Optimal. Leaf size=44 \[ -\frac {\sqrt {1-x} \sqrt {1+x}}{x}+\sin ^{-1}(x)-2 \tanh ^{-1}\left (\sqrt {1-x} \sqrt {1+x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {100, 163, 41,
222, 94, 212} \begin {gather*} \text {ArcSin}(x)-\frac {\sqrt {1-x} \sqrt {x+1}}{x}-2 \tanh ^{-1}\left (\sqrt {1-x} \sqrt {x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 94
Rule 100
Rule 163
Rule 212
Rule 222
Rubi steps
\begin {align*} \int \frac {(1+x)^{3/2}}{\sqrt {1-x} x^2} \, dx &=-\frac {\sqrt {1-x} \sqrt {1+x}}{x}-\int \frac {-2-x}{\sqrt {1-x} x \sqrt {1+x}} \, dx\\ &=-\frac {\sqrt {1-x} \sqrt {1+x}}{x}+2 \int \frac {1}{\sqrt {1-x} x \sqrt {1+x}} \, dx+\int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\frac {\sqrt {1-x} \sqrt {1+x}}{x}-2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x} \sqrt {1+x}\right )+\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x} \sqrt {1+x}}{x}+\sin ^{-1}(x)-2 \tanh ^{-1}\left (\sqrt {1-x} \sqrt {1+x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 59, normalized size = 1.34 \begin {gather*} -\frac {\sqrt {1-x^2}+2 x \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right )+4 x \tanh ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 55, normalized size = 1.25
method | result | size |
default | \(\frac {\sqrt {1+x}\, \sqrt {1-x}\, \left (\arcsin \left (x \right ) x -2 \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x -\sqrt {-x^{2}+1}\right )}{x \sqrt {-x^{2}+1}}\) | \(55\) |
risch | \(\frac {\sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{x \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {\left (\arcsin \left (x \right )-2 \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {1-x}\, \sqrt {1+x}}\) | \(81\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 42, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {-x^{2} + 1}}{x} + \arcsin \left (x\right ) - 2 \, \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.33, size = 65, normalized size = 1.48 \begin {gather*} -\frac {2 \, x \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - 2 \, x \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + \sqrt {x + 1} \sqrt {-x + 1}}{x} \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 {\left (x + 1\right )^{\frac {3}{2}}}{x^{2} \sqrt {1 - x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \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 {{\left (x+1\right )}^{3/2}}{x^2\,\sqrt {1-x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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