Optimal. Leaf size=23 \[ \frac {2 \sqrt {1+x}}{\sqrt {1-x}}-\sin ^{-1}(x) \]
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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {49, 41, 222}
\begin {gather*} \frac {2 \sqrt {x+1}}{\sqrt {1-x}}-\text {ArcSin}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 49
Rule 222
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x}}{(1-x)^{3/2}} \, dx &=\frac {2 \sqrt {1+x}}{\sqrt {1-x}}-\int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {2 \sqrt {1+x}}{\sqrt {1-x}}-\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {2 \sqrt {1+x}}{\sqrt {1-x}}-\sin ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 39, normalized size = 1.70 \begin {gather*} \frac {2 \sqrt {1+x}}{\sqrt {1-x}}-2 \tan ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs.
\(2(19)=38\).
time = 0.19, size = 64, normalized size = 2.78
method | result | size |
risch | \(\frac {2 \sqrt {1+x}\, \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 21, normalized size = 0.91 \begin {gather*} -\frac {2 \, \sqrt {-x^{2} + 1}}{x - 1} - \arcsin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs.
\(2 (19) = 38\).
time = 1.29, size = 48, normalized size = 2.09 \begin {gather*} \frac {2 \, {\left ({\left (x - 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + x - \sqrt {x + 1} \sqrt {-x + 1} - 1\right )}}{x - 1} \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 = 0.79, size = 70, normalized size = 3.04 \begin {gather*} \begin {cases} 2 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {2 i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- 2 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {2 \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 = 0.91, size = 33, normalized size = 1.43 \begin {gather*} -\frac {2 \, \sqrt {x + 1} \sqrt {-x + 1}}{x - 1} - 2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\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 = -0.04 \begin {gather*} \int \frac {\sqrt {x+1}}{{\left (1-x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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