Optimal. Leaf size=24 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {5-x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {455, 65, 213}
\begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {5-x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 213
Rule 455
Rubi steps
\begin {align*} \int \frac {x}{\left (3-x^2\right ) \sqrt {5-x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{(3-x) \sqrt {5-x}} \, dx,x,x^2\right )\\ &=-\text {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\sqrt {5-x^2}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {5-x^2}}{\sqrt {2}}\right )}{\sqrt {2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 24, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {5-x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 5.62, size = 63, normalized size = 2.62 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\sqrt {2} \text {ArcCoth}\left [\frac {\sqrt {2}}{\sqrt {5-x^2}}\right ]}{2},\frac {1}{-5+x^2}<-\frac {1}{2}\right \},\left \{\frac {\sqrt {2} \text {ArcTanh}\left [\frac {\sqrt {2}}{\sqrt {5-x^2}}\right ]}{2},\frac {1}{-5+x^2}>-\frac {1}{2}\right \},\left \{0,\text {True}\right \}\right \}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(99\) vs.
\(2(20)=40\).
time = 0.12, size = 100, normalized size = 4.17
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}-7 \RootOf \left (\textit {\_Z}^{2}-2\right )-4 \sqrt {-x^{2}+5}}{x^{2}-3}\right )}{4}\) | \(48\) |
default | \(\frac {\sqrt {2}\, \arctanh \left (\frac {\left (4-2 \sqrt {3}\, \left (x -\sqrt {3}\right )\right ) \sqrt {2}}{4 \sqrt {-\left (x -\sqrt {3}\right )^{2}-2 \sqrt {3}\, \left (x -\sqrt {3}\right )+2}}\right )}{4}+\frac {\sqrt {2}\, \arctanh \left (\frac {\left (4+2 \sqrt {3}\, \left (x +\sqrt {3}\right )\right ) \sqrt {2}}{4 \sqrt {-\left (x +\sqrt {3}\right )^{2}+2 \sqrt {3}\, \left (x +\sqrt {3}\right )+2}}\right )}{4}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs.
\(2 (20) = 40\).
time = 0.35, size = 112, normalized size = 4.67 \begin {gather*} \frac {1}{12} \, \sqrt {3} {\left (\sqrt {3} \sqrt {2} \log \left (\sqrt {3} + \frac {2 \, \sqrt {2} \sqrt {-x^{2} + 5}}{{\left | 2 \, x + 2 \, \sqrt {3} \right |}} + \frac {4}{{\left | 2 \, x + 2 \, \sqrt {3} \right |}}\right ) + \sqrt {3} \sqrt {2} \log \left (-\sqrt {3} + \frac {2 \, \sqrt {2} \sqrt {-x^{2} + 5}}{{\left | 2 \, x - 2 \, \sqrt {3} \right |}} + \frac {4}{{\left | 2 \, x - 2 \, \sqrt {3} \right |}}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs.
\(2 (20) = 40\).
time = 0.32, size = 48, normalized size = 2.00 \begin {gather*} \frac {1}{8} \, \sqrt {2} \log \left (\frac {x^{4} - 4 \, \sqrt {2} {\left (x^{2} - 7\right )} \sqrt {-x^{2} + 5} - 22 \, x^{2} + 89}{x^{4} - 6 \, x^{2} + 9}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 2.31, size = 61, normalized size = 2.54 \begin {gather*} - \begin {cases} - \frac {\sqrt {2} \operatorname {acoth}{\left (\frac {\sqrt {2}}{\sqrt {5 - x^{2}}} \right )}}{2} & \text {for}\: \frac {1}{5 - x^{2}} > \frac {1}{2} \\- \frac {\sqrt {2} \operatorname {atanh}{\left (\frac {\sqrt {2}}{\sqrt {5 - x^{2}}} \right )}}{2} & \text {for}\: \frac {1}{5 - x^{2}} < \frac {1}{2} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 42 vs.
\(2 (20) = 40\).
time = 0.00, size = 54, normalized size = 2.25 \begin {gather*} -\frac {1}{4} \sqrt {2} \ln \left |\sqrt {-x^{2}+5}-\sqrt {2}\right |+\frac {1}{4} \sqrt {2} \ln \left (\sqrt {-x^{2}+5}+\sqrt {2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.79, size = 78, normalized size = 3.25 \begin {gather*} \frac {\sqrt {2}\,\left (\ln \left (\frac {\frac {\sqrt {2}\,\left (\sqrt {3}\,x+5\right )\,1{}\mathrm {i}}{2}+\sqrt {5-x^2}\,1{}\mathrm {i}}{x+\sqrt {3}}\right )+\ln \left (\frac {\frac {\sqrt {2}\,\left (\sqrt {3}\,x-5\right )\,1{}\mathrm {i}}{2}-\sqrt {5-x^2}\,1{}\mathrm {i}}{x-\sqrt {3}}\right )\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________