Optimal. Leaf size=41 \[ -\frac{2 (1-x)^{3/2}}{\sqrt{x+1}}-3 \sqrt{x+1} \sqrt{1-x}-3 \sin ^{-1}(x) \]
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Rubi [A] time = 0.0065795, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {47, 50, 41, 216} \[ -\frac{2 (1-x)^{3/2}}{\sqrt{x+1}}-3 \sqrt{x+1} \sqrt{1-x}-3 \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{(1-x)^{3/2}}{(1+x)^{3/2}} \, dx &=-\frac{2 (1-x)^{3/2}}{\sqrt{1+x}}-3 \int \frac{\sqrt{1-x}}{\sqrt{1+x}} \, dx\\ &=-\frac{2 (1-x)^{3/2}}{\sqrt{1+x}}-3 \sqrt{1-x} \sqrt{1+x}-3 \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-\frac{2 (1-x)^{3/2}}{\sqrt{1+x}}-3 \sqrt{1-x} \sqrt{1+x}-3 \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{2 (1-x)^{3/2}}{\sqrt{1+x}}-3 \sqrt{1-x} \sqrt{1+x}-3 \sin ^{-1}(x)\\ \end{align*}
Mathematica [C] time = 0.0085057, size = 37, normalized size = 0.9 \[ -\frac{(1-x)^{5/2} \, _2F_1\left (\frac{3}{2},\frac{5}{2};\frac{7}{2};\frac{1-x}{2}\right )}{5 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 71, normalized size = 1.7 \begin{align*}{({x}^{2}+4\,x-5)\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}}-3\,{\frac{\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }\arcsin \left ( x \right ) }{\sqrt{1-x}\sqrt{1+x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57728, size = 55, normalized size = 1.34 \begin{align*} \frac{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{x^{2} + 2 \, x + 1} - \frac{6 \, \sqrt{-x^{2} + 1}}{x + 1} - 3 \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85204, size = 146, normalized size = 3.56 \begin{align*} -\frac{{\left (x + 5\right )} \sqrt{x + 1} \sqrt{-x + 1} - 6 \,{\left (x + 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 5 \, x + 5}{x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.81974, size = 133, normalized size = 3.24 \begin{align*} \begin{cases} 6 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{i \left (x + 1\right )^{\frac{3}{2}}}{\sqrt{x - 1}} - \frac{2 i \sqrt{x + 1}}{\sqrt{x - 1}} + \frac{8 i}{\sqrt{x - 1} \sqrt{x + 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\- 6 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{\left (x + 1\right )^{\frac{3}{2}}}{\sqrt{1 - x}} + \frac{2 \sqrt{x + 1}}{\sqrt{1 - x}} - \frac{8}{\sqrt{1 - x} \sqrt{x + 1}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.0769, size = 95, normalized size = 2.32 \begin{align*} -\sqrt{x + 1} \sqrt{-x + 1} + \frac{2 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{\sqrt{x + 1}} - \frac{2 \, \sqrt{x + 1}}{\sqrt{2} - \sqrt{-x + 1}} - 6 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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