Integrand size = 15, antiderivative size = 64 \[ \int \frac {\sqrt {x}}{\text {arctanh}(\tanh (a+b x))} \, dx=\frac {2 \sqrt {x}}{b}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x-\text {arctanh}(\tanh (a+b x))}}\right ) \sqrt {b x-\text {arctanh}(\tanh (a+b x))}}{b^{3/2}} \] Output:
2*x^(1/2)/b-2*arctanh(b^(1/2)*x^(1/2)/(b*x-arctanh(tanh(b*x+a)))^(1/2))*(b *x-arctanh(tanh(b*x+a)))^(1/2)/b^(3/2)
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {x}}{\text {arctanh}(\tanh (a+b x))} \, dx=\frac {2 \sqrt {x}}{b}-\frac {2 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-b x+\text {arctanh}(\tanh (a+b x))}}\right ) \sqrt {-b x+\text {arctanh}(\tanh (a+b x))}}{b^{3/2}} \] Input:
Integrate[Sqrt[x]/ArcTanh[Tanh[a + b*x]],x]
Output:
(2*Sqrt[x])/b - (2*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[-(b*x) + ArcTanh[Tanh[a + b*x]]]]*Sqrt[-(b*x) + ArcTanh[Tanh[a + b*x]]])/b^(3/2)
Time = 0.36 (sec) , antiderivative size = 64, 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 = {2590, 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.
\(\displaystyle \int \frac {\sqrt {x}}{\text {arctanh}(\tanh (a+b x))} \, dx\) |
\(\Big \downarrow \) 2590 |
\(\displaystyle \frac {(b x-\text {arctanh}(\tanh (a+b x))) \int \frac {1}{\sqrt {x} \text {arctanh}(\tanh (a+b x))}dx}{b}+\frac {2 \sqrt {x}}{b}\) |
\(\Big \downarrow \) 2593 |
\(\displaystyle \frac {2 \sqrt {x}}{b}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x-\text {arctanh}(\tanh (a+b x))}}\right ) \sqrt {b x-\text {arctanh}(\tanh (a+b x))}}{b^{3/2}}\) |
Input:
Int[Sqrt[x]/ArcTanh[Tanh[a + b*x]],x]
Output:
(2*Sqrt[x])/b - (2*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b *x]]]]*Sqrt[b*x - ArcTanh[Tanh[a + b*x]]])/b^(3/2)
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*(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]
Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(\frac {2 \sqrt {x}}{b}+\frac {2 \left (b x -\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )\right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\right )}{b \sqrt {\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\) | \(66\) |
default | \(\frac {2 \sqrt {x}}{b}+\frac {2 \left (b x -\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )\right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\right )}{b \sqrt {\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) b}}\) | \(66\) |
Input:
int(x^(1/2)/arctanh(tanh(b*x+a)),x,method=_RETURNVERBOSE)
Output:
2*x^(1/2)/b+2*(b*x-arctanh(tanh(b*x+a)))/b/((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 = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {x}}{\text {arctanh}(\tanh (a+b x))} \, dx=\left [\frac {\sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) + 2 \, \sqrt {x}}{b}, -\frac {2 \, {\left (\sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) - \sqrt {x}\right )}}{b}\right ] \] Input:
integrate(x^(1/2)/arctanh(tanh(b*x+a)),x, algorithm="fricas")
Output:
[(sqrt(-a/b)*log((b*x - 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*x + a)) + 2*sqrt(x) )/b, -2*(sqrt(a/b)*arctan(b*sqrt(x)*sqrt(a/b)/a) - sqrt(x))/b]
\[ \int \frac {\sqrt {x}}{\text {arctanh}(\tanh (a+b x))} \, dx=\int \frac {\sqrt {x}}{\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \] Input:
integrate(x**(1/2)/atanh(tanh(b*x+a)),x)
Output:
Integral(sqrt(x)/atanh(tanh(a + b*x)), x)
Time = 0.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.48 \[ \int \frac {\sqrt {x}}{\text {arctanh}(\tanh (a+b x))} \, dx=-\frac {2 \, a \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {2 \, \sqrt {x}}{b} \] Input:
integrate(x^(1/2)/arctanh(tanh(b*x+a)),x, algorithm="maxima")
Output:
-2*a*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b) + 2*sqrt(x)/b
Time = 0.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.48 \[ \int \frac {\sqrt {x}}{\text {arctanh}(\tanh (a+b x))} \, dx=-\frac {2 \, a \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {2 \, \sqrt {x}}{b} \] Input:
integrate(x^(1/2)/arctanh(tanh(b*x+a)),x, algorithm="giac")
Output:
-2*a*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b) + 2*sqrt(x)/b
Time = 4.09 (sec) , antiderivative size = 296, normalized size of antiderivative = 4.62 \[ \int \frac {\sqrt {x}}{\text {arctanh}(\tanh (a+b x))} \, dx=\frac {2\,\sqrt {x}}{b}+\frac {\sqrt {2}\,\ln \left (\frac {b^{7/2}\,\left (\sqrt {2}\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )-4\,\sqrt {b}\,\sqrt {x}\,\sqrt {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x}+2\,\sqrt {2}\,b\,x\right )}{\left (\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\right )\,\sqrt {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x}}\right )\,\sqrt {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x}}{2\,b^{3/2}} \] Input:
int(x^(1/2)/atanh(tanh(a + b*x)),x)
Output:
(2*x^(1/2))/b + (2^(1/2)*log((b^(7/2)*(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))/((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)^(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*b^(3/2))
\[ \int \frac {\sqrt {x}}{\text {arctanh}(\tanh (a+b x))} \, dx=\int \frac {\sqrt {x}}{\mathit {atanh} \left (\tanh \left (b x +a \right )\right )}d x \] Input:
int(x^(1/2)/atanh(tanh(b*x+a)),x)
Output:
int(sqrt(x)/atanh(tanh(a + b*x)),x)