Optimal. Leaf size=35 \[ \frac {4 (x+1)}{\sqrt {1-x^2}}-\tanh ^{-1}\left (\sqrt {1-x^2}\right )-\sin ^{-1}(x) \]
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Rubi [A] time = 0.06, antiderivative size = 35, 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 = {1805, 844, 216, 266, 63, 206} \[ \frac {4 (x+1)}{\sqrt {1-x^2}}-\tanh ^{-1}\left (\sqrt {1-x^2}\right )-\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 216
Rule 266
Rule 844
Rule 1805
Rubi steps
\begin {align*} \int \frac {(1+x)^3}{x \left (1-x^2\right )^{3/2}} \, dx &=\frac {4 (1+x)}{\sqrt {1-x^2}}-\int \frac {-1+x}{x \sqrt {1-x^2}} \, dx\\ &=\frac {4 (1+x)}{\sqrt {1-x^2}}-\int \frac {1}{\sqrt {1-x^2}} \, dx+\int \frac {1}{x \sqrt {1-x^2}} \, dx\\ &=\frac {4 (1+x)}{\sqrt {1-x^2}}-\sin ^{-1}(x)+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^2\right )\\ &=\frac {4 (1+x)}{\sqrt {1-x^2}}-\sin ^{-1}(x)-\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^2}\right )\\ &=\frac {4 (1+x)}{\sqrt {1-x^2}}-\sin ^{-1}(x)-\tanh ^{-1}\left (\sqrt {1-x^2}\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 47, normalized size = 1.34 \[ \frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1-x^2\right )-\sqrt {1-x^2} \sin ^{-1}(x)+4 x+3}{\sqrt {1-x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 63, normalized size = 1.80 \[ \frac {2 \, {\left (x - 1\right )} \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) + {\left (x - 1\right )} \log \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) + 4 \, x - 4 \, \sqrt {-x^{2} + 1} - 4}{x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 44, normalized size = 1.26 \[ \frac {8}{\frac {\sqrt {-x^{2} + 1} - 1}{x} + 1} - \arcsin \relax (x) + \log \left (-\frac {\sqrt {-x^{2} + 1} - 1}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 41, normalized size = 1.17 \[ \frac {4 x}{\sqrt {-x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )-\arcsin \relax (x )+\frac {4}{\sqrt {-x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.99, size = 53, normalized size = 1.51 \[ \frac {4 \, x}{\sqrt {-x^{2} + 1}} + \frac {4}{\sqrt {-x^{2} + 1}} - \arcsin \relax (x) - \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.20, size = 37, normalized size = 1.06 \[ \ln \left (\sqrt {\frac {1}{x^2}-1}-\sqrt {\frac {1}{x^2}}\right )-\mathrm {asin}\relax (x)-\frac {4\,\sqrt {1-x^2}}{x-1} \]
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 \[ \int \frac {\left (x + 1\right )^{3}}{x \left (- \left (x - 1\right ) \left (x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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