Optimal. Leaf size=68 \[ \tan ^{-1}\left (\sqrt {-1+x}\right )+\frac {\sqrt {(-1+x)^3} \tan ^{-1}\left (\sqrt {-1+x}\right )}{(-1+x)^{3/2}}+\tanh ^{-1}\left (\sqrt {-1+x}\right )-\frac {\sqrt {(-1+x)^3} \tanh ^{-1}\left (\sqrt {-1+x}\right )}{(-1+x)^{3/2}} \]
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Rubi [A]
time = 0.10, 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 = {6861, 1607,
6857, 335, 218, 212, 209, 15, 304} \begin {gather*} \frac {\sqrt {(x-1)^3} \text {ArcTan}\left (\sqrt {x-1}\right )}{(x-1)^{3/2}}+\text {ArcTan}\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.
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Rule 15
Rule 209
Rule 212
Rule 218
Rule 304
Rule 335
Rule 1607
Rule 6857
Rule 6861
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+x}+\sqrt {(-1+x)^3}} \, dx &=\text {Subst}\left (\int \frac {1}{\sqrt {x}+\sqrt {x^3}} \, dx,x,-1+x\right )\\ &=\text {Subst}\left (\int \frac {\sqrt {x}-\sqrt {x^3}}{x-x^3} \, dx,x,-1+x\right )\\ &=\text {Subst}\left (\int \frac {\sqrt {x}-\sqrt {x^3}}{x \left (1-x^2\right )} \, dx,x,-1+x\right )\\ &=\text {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 )\\ &=-\text {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+x^2\right )} \, dx,x,-1+x\right )+\text {Subst}\left (\int \frac {\sqrt {x^3}}{x \left (-1+x^2\right )} \, dx,x,-1+x\right )\\ &=-\left (2 \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt {-1+x}\right )\right )+\frac {\sqrt {(-1+x)^3} \text {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 ) \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt {-1+x}\right )}{(-1+x)^{3/2}}+\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {-1+x}\right )+\text {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} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {-1+x}\right )}{(-1+x)^{3/2}}+\frac {\sqrt {(-1+x)^3} \text {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*}
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Mathematica [A]
time = 1.76, size = 67, normalized size = 0.99 \begin {gather*} \tan ^{-1}\left (\sqrt {-1+x}\right )+\tan ^{-1}\left (\frac {\sqrt {-1+3 x-3 x^2+x^3}}{-1+x}\right )+\tanh ^{-1}\left (\sqrt {-1+x}\right )-\tanh ^{-1}\left (\frac {\sqrt {-1+3 x-3 x^2+x^3}}{-1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 40, normalized size = 0.59
method | result | size |
default | \(\frac {2 \arctan \left (\sqrt {\frac {\sqrt {\left (-1+x \right )^{3}}}{\left (-1+x \right )^{\frac {3}{2}}}}\, \sqrt {-1+x}\right )}{\sqrt {\frac {\sqrt {\left (-1+x \right )^{3}}}{\left (-1+x \right )^{\frac {3}{2}}}}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, 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.
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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.
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Giac [A]
time = 3.01, 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.
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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.
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