) + 2*2^(1/2)*b*x)*(16*a^4*b + 
b*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + log(2/(e 
xp(2*a)*exp(2*b*x) + 1)) + 2*b*x)^4 - 8*a*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 - 32*a^3*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) + 24*a^2*b*(2*a - log((2*ex 
p(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))/(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^(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...
 

Reduce [F]

\[ \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))^3} \, dx=\int \frac {1}{\sqrt {x}\, \mathit {atanh} \left (\tanh \left (b x +a \right )\right )^{3}}d x \] Input:

int(1/x^(1/2)/atanh(tanh(b*x+a))^3,x)
 

Output:

int(1/(sqrt(x)*atanh(tanh(a + b*x))**3),x)