Optimal. Leaf size=52 \[ \tan ^{-1}\left (\sqrt {x}\right )+\frac {\sqrt {x^3} \tan ^{-1}\left (\sqrt {x}\right )}{x^{3/2}}+\tanh ^{-1}\left (\sqrt {x}\right )-\frac {\sqrt {x^3} \tanh ^{-1}\left (\sqrt {x}\right )}{x^{3/2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1607, 6857,
335, 218, 212, 209, 15, 304} \begin {gather*} \frac {\sqrt {x^3} \text {ArcTan}\left (\sqrt {x}\right )}{x^{3/2}}+\text {ArcTan}\left (\sqrt {x}\right )-\frac {\sqrt {x^3} \tanh ^{-1}\left (\sqrt {x}\right )}{x^{3/2}}+\tanh ^{-1}\left (\sqrt {x}\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
Rubi steps
\begin {align*} \int \frac {\sqrt {x}-\sqrt {x^3}}{x-x^3} \, dx &=\int \frac {\sqrt {x}-\sqrt {x^3}}{x \left (1-x^2\right )} \, dx\\ &=\int \left (-\frac {1}{\sqrt {x} \left (-1+x^2\right )}+\frac {\sqrt {x^3}}{x \left (-1+x^2\right )}\right ) \, dx\\ &=-\int \frac {1}{\sqrt {x} \left (-1+x^2\right )} \, dx+\int \frac {\sqrt {x^3}}{x \left (-1+x^2\right )} \, dx\\ &=-\left (2 \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt {x}\right )\right )+\frac {\sqrt {x^3} \int \frac {\sqrt {x}}{-1+x^2} \, dx}{x^{3/2}}\\ &=\frac {\left (2 \sqrt {x^3}\right ) \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt {x}\right )}{x^{3/2}}+\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {x}\right )+\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right )\\ &=\tan ^{-1}\left (\sqrt {x}\right )+\tanh ^{-1}\left (\sqrt {x}\right )-\frac {\sqrt {x^3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {x}\right )}{x^{3/2}}+\frac {\sqrt {x^3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right )}{x^{3/2}}\\ &=\tan ^{-1}\left (\sqrt {x}\right )+\frac {\sqrt {x^3} \tan ^{-1}\left (\sqrt {x}\right )}{x^{3/2}}+\tanh ^{-1}\left (\sqrt {x}\right )-\frac {\sqrt {x^3} \tanh ^{-1}\left (\sqrt {x}\right )}{x^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 1.63, size = 39, normalized size = 0.75 \begin {gather*} \tan ^{-1}\left (\sqrt {x}\right )+\tan ^{-1}\left (\frac {\sqrt {x^3}}{x}\right )+\tanh ^{-1}\left (\sqrt {x}\right )-\tanh ^{-1}\left (\frac {\sqrt {x^3}}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 41, normalized size = 0.79
method | result | size |
default | \(\arctan \left (\sqrt {x}\right )+\arctanh \left (\sqrt {x}\right )+\frac {\sqrt {x^{3}}\, \left (\ln \left (-1+\sqrt {x}\right )-\ln \left (1+\sqrt {x}\right )+2 \arctan \left (\sqrt {x}\right )\right )}{2 x^{\frac {3}{2}}}\) | \(41\) |
meijerg | \(-\frac {\sqrt {x}\, \left (\ln \left (1-\left (x^{2}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (x^{2}\right )^{\frac {1}{4}}\right )-2 \arctan \left (\left (x^{2}\right )^{\frac {1}{4}}\right )\right )}{2 \left (x^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {x^{3}}\, \left (\ln \left (1-\left (x^{2}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (x^{2}\right )^{\frac {1}{4}}\right )+2 \arctan \left (\left (x^{2}\right )^{\frac {1}{4}}\right )\right )}{2 \left (x^{2}\right )^{\frac {3}{4}}}\) | \(82\) |
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 = 0.34, size = 6, normalized size = 0.12 \begin {gather*} 2 \, \arctan \left (\sqrt {x}\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 {\sqrt {x}}{x^{3} - x}\, dx - \int \left (- \frac {\sqrt {x^{3}}}{x^{3} - x}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.10, size = 6, normalized size = 0.12 \begin {gather*} 2 \, \arctan \left (\sqrt {x}\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 = -0.02 \begin {gather*} -\int \frac {\sqrt {x^3}-\sqrt {x}}{x-x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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