Integrand size = 19, antiderivative size = 56 \[ \int \frac {1}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=-\frac {2 \sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {-\sqrt {x}+x}}{\sqrt {2} \sqrt {x}}\right ) \] Output:
-2*(-x^(1/2)+x)^(1/2)/(-1+x^(1/2))+2^(1/2)*arctanh(1/2*(-x^(1/2)+x)^(1/2)* 2^(1/2)/x^(1/2))
Time = 0.14 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=-\frac {2 \sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right ) \] Input:
Integrate[1/((-1 + x)*Sqrt[-Sqrt[x] + x]),x]
Output:
(-2*Sqrt[-Sqrt[x] + x])/(-1 + Sqrt[x]) + Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[-Sq rt[x] + x])/(-1 + Sqrt[x])]
Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.41, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2468, 25, 1388, 1000, 25, 105, 104, 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.
\(\displaystyle \int \frac {1}{(x-1) \sqrt {x-\sqrt {x}}} \, dx\) |
\(\Big \downarrow \) 2468 |
\(\displaystyle \frac {\sqrt {\sqrt {x}-1} \sqrt [4]{x} \int -\frac {1}{\sqrt {\sqrt {x}-1} (1-x) \sqrt [4]{x}}dx}{\sqrt {x-\sqrt {x}}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {\sqrt {x}-1} \sqrt [4]{x} \int \frac {1}{\sqrt {\sqrt {x}-1} (1-x) \sqrt [4]{x}}dx}{\sqrt {x-\sqrt {x}}}\) |
\(\Big \downarrow \) 1388 |
\(\displaystyle -\frac {\sqrt {\sqrt {x}-1} \sqrt [4]{x} \int \frac {1}{\left (-\sqrt {x}-1\right ) \left (\sqrt {x}-1\right )^{3/2} \sqrt [4]{x}}dx}{\sqrt {x-\sqrt {x}}}\) |
\(\Big \downarrow \) 1000 |
\(\displaystyle -\frac {2 \sqrt {\sqrt {x}-1} \sqrt [4]{x} \int -\frac {\sqrt [4]{x}}{\left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )}d\sqrt {x}}{\sqrt {x-\sqrt {x}}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 \sqrt {\sqrt {x}-1} \sqrt [4]{x} \int \frac {\sqrt [4]{x}}{\left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )}d\sqrt {x}}{\sqrt {x-\sqrt {x}}}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle -\frac {2 \sqrt {\sqrt {x}-1} \sqrt [4]{x} \left (\frac {\sqrt [4]{x}}{\sqrt {\sqrt {x}-1}}-\frac {1}{2} \int \frac {1}{\sqrt {\sqrt {x}-1} \left (\sqrt {x}+1\right ) \sqrt [4]{x}}d\sqrt {x}\right )}{\sqrt {x-\sqrt {x}}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle -\frac {2 \sqrt {\sqrt {x}-1} \sqrt [4]{x} \left (\frac {\sqrt [4]{x}}{\sqrt {\sqrt {x}-1}}-\int \frac {1}{1-2 x}d\frac {\sqrt [4]{x}}{\sqrt {\sqrt {x}-1}}\right )}{\sqrt {x-\sqrt {x}}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {2 \sqrt {\sqrt {x}-1} \sqrt [4]{x} \left (\frac {\sqrt [4]{x}}{\sqrt {\sqrt {x}-1}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt {\sqrt {x}-1}}\right )}{\sqrt {2}}\right )}{\sqrt {x-\sqrt {x}}}\) |
Input:
Int[1/((-1 + x)*Sqrt[-Sqrt[x] + x]),x]
Output:
(-2*Sqrt[-1 + Sqrt[x]]*x^(1/4)*(x^(1/4)/Sqrt[-1 + Sqrt[x]] - ArcTanh[(Sqrt [2]*x^(1/4))/Sqrt[-1 + Sqrt[x]]]/Sqrt[2]))/Sqrt[-Sqrt[x] + x]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{g = Denominator[n]}, Simp[g Subst[Int[x^(g*(m + 1) - 1)*(a + b*x^(g*n))^p*(c + d*x^(g*n))^q, x], x, x^(1/g)], x]] /; FreeQ[{a, b , c, d, m, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[n]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_), x_Symbol] :> Simp[(a *x^r + b*x^s)^p/(x^(p*r)*(a + b*x^(s - r))^p) Int[x^(p*r)*(a + b*x^(s - r ))^p*Fx, x], x] /; FreeQ[{a, b, p, r, s}, x] && !IntegerQ[p] && PosQ[s - r ] && !(EqQ[p, 1] && EqQ[Fx, 1])
Time = 0.12 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(-\frac {2 \sqrt {\left (-1+\sqrt {x}\right )^{2}+\sqrt {x}-1}}{-1+\sqrt {x}}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (1-3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {\left (1+\sqrt {x}\right )^{2}-3 \sqrt {x}-1}}\right )}{2}\) | \(59\) |
default | \(-\frac {\sqrt {-\sqrt {x}+x}\, \left (2 \,\operatorname {arctanh}\left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right ) \sqrt {x}\, \sqrt {2}-\operatorname {arctanh}\left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right ) x \sqrt {2}+4 \left (-\sqrt {x}+x \right )^{\frac {3}{2}}-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right )+8 \sqrt {x}\, \sqrt {-\sqrt {x}+x}-4 x \sqrt {-\sqrt {x}+x}-4 \sqrt {-\sqrt {x}+x}\right )}{2 \sqrt {\sqrt {x}\, \left (-1+\sqrt {x}\right )}\, \left (-1+\sqrt {x}\right )^{2}}\) | \(164\) |
Input:
int(1/(-1+x)/(-x^(1/2)+x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/(-1+x^(1/2))*((-1+x^(1/2))^2+x^(1/2)-1)^(1/2)-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. 98 vs. \(2 (41) = 82\).
Time = 0.96 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.75 \[ \int \frac {1}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=\frac {\sqrt {2} {\left (x - 1\right )} \log \left (-\frac {17 \, x^{2} + 4 \, {\left (\sqrt {2} {\left (3 \, x + 5\right )} \sqrt {x} - \sqrt {2} {\left (7 \, x + 1\right )}\right )} \sqrt {x - \sqrt {x}} - 16 \, {\left (3 \, x + 1\right )} \sqrt {x} + 46 \, x + 1}{x^{2} - 2 \, x + 1}\right ) - 8 \, \sqrt {x - \sqrt {x}} {\left (\sqrt {x} + 1\right )}}{4 \, {\left (x - 1\right )}} \] Input:
integrate(1/(-1+x)/(-x^(1/2)+x)^(1/2),x, algorithm="fricas")
Output:
1/4*(sqrt(2)*(x - 1)*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)) - 8*sqrt(x - sqrt(x))*(sqrt(x) + 1))/(x - 1)
\[ \int \frac {1}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=\int \frac {1}{\sqrt {- \sqrt {x} + x} \left (x - 1\right )}\, dx \] Input:
integrate(1/(-1+x)/(-x**(1/2)+x)**(1/2),x)
Output:
Integral(1/(sqrt(-sqrt(x) + x)*(x - 1)), x)
\[ \int \frac {1}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=\int { \frac {1}{\sqrt {x - \sqrt {x}} {\left (x - 1\right )}} \,d x } \] Input:
integrate(1/(-1+x)/(-x^(1/2)+x)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(x - sqrt(x))*(x - 1)), x)
Time = 0.92 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.32 \[ \int \frac {1}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (\frac {2 \, {\left (\sqrt {2} - \sqrt {x - \sqrt {x}} + \sqrt {x} + 1\right )}}{{\left | 2 \, \sqrt {2} + 2 \, \sqrt {x - \sqrt {x}} - 2 \, \sqrt {x} - 2 \right |}}\right ) - \frac {2}{\sqrt {x - \sqrt {x}} - \sqrt {x} + 1} \] Input:
integrate(1/(-1+x)/(-x^(1/2)+x)^(1/2),x, algorithm="giac")
Output:
1/2*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/(sqrt(x - sqrt(x)) - sqrt(x) + 1)
Timed out. \[ \int \frac {1}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=\int \frac {1}{\sqrt {x-\sqrt {x}}\,\left (x-1\right )} \,d x \] Input:
int(1/((x - x^(1/2))^(1/2)*(x - 1)),x)
Output:
int(1/((x - x^(1/2))^(1/2)*(x - 1)), x)
Time = 0.15 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.86 \[ \int \frac {1}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx=\frac {-4 x^{\frac {1}{4}} \sqrt {\sqrt {x}-1}+\sqrt {x}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x}-1}+x^{\frac {1}{4}}-\sqrt {2}\, i +i \right )+\sqrt {x}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x}-1}+x^{\frac {1}{4}}+\sqrt {2}\, i -i \right )-\sqrt {x}\, \sqrt {2}\, \mathrm {log}\left (2 x^{\frac {1}{4}} \sqrt {\sqrt {x}-1}+2 \sqrt {x}+2 \sqrt {2}+2\right )-4 \sqrt {x}-\sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x}-1}+x^{\frac {1}{4}}-\sqrt {2}\, i +i \right )-\sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x}-1}+x^{\frac {1}{4}}+\sqrt {2}\, i -i \right )+\sqrt {2}\, \mathrm {log}\left (2 x^{\frac {1}{4}} \sqrt {\sqrt {x}-1}+2 \sqrt {x}+2 \sqrt {2}+2\right )+4}{2 \sqrt {x}-2} \] Input:
int(1/(-1+x)/(-x^(1/2)+x)^(1/2),x)
Output:
( - 4*x**(1/4)*sqrt(sqrt(x) - 1) + sqrt(x)*sqrt(2)*log(sqrt(sqrt(x) - 1) + x**(1/4) - sqrt(2)*i + i) + sqrt(x)*sqrt(2)*log(sqrt(sqrt(x) - 1) + x**(1 /4) + sqrt(2)*i - i) - sqrt(x)*sqrt(2)*log(2*x**(1/4)*sqrt(sqrt(x) - 1) + 2*sqrt(x) + 2*sqrt(2) + 2) - 4*sqrt(x) - 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)*sqrt(sqrt(x) - 1) + 2*sqrt(x) + 2*sqrt (2) + 2) + 4)/(2*(sqrt(x) - 1))