Integrand size = 23, antiderivative size = 63 \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {-\sqrt {x}+x}} \, dx=4 \text {arctanh}\left (\frac {\sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right ) \] Output:
4*arctanh((-x^(1/2)+x)^(1/2)/(-1+x^(1/2)))-2*2^(1/2)*arctanh(2^(1/2)*(-x^( 1/2)+x)^(1/2)/(-1+x^(1/2)))
Time = 0.13 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {-\sqrt {x}+x}} \, dx=4 \text {arctanh}\left (\frac {\sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right ) \] Input:
Integrate[1/((1 + Sqrt[x])*Sqrt[-Sqrt[x] + x]),x]
Output:
4*ArcTanh[Sqrt[-Sqrt[x] + x]/(-1 + Sqrt[x])] - 2*Sqrt[2]*ArcTanh[(Sqrt[2]* Sqrt[-Sqrt[x] + x])/(-1 + Sqrt[x])]
Time = 0.23 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1731, 1269, 1091, 219, 1154, 219}
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.
\(\displaystyle \int \frac {1}{\left (\sqrt {x}+1\right ) \sqrt {x-\sqrt {x}}} \, dx\) |
\(\Big \downarrow \) 1731 |
\(\displaystyle 2 \int \frac {\sqrt {x}}{\left (\sqrt {x}+1\right ) \sqrt {x-\sqrt {x}}}d\sqrt {x}\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle 2 \left (\int \frac {1}{\sqrt {x-\sqrt {x}}}d\sqrt {x}-\int \frac {1}{\left (\sqrt {x}+1\right ) \sqrt {x-\sqrt {x}}}d\sqrt {x}\right )\) |
\(\Big \downarrow \) 1091 |
\(\displaystyle 2 \left (2 \int \frac {1}{1-x}d\frac {\sqrt {x}}{\sqrt {x-\sqrt {x}}}-\int \frac {1}{\left (\sqrt {x}+1\right ) \sqrt {x-\sqrt {x}}}d\sqrt {x}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 2 \left (2 \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt {x-\sqrt {x}}}\right )-\int \frac {1}{\left (\sqrt {x}+1\right ) \sqrt {x-\sqrt {x}}}d\sqrt {x}\right )\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle 2 \left (2 \int \frac {1}{8-x}d\frac {1-3 \sqrt {x}}{\sqrt {x-\sqrt {x}}}+2 \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt {x-\sqrt {x}}}\right )\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 2 \left (\frac {\text {arctanh}\left (\frac {1-3 \sqrt {x}}{2 \sqrt {2} \sqrt {x-\sqrt {x}}}\right )}{\sqrt {2}}+2 \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt {x-\sqrt {x}}}\right )\right )\) |
Input:
Int[1/((1 + Sqrt[x])*Sqrt[-Sqrt[x] + x]),x]
Output:
2*(ArcTanh[(1 - 3*Sqrt[x])/(2*Sqrt[2]*Sqrt[-Sqrt[x] + x])]/Sqrt[2] + 2*Arc Tanh[Sqrt[x]/Sqrt[-Sqrt[x] + x]])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[((a_.) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^ (n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, Simp[g Subst[Int[x^( g - 1)*(d + e*x^(g*n))^q*(a + b*x^(g*n) + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 2*n] && FractionQ[n]
Time = 0.11 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(2 \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (1-3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {\left (1+\sqrt {x}\right )^{2}-3 \sqrt {x}-1}}\right )\) | \(52\) |
default | \(-\frac {\sqrt {-\sqrt {x}+x}\, \left (\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right )-2 \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )\right )}{\sqrt {\sqrt {x}\, \left (-1+\sqrt {x}\right )}}\) | \(67\) |
Input:
int(1/(1+x^(1/2))/(-x^(1/2)+x)^(1/2),x,method=_RETURNVERBOSE)
Output:
2*ln(-1/2+x^(1/2)+(-x^(1/2)+x)^(1/2))+2^(1/2)*arctanh(1/4*(1-3*x^(1/2))*2^ (1/2)/((1+x^(1/2))^2-3*x^(1/2)-1)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (47) = 94\).
Time = 0.77 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.62 \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {-\sqrt {x}+x}} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (-\frac {17 \, x^{2} - 4 \, {\left (\sqrt {2} {\left (3 \, x + 5\right )} \sqrt {x} - \sqrt {2} {\left (7 \, x + 1\right )}\right )} \sqrt {x - \sqrt {x}} - 16 \, {\left (3 \, x + 1\right )} \sqrt {x} + 46 \, x + 1}{x^{2} - 2 \, x + 1}\right ) + \log \left (-4 \, \sqrt {x - \sqrt {x}} {\left (2 \, \sqrt {x} - 1\right )} - 8 \, x + 8 \, \sqrt {x} - 1\right ) \] Input:
integrate(1/(1+x^(1/2))/(-x^(1/2)+x)^(1/2),x, algorithm="fricas")
Output:
1/2*sqrt(2)*log(-(17*x^2 - 4*(sqrt(2)*(3*x + 5)*sqrt(x) - sqrt(2)*(7*x + 1 ))*sqrt(x - sqrt(x)) - 16*(3*x + 1)*sqrt(x) + 46*x + 1)/(x^2 - 2*x + 1)) + log(-4*sqrt(x - sqrt(x))*(2*sqrt(x) - 1) - 8*x + 8*sqrt(x) - 1)
\[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {-\sqrt {x}+x}} \, dx=\int \frac {1}{\sqrt {- \sqrt {x} + x} \left (\sqrt {x} + 1\right )}\, dx \] Input:
integrate(1/(1+x**(1/2))/(-x**(1/2)+x)**(1/2),x)
Output:
Integral(1/(sqrt(-sqrt(x) + x)*(sqrt(x) + 1)), x)
\[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {-\sqrt {x}+x}} \, dx=\int { \frac {1}{\sqrt {x - \sqrt {x}} {\left (\sqrt {x} + 1\right )}} \,d x } \] Input:
integrate(1/(1+x^(1/2))/(-x^(1/2)+x)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(x - sqrt(x))*(sqrt(x) + 1)), x)
Time = 0.68 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {-\sqrt {x}+x}} \, dx=-\sqrt {2} \log \left (\frac {2 \, {\left (\sqrt {2} - \sqrt {x - \sqrt {x}} + \sqrt {x} + 1\right )}}{{\left | 2 \, \sqrt {2} + 2 \, \sqrt {x - \sqrt {x}} - 2 \, \sqrt {x} - 2 \right |}}\right ) - 2 \, \log \left ({\left | 2 \, \sqrt {x - \sqrt {x}} - 2 \, \sqrt {x} + 1 \right |}\right ) \] Input:
integrate(1/(1+x^(1/2))/(-x^(1/2)+x)^(1/2),x, algorithm="giac")
Output:
-sqrt(2)*log(2*(sqrt(2) - sqrt(x - sqrt(x)) + sqrt(x) + 1)/abs(2*sqrt(2) + 2*sqrt(x - sqrt(x)) - 2*sqrt(x) - 2)) - 2*log(abs(2*sqrt(x - sqrt(x)) - 2 *sqrt(x) + 1))
Timed out. \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {-\sqrt {x}+x}} \, dx=\int \frac {1}{\sqrt {x-\sqrt {x}}\,\left (\sqrt {x}+1\right )} \,d x \] Input:
int(1/((x - x^(1/2))^(1/2)*(x^(1/2) + 1)),x)
Output:
int(1/((x - x^(1/2))^(1/2)*(x^(1/2) + 1)), x)
Time = 0.17 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.24 \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {-\sqrt {x}+x}} \, dx=-\sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x}-1}+x^{\frac {1}{4}}-\sqrt {2}\, i +i \right )-\sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x}-1}+x^{\frac {1}{4}}+\sqrt {2}\, i -i \right )+\sqrt {2}\, \mathrm {log}\left (2 x^{\frac {1}{4}} \sqrt {\sqrt {x}-1}+2 \sqrt {x}+2 \sqrt {2}+2\right )+4 \,\mathrm {log}\left (\sqrt {\sqrt {x}-1}+x^{\frac {1}{4}}\right ) \] Input:
int(1/(1+x^(1/2))/(-x^(1/2)+x)^(1/2),x)
Output:
- sqrt(2)*log(sqrt(sqrt(x) - 1) + x**(1/4) - sqrt(2)*i + i) - sqrt(2)*log (sqrt(sqrt(x) - 1) + x**(1/4) + sqrt(2)*i - i) + sqrt(2)*log(2*x**(1/4)*sq rt(sqrt(x) - 1) + 2*sqrt(x) + 2*sqrt(2) + 2) + 4*log(sqrt(sqrt(x) - 1) + x **(1/4))