Integrand size = 15, antiderivative size = 97 \[ \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))^2} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x-\text {arctanh}(\tanh (a+b x))}}\right )}{\sqrt {b} (b x-\text {arctanh}(\tanh (a+b x)))^{3/2}}-\frac {1}{b \sqrt {x} (b x-\text {arctanh}(\tanh (a+b x)))}-\frac {1}{b \sqrt {x} \text {arctanh}(\tanh (a+b x))} \] Output:
arctanh(b^(1/2)*x^(1/2)/(b*x-arctanh(tanh(b*x+a)))^(1/2))/b^(1/2)/(b*x-arc tanh(tanh(b*x+a)))^(3/2)-1/b/x^(1/2)/(b*x-arctanh(tanh(b*x+a)))-1/b/x^(1/2 )/arctanh(tanh(b*x+a))
Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))^2} \, dx=\frac {\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-b x+\text {arctanh}(\tanh (a+b x))}}\right )}{\sqrt {b} (-b x+\text {arctanh}(\tanh (a+b x)))^{3/2}}+\frac {\sqrt {x}}{\text {arctanh}(\tanh (a+b x)) (-b x+\text {arctanh}(\tanh (a+b x)))} \] Input:
Integrate[1/(Sqrt[x]*ArcTanh[Tanh[a + b*x]]^2),x]
Output:
ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[-(b*x) + ArcTanh[Tanh[a + b*x]]]]/(Sqrt[b]*( -(b*x) + ArcTanh[Tanh[a + b*x]])^(3/2)) + Sqrt[x]/(ArcTanh[Tanh[a + b*x]]* (-(b*x) + ArcTanh[Tanh[a + b*x]]))
Time = 0.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2599, 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.
\(\displaystyle \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))^2} \, dx\) |
\(\Big \downarrow \) 2599 |
\(\displaystyle -\frac {\int \frac {1}{x^{3/2} \text {arctanh}(\tanh (a+b x))}dx}{2 b}-\frac {1}{b \sqrt {x} \text {arctanh}(\tanh (a+b x))}\) |
\(\Big \downarrow \) 2594 |
\(\displaystyle -\frac {\frac {b \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))}dx}{b x-\text {arctanh}(\tanh (a+b x))}+\frac {2}{\sqrt {x} (b x-\text {arctanh}(\tanh (a+b x)))}}{2 b}-\frac {1}{b \sqrt {x} \text {arctanh}(\tanh (a+b x))}\) |
\(\Big \downarrow \) 2593 |
\(\displaystyle -\frac {\frac {2}{\sqrt {x} (b x-\text {arctanh}(\tanh (a+b x)))}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x-\text {arctanh}(\tanh (a+b x))}}\right )}{(b x-\text {arctanh}(\tanh (a+b x)))^{3/2}}}{2 b}-\frac {1}{b \sqrt {x} \text {arctanh}(\tanh (a+b x))}\) |
Input:
Int[1/(Sqrt[x]*ArcTanh[Tanh[a + b*x]]^2),x]
Output:
-1/2*((-2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b* x]]]])/(b*x - ArcTanh[Tanh[a + b*x]])^(3/2) + 2/(Sqrt[x]*(b*x - ArcTanh[Ta nh[a + b*x]])))/b - 1/(b*Sqrt[x]*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*(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]
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]
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 = 1.50 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {\sqrt {x}}{\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\right )}{\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) \sqrt {\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\) | \(82\) |
default | \(\frac {\sqrt {x}}{\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\right )}{\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) \sqrt {\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\) | \(82\) |
risch | \(\text {Expression too large to display}\) | \(1324\) |
Input:
int(1/x^(1/2)/arctanh(tanh(b*x+a))^2,x,method=_RETURNVERBOSE)
Output:
x^(1/2)/(arctanh(tanh(b*x+a))-b*x)/arctanh(tanh(b*x+a))+1/(arctanh(tanh(b* x+a))-b*x)/((arctanh(tanh(b*x+a))-b*x)*b)^(1/2)*arctan(b*x^(1/2)/((arctanh (tanh(b*x+a))-b*x)*b)^(1/2))
Time = 0.09 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))^2} \, dx=\left [\frac {2 \, a b \sqrt {x} - \sqrt {-a b} {\left (b x + a\right )} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right )}{2 \, {\left (a^{2} b^{2} x + a^{3} b\right )}}, \frac {a b \sqrt {x} - \sqrt {a b} {\left (b x + a\right )} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right )}{a^{2} b^{2} x + a^{3} b}\right ] \] Input:
integrate(1/x^(1/2)/arctanh(tanh(b*x+a))^2,x, algorithm="fricas")
Output:
[1/2*(2*a*b*sqrt(x) - sqrt(-a*b)*(b*x + a)*log((b*x - a - 2*sqrt(-a*b)*sqr t(x))/(b*x + a)))/(a^2*b^2*x + a^3*b), (a*b*sqrt(x) - sqrt(a*b)*(b*x + a)* arctan(sqrt(a*b)/(b*sqrt(x))))/(a^2*b^2*x + a^3*b)]
\[ \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))^2} \, dx=\int \frac {1}{\sqrt {x} \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \] Input:
integrate(1/x**(1/2)/atanh(tanh(b*x+a))**2,x)
Output:
Integral(1/(sqrt(x)*atanh(tanh(a + b*x))**2), x)
Time = 0.12 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.36 \[ \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))^2} \, dx=\frac {\sqrt {x}}{a b x + a^{2}} + \frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a} \] Input:
integrate(1/x^(1/2)/arctanh(tanh(b*x+a))^2,x, algorithm="maxima")
Output:
sqrt(x)/(a*b*x + a^2) + arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a)
Time = 0.11 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.36 \[ \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))^2} \, dx=\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a} + \frac {\sqrt {x}}{{\left (b x + a\right )} a} \] Input:
integrate(1/x^(1/2)/arctanh(tanh(b*x+a))^2,x, algorithm="giac")
Output:
arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a) + sqrt(x)/((b*x + a)*a)
Time = 4.30 (sec) , antiderivative size = 516, normalized size of antiderivative = 5.32 \[ \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))^2} \, dx =\text {Too large to display} \] Input:
int(1/(x^(1/2)*atanh(tanh(a + b*x))^2),x)
Output:
(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/(ex p(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)) - lo g((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)*(4*a^2*b + 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)^2 - 4*a*b*(2*a - 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)) + 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))))))/(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)^(3/2)) - (4*x^(1/2))/((log((2*exp(2*a)*exp(2*b*x))/(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) + 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + 2*b* x))
\[ \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))^2} \, dx=\int \frac {1}{\sqrt {x}\, \mathit {atanh} \left (\tanh \left (b x +a \right )\right )^{2}}d x \] Input:
int(1/x^(1/2)/atanh(tanh(b*x+a))^2,x)
Output:
int(1/(sqrt(x)*atanh(tanh(a + b*x))**2),x)