Optimal. Leaf size=68 \[ \frac {\sqrt {(x-1)^3} \tan ^{-1}\left (\sqrt {x-1}\right )}{(x-1)^{3/2}}+\tan ^{-1}\left (\sqrt {x-1}\right )-\frac {\sqrt {(x-1)^3} \tanh ^{-1}\left (\sqrt {x-1}\right )}{(x-1)^{3/2}}+\tanh ^{-1}\left (\sqrt {x-1}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {6729, 1593, 6725, 329, 212, 206, 203, 15, 298} \begin {gather*} \frac {\sqrt {(x-1)^3} \tan ^{-1}\left (\sqrt {x-1}\right )}{(x-1)^{3/2}}+\tan ^{-1}\left (\sqrt {x-1}\right )-\frac {\sqrt {(x-1)^3} \tanh ^{-1}\left (\sqrt {x-1}\right )}{(x-1)^{3/2}}+\tanh ^{-1}\left (\sqrt {x-1}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 203
Rule 206
Rule 212
Rule 298
Rule 329
Rule 1593
Rule 6725
Rule 6729
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+x}+\sqrt {(-1+x)^3}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {x}+\sqrt {x^3}} \, dx,x,-1+x\right )\\ &=\operatorname {Subst}\left (\int \frac {\sqrt {x}-\sqrt {x^3}}{x-x^3} \, dx,x,-1+x\right )\\ &=\operatorname {Subst}\left (\int \frac {\sqrt {x}-\sqrt {x^3}}{x \left (1-x^2\right )} \, dx,x,-1+x\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {x} \left (-1+x^2\right )}+\frac {\sqrt {x^3}}{x \left (-1+x^2\right )}\right ) \, dx,x,-1+x\right )\\ &=-\operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+x^2\right )} \, dx,x,-1+x\right )+\operatorname {Subst}\left (\int \frac {\sqrt {x^3}}{x \left (-1+x^2\right )} \, dx,x,-1+x\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt {-1+x}\right )\right )+\frac {\sqrt {(-1+x)^3} \operatorname {Subst}\left (\int \frac {\sqrt {x}}{-1+x^2} \, dx,x,-1+x\right )}{(-1+x)^{3/2}}\\ &=\frac {\left (2 \sqrt {(-1+x)^3}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt {-1+x}\right )}{(-1+x)^{3/2}}+\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {-1+x}\right )+\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x}\right )\\ &=\tan ^{-1}\left (\sqrt {-1+x}\right )+\tanh ^{-1}\left (\sqrt {-1+x}\right )-\frac {\sqrt {(-1+x)^3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {-1+x}\right )}{(-1+x)^{3/2}}+\frac {\sqrt {(-1+x)^3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x}\right )}{(-1+x)^{3/2}}\\ &=\tan ^{-1}\left (\sqrt {-1+x}\right )+\frac {\sqrt {(-1+x)^3} \tan ^{-1}\left (\sqrt {-1+x}\right )}{(-1+x)^{3/2}}+\tanh ^{-1}\left (\sqrt {-1+x}\right )-\frac {\sqrt {(-1+x)^3} \tanh ^{-1}\left (\sqrt {-1+x}\right )}{(-1+x)^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.16, size = 64, normalized size = 0.94 \begin {gather*} \left (\frac {\sqrt {(x-1)^3}}{(x-1)^{3/2}}+1\right ) \tan ^{-1}\left (\sqrt {x-1}\right )+\frac {\left ((x-1)^{3/2}-\sqrt {(x-1)^3}\right ) \tanh ^{-1}\left (\sqrt {x-1}\right )}{(x-1)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 2.93, size = 67, normalized size = 0.99 \begin {gather*} \tan ^{-1}\left (\frac {\sqrt {x^3-3 x^2+3 x-1}}{x-1}\right )-\tanh ^{-1}\left (\frac {\sqrt {x^3-3 x^2+3 x-1}}{x-1}\right )+\tan ^{-1}\left (\sqrt {x-1}\right )+\tanh ^{-1}\left (\sqrt {x-1}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 8, normalized size = 0.12 \begin {gather*} 2 \, \arctan \left (\sqrt {x - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.41, size = 8, normalized size = 0.12 \begin {gather*} 2 \, \arctan \left (\sqrt {x - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 40, normalized size = 0.59 \begin {gather*} \frac {2 \arctan \left (\sqrt {\frac {\sqrt {\left (x -1\right )^{3}}}{\left (x -1\right )^{\frac {3}{2}}}}\, \sqrt {x -1}\right )}{\sqrt {\frac {\sqrt {\left (x -1\right )^{3}}}{\left (x -1\right )^{\frac {3}{2}}}}} \end {gather*}
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*} 2 \, \sqrt {x - 1} - \int \frac {\sqrt {x - 1}}{x}\,{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 = -0.01 \begin {gather*} \int -\frac {\sqrt {x-1}-\sqrt {{\left (x-1\right )}^3}}{{\left (x-1\right )}^3-x+1} \,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 {1}{\sqrt {x - 1} + \sqrt {\left (x - 1\right )^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________