Optimal. Leaf size=213 \[ \sqrt {\frac {x-1}{x}} \left (\frac {8}{63} \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}} \sqrt {1-\sqrt {\frac {x-1}{x}}}-\frac {8}{315} \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}\right )-\frac {64}{315} \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}+\frac {64}{315} \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}} \sqrt {1-\sqrt {\frac {x-1}{x}}}+\frac {8 \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}} (x-1)}{9 x} \]
________________________________________________________________________________________
Rubi [A] time = 0.35, antiderivative size = 94, normalized size of antiderivative = 0.44, number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {6715, 371, 1398, 772} \begin {gather*} \frac {8}{9} \left (1-\sqrt {1-\sqrt {\frac {x-1}{x}}}\right )^{9/2}-\frac {24}{7} \left (1-\sqrt {1-\sqrt {\frac {x-1}{x}}}\right )^{7/2}+\frac {16}{5} \left (1-\sqrt {1-\sqrt {\frac {x-1}{x}}}\right )^{5/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rule 772
Rule 1398
Rule 6715
Rubi steps
\begin {align*} \int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{x^2} \, dx &=-\operatorname {Subst}\left (\int \sqrt {1-\sqrt {1-\sqrt {1-x}}} \, dx,x,\frac {1}{x}\right )\\ &=2 \operatorname {Subst}\left (\int \sqrt {1-\sqrt {1-x}} x \, dx,x,\sqrt {\frac {-1+x}{x}}\right )\\ &=2 \operatorname {Subst}\left (\int \sqrt {1-\sqrt {x}} (-1+x) \, dx,x,1-\sqrt {\frac {-1+x}{x}}\right )\\ &=4 \operatorname {Subst}\left (\int \sqrt {1-x} x \left (-1+x^2\right ) \, dx,x,\sqrt {1-\sqrt {\frac {-1+x}{x}}}\right )\\ &=4 \operatorname {Subst}\left (\int \left (-2 (1-x)^{3/2}+3 (1-x)^{5/2}-(1-x)^{7/2}\right ) \, dx,x,\sqrt {1-\sqrt {\frac {-1+x}{x}}}\right )\\ &=\frac {16}{5} \left (1-\sqrt {1-\sqrt {-\frac {1-x}{x}}}\right )^{5/2}-\frac {24}{7} \left (1-\sqrt {1-\sqrt {-\frac {1-x}{x}}}\right )^{7/2}+\frac {8}{9} \left (1-\sqrt {1-\sqrt {-\frac {1-x}{x}}}\right )^{9/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 67, normalized size = 0.31 \begin {gather*} \frac {8}{315} \left (1-\sqrt {1-\sqrt {\frac {x-1}{x}}}\right )^{5/2} \left (65 \sqrt {1-\sqrt {\frac {x-1}{x}}}-35 \sqrt {\frac {x-1}{x}}+61\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 2.20, size = 184, normalized size = 0.86 \begin {gather*} \frac {64}{315} \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}} \sqrt {1-\sqrt {\frac {-1+x}{x}}}+\left (-\frac {8}{315} \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}+\frac {8}{63} \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}} \sqrt {1-\sqrt {\frac {-1+x}{x}}}\right ) \sqrt {\frac {-1+x}{x}}+\frac {8 \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}} (-35+27 x)}{315 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.93, size = 75, normalized size = 0.35 \begin {gather*} \frac {8 \, {\left ({\left (5 \, x \sqrt {\frac {x - 1}{x}} + 8 \, x\right )} \sqrt {-\sqrt {\frac {x - 1}{x}} + 1} - x \sqrt {\frac {x - 1}{x}} + 27 \, x - 35\right )} \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}}{315 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-\sqrt {-\sqrt {-\frac {1}{x} + 1} + 1} + 1}}{x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.67, size = 71, normalized size = 0.33
method | result | size |
derivativedivides | \(\frac {8 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{\frac {9}{2}}}{9}-\frac {24 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{\frac {7}{2}}}{7}+\frac {16 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{\frac {5}{2}}}{5}\) | \(71\) |
default | \(\frac {8 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{\frac {9}{2}}}{9}-\frac {24 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{\frac {7}{2}}}{7}+\frac {16 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{\frac {5}{2}}}{5}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-\sqrt {-\sqrt {-\frac {1}{x} + 1} + 1} + 1}}{x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1 - \sqrt {1 - \sqrt {1 - \frac {1}{x}}}}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________