Optimal. Leaf size=89 \[ \tan ^{-1}\left (\frac {3-\sqrt {x}}{2 \sqrt {x-\sqrt {x}-1}}\right )-2 \tanh ^{-1}\left (\frac {1-2 \sqrt {x}}{2 \sqrt {x-\sqrt {x}-1}}\right )-\tanh ^{-1}\left (\frac {3 \sqrt {x}+1}{2 \sqrt {x-\sqrt {x}-1}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.26, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {990, 621, 206, 1033, 724, 204} \[ \tan ^{-1}\left (\frac {3-\sqrt {x}}{2 \sqrt {x-\sqrt {x}-1}}\right )-2 \tanh ^{-1}\left (\frac {1-2 \sqrt {x}}{2 \sqrt {x-\sqrt {x}-1}}\right )-\tanh ^{-1}\left (\frac {3 \sqrt {x}+1}{2 \sqrt {x-\sqrt {x}-1}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 206
Rule 621
Rule 724
Rule 990
Rule 1033
Rubi steps
\begin {align*} \int \frac {\sqrt {-1-\sqrt {x}+x}}{(-1+x) \sqrt {x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {\sqrt {-1-x+x^2}}{-1+x^2} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x+x^2}} \, dx,x,\sqrt {x}\right )-2 \operatorname {Subst}\left (\int \frac {x}{\left (-1+x^2\right ) \sqrt {-1-x+x^2}} \, dx,x,\sqrt {x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+2 \sqrt {x}}{\sqrt {-1-\sqrt {x}+x}}\right )-\operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {-1-x+x^2}} \, dx,x,\sqrt {x}\right )-\operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt {-1-x+x^2}} \, dx,x,\sqrt {x}\right )\\ &=-2 \tanh ^{-1}\left (\frac {1-2 \sqrt {x}}{2 \sqrt {-1-\sqrt {x}+x}}\right )+2 \operatorname {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,\frac {-3+\sqrt {x}}{\sqrt {-1-\sqrt {x}+x}}\right )+2 \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1-3 \sqrt {x}}{\sqrt {-1-\sqrt {x}+x}}\right )\\ &=\tan ^{-1}\left (\frac {3-\sqrt {x}}{2 \sqrt {-1-\sqrt {x}+x}}\right )-2 \tanh ^{-1}\left (\frac {1-2 \sqrt {x}}{2 \sqrt {-1-\sqrt {x}+x}}\right )-\tanh ^{-1}\left (\frac {1+3 \sqrt {x}}{2 \sqrt {-1-\sqrt {x}+x}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 89, normalized size = 1.00 \[ \tan ^{-1}\left (\frac {3-\sqrt {x}}{2 \sqrt {x-\sqrt {x}-1}}\right )-2 \tanh ^{-1}\left (\frac {1-2 \sqrt {x}}{2 \sqrt {x-\sqrt {x}-1}}\right )-\tanh ^{-1}\left (\frac {3 \sqrt {x}+1}{2 \sqrt {x-\sqrt {x}-1}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 4.14, size = 87, normalized size = 0.98 \[ -\arctan \left (\frac {{\left ({\left (x - 4\right )} \sqrt {x} - 2 \, x + 3\right )} \sqrt {x - \sqrt {x} - 1}}{2 \, {\left (x^{2} - 3 \, x + 1\right )}}\right ) + \log \left (-\frac {8 \, x^{2} + 2 \, {\left ({\left (4 \, x - 5\right )} \sqrt {x} + 2 \, x - 1\right )} \sqrt {x - \sqrt {x} - 1} - 17 \, x - 2 \, \sqrt {x} + 11}{x - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.24, size = 81, normalized size = 0.91 \[ -2 \, \arctan \left (\sqrt {x - \sqrt {x} - 1} - \sqrt {x} + 1\right ) - \log \left (-\sqrt {x - \sqrt {x} - 1} + \sqrt {x} + 2\right ) + \log \left (-\sqrt {x - \sqrt {x} - 1} + \sqrt {x}\right ) - 2 \, \log \left ({\left | 2 \, \sqrt {x - \sqrt {x} - 1} - 2 \, \sqrt {x} + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 130, normalized size = 1.46 \[ \arctanh \left (\frac {-3 \sqrt {x}-1}{2 \sqrt {-3 \sqrt {x}+\left (\sqrt {x}+1\right )^{2}-2}}\right )-\arctan \left (\frac {\sqrt {x}-3}{2 \sqrt {\sqrt {x}+\left (\sqrt {x}-1\right )^{2}-2}}\right )+\frac {3 \ln \left (\sqrt {x}-\frac {1}{2}+\sqrt {-3 \sqrt {x}+\left (\sqrt {x}+1\right )^{2}-2}\right )}{2}+\frac {\ln \left (\sqrt {x}-\frac {1}{2}+\sqrt {\sqrt {x}+\left (\sqrt {x}-1\right )^{2}-2}\right )}{2}+\sqrt {\sqrt {x}+\left (\sqrt {x}-1\right )^{2}-2}-\sqrt {-3 \sqrt {x}+\left (\sqrt {x}+1\right )^{2}-2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x - \sqrt {x} - 1}}{{\left (x - 1\right )} \sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {x-\sqrt {x}-1}}{\sqrt {x}\,\left (x-1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \sqrt {x} + x - 1}}{\sqrt {x} \left (x - 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________