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 = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, 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 41
Rule 47
Rule 50
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*}
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Mathematica [C] time = 0.01, size = 37, normalized size = 0.90 \[ -\frac {(1-x)^{5/2} \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};\frac {1-x}{2}\right )}{5 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 53, normalized size = 1.29 \[ -\frac {{\left (x + 5\right )} \sqrt {x + 1} \sqrt {-x + 1} - 6 \, {\left (x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 5 \, x + 5}{x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.73, size = 70, normalized size = 1.71 \[ -\sqrt {x + 1} \sqrt {-x + 1} + \frac {2 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{\sqrt {x + 1}} - \frac {2 \, \sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} - 6 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 71, normalized size = 1.73 \[ -\frac {3 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{\sqrt {x +1}\, \sqrt {-x +1}}+\frac {\left (x^{2}+4 x -5\right ) \sqrt {\left (x +1\right ) \left (-x +1\right )}}{\sqrt {-\left (x +1\right ) \left (x -1\right )}\, \sqrt {-x +1}\, \sqrt {x +1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.86, size = 41, normalized size = 1.00 \[ \frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{x^{2} + 2 \, x + 1} - \frac {6 \, \sqrt {-x^{2} + 1}}{x + 1} - 3 \, \arcsin \relax (x) \]
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 \[ \int \frac {{\left (1-x\right )}^{3/2}}{{\left (x+1\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.48, size = 133, normalized size = 3.24 \[ \begin {cases} 6 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {i \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {x - 1}} - \frac {2 i \sqrt {x + 1}}{\sqrt {x - 1}} + \frac {8 i}{\sqrt {x - 1} \sqrt {x + 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\- 6 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {3}{2}}}{\sqrt {1 - x}} + \frac {2 \sqrt {x + 1}}{\sqrt {1 - x}} - \frac {8}{\sqrt {1 - x} \sqrt {x + 1}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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