Optimal. Leaf size=41 \[ 3 \sqrt {1-x} \sqrt {1+x}+\frac {2 (1+x)^{3/2}}{\sqrt {1-x}}-3 \sin ^{-1}(x) \]
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Rubi [A]
time = 0.00, 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 = {49, 52, 41, 222}
\begin {gather*} \frac {2 (x+1)^{3/2}}{\sqrt {1-x}}+3 \sqrt {1-x} \sqrt {x+1}-3 \sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 49
Rule 52
Rule 222
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\\ &=3 \sqrt {1-x} \sqrt {1+x}+\frac {2 (1+x)^{3/2}}{\sqrt {1-x}}-3 \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=3 \sqrt {1-x} \sqrt {1+x}+\frac {2 (1+x)^{3/2}}{\sqrt {1-x}}-3 \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=3 \sqrt {1-x} \sqrt {1+x}+\frac {2 (1+x)^{3/2}}{\sqrt {1-x}}-3 \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 41, normalized size = 1.00 \begin {gather*} \frac {(-5+x) \sqrt {1-x^2}}{-1+x}-6 \tan ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 3.67, size = 89, normalized size = 2.17 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (\left (1+x\right )^{\frac {3}{2}}-6 \sqrt {1+x}+6 \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \sqrt {-1+x}\right )}{\sqrt {-1+x}},\text {Abs}\left [1+x\right ]>2\right \}\right \},-6 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]-\frac {\left (1+x\right )^{\frac {3}{2}}}{\sqrt {1-x}}+\frac {6 \sqrt {1+x}}{\sqrt {1-x}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs.
\(2(33)=66\).
time = 0.16, size = 72, normalized size = 1.76
method | result | size |
risch | \(-\frac {\left (x^{2}-4 x -5\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}\, \sqrt {1+x}}-\frac {3 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 42, normalized size = 1.02 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 52, normalized size = 1.27 \begin {gather*} \frac {\sqrt {x + 1} {\left (x - 5\right )} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.67, size = 99, normalized size = 2.41 \begin {gather*} \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 {6 i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- 6 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {\left (x + 1\right )^{\frac {3}{2}}}{\sqrt {1 - x}} + \frac {6 \sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 61, normalized size = 1.49 \begin {gather*} \frac {2 \left (-\frac {1}{2} \sqrt {x+1} \sqrt {x+1}+3\right ) \sqrt {x+1} \sqrt {-x+1}}{-x+1}-6 \arcsin \left (\frac {\sqrt {x+1}}{\sqrt {2}}\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 {{\left (x+1\right )}^{3/2}}{{\left (1-x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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