Optimal. Leaf size=49 \[ \frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{8} \sin ^{-1}(x) \]
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Rubi [A]
time = 0.00, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {38, 41, 222}
\begin {gather*} \frac {1}{4} (1-x)^{3/2} x (x+1)^{3/2}+\frac {3}{8} \sqrt {1-x} x \sqrt {x+1}+\frac {3}{8} \sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 38
Rule 41
Rule 222
Rubi steps
\begin {align*} \int (1-x)^{3/2} (1+x)^{3/2} \, dx &=\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{4} \int \sqrt {1-x} \sqrt {1+x} \, dx\\ &=\frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{8} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{8} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{8} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 46, normalized size = 0.94 \begin {gather*} -\frac {1}{8} x \sqrt {1-x^2} \left (-5+2 x^2\right )-\frac {3}{4} \tan ^{-1}\left (\frac {\sqrt {1-x^2}}{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 = 13.50, size = 154, normalized size = 3.14 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-13 \left (1+x\right )^{\frac {5}{2}}-6 \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \sqrt {-1+x}-2 \left (1+x\right )^{\frac {9}{2}}-\left (1+x\right )^{\frac {3}{2}}+6 \sqrt {1+x}+10 \left (1+x\right )^{\frac {7}{2}}\right )}{8 \sqrt {-1+x}},\text {Abs}\left [1+x\right ]>2\right \}\right \},\frac {-5 \left (1+x\right )^{\frac {7}{2}}}{4 \sqrt {1-x}}-\frac {3 \sqrt {1+x}}{4 \sqrt {1-x}}+\frac {\left (1+x\right )^{\frac {3}{2}}}{8 \sqrt {1-x}}+\frac {\left (1+x\right )^{\frac {9}{2}}}{4 \sqrt {1-x}}+\frac {3 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]}{4}+\frac {13 \left (1+x\right )^{\frac {5}{2}}}{8 \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. \(84\) vs.
\(2(35)=70\).
time = 0.14, size = 85, normalized size = 1.73
method | result | size |
risch | \(\frac {x \left (2 x^{2}-5\right ) \sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{8 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {3 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\) | \(75\) |
default | \(\frac {\left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {5}{2}}}{4}+\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {5}{2}}}{4}-\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{8}-\frac {3 \sqrt {1-x}\, \sqrt {1+x}}{8}+\frac {3 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 29, normalized size = 0.59 \begin {gather*} \frac {1}{4} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {3}{8} \, \sqrt {-x^{2} + 1} x + \frac {3}{8} \, \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 = 46, normalized size = 0.94 \begin {gather*} -\frac {1}{8} \, {\left (2 \, x^{3} - 5 \, x\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {3}{4} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 11.65, size = 212, normalized size = 4.33 \begin {gather*} \begin {cases} - \frac {3 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} - \frac {i \left (x + 1\right )^{\frac {9}{2}}}{4 \sqrt {x - 1}} + \frac {5 i \left (x + 1\right )^{\frac {7}{2}}}{4 \sqrt {x - 1}} - \frac {13 i \left (x + 1\right )^{\frac {5}{2}}}{8 \sqrt {x - 1}} - \frac {i \left (x + 1\right )^{\frac {3}{2}}}{8 \sqrt {x - 1}} + \frac {3 i \sqrt {x + 1}}{4 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {3 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} + \frac {\left (x + 1\right )^{\frac {9}{2}}}{4 \sqrt {1 - x}} - \frac {5 \left (x + 1\right )^{\frac {7}{2}}}{4 \sqrt {1 - x}} + \frac {13 \left (x + 1\right )^{\frac {5}{2}}}{8 \sqrt {1 - x}} + \frac {\left (x + 1\right )^{\frac {3}{2}}}{8 \sqrt {1 - x}} - \frac {3 \sqrt {x + 1}}{4 \sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 104 vs.
\(2 (35) = 70\).
time = 0.02, size = 298, normalized size = 6.08 \begin {gather*} -2 \left (2 \left (\left (\left (\frac {13}{48}-\frac {1}{16} \sqrt {-x+1} \sqrt {-x+1}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {43}{96}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {13}{32}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {3}{8} \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )\right )+2 \left (2 \left (\left (\frac {1}{12} \sqrt {-x+1} \sqrt {-x+1}-\frac {7}{24}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {3}{8}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {\arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )}{2}\right )+2 \left (2 \left (\frac {3}{8}-\frac {1}{8} \sqrt {-x+1} \sqrt {-x+1}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {\arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )}{2}\right )-2 \left (\frac {1}{2} \sqrt {-x+1} \sqrt {x+1}+\arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )\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 {\left (1-x\right )}^{3/2}\,{\left (x+1\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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