Optimal. Leaf size=20 \[ -\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-b x}}{\sqrt {2}}\right )}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {65, 221}
\begin {gather*} -\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-b x}}{\sqrt {2}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 221
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-b x} \sqrt {2-b x}} \, dx &=-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {2+x^2}} \, dx,x,\sqrt {-b x}\right )}{b}\\ &=-\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-b x}}{\sqrt {2}}\right )}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(48\) vs. \(2(20)=40\).
time = 0.01, size = 48, normalized size = 2.40 \begin {gather*} -\frac {2 \sqrt {x} \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {2-b x}\right )}{\sqrt {-b} \sqrt {-b x}} \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 = 2.61, size = 43, normalized size = 2.15 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {-2 \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2}\right ]}{b},\text {Abs}\left [b x\right ]>2\right \}\right \},\frac {-2 I \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2}\right ]}{b}\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. \(63\) vs.
\(2(17)=34\).
time = 0.14, size = 64, normalized size = 3.20
method | result | size |
meijerg | \(\frac {2 \sqrt {x}\, \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{\sqrt {b}\, \sqrt {-b x}}\) | \(27\) |
default | \(\frac {\sqrt {-b x \left (-b x +2\right )}\, \ln \left (\frac {b^{2} x -b}{\sqrt {b^{2}}}+\sqrt {x^{2} b^{2}-2 b x}\right )}{\sqrt {-b x}\, \sqrt {-b x +2}\, \sqrt {b^{2}}}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 32, normalized size = 1.60 \begin {gather*} \frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} - 2 \, b x} b - 2 \, b\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.29, size = 27, normalized size = 1.35 \begin {gather*} -\frac {\log \left (-b x + \sqrt {-b x + 2} \sqrt {-b x} + 1\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.76, size = 51, normalized size = 2.55 \begin {gather*} \begin {cases} - \frac {2 \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b} & \text {for}\: \left |{b x}\right | > 2 \\- \frac {2 i \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 25, normalized size = 1.25 \begin {gather*} \frac {2 \ln \left (\sqrt {-b x+2}-\sqrt {-b x}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.28, size = 39, normalized size = 1.95 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {2}-\sqrt {2-b\,x}\right )}{\sqrt {-b\,x}\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________