Optimal. Leaf size=4 \[ \sin ^{-1}(4+x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1976, 633, 222}
\begin {gather*} \text {ArcSin}(x+4) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 222
Rule 633
Rule 1976
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {(-3-x) (5+x)}} \, dx &=\int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4}}} \, dx,x,-8-2 x\right )\right )\\ &=\sin ^{-1}(4+x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(44\) vs. \(2(4)=8\).
time = 0.00, size = 44, normalized size = 11.00 \begin {gather*} \frac {2 \sqrt {3+x} \sqrt {5+x} \tanh ^{-1}\left (\frac {\sqrt {3+x}}{\sqrt {5+x}}\right )}{\sqrt {-((3+x) (5+x))}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.58, size = 5, normalized size = 1.25
method | result | size |
default | \(\arcsin \left (x +4\right )\) | \(5\) |
trager | \(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-x \RootOf \left (\textit {\_Z}^{2}+1\right )-4 \RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {-x^{2}-8 x -15}\right )\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 8, normalized size = 2.00 \begin {gather*} -\arcsin \left (-x - 4\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. 29 vs.
\(2 (4) = 8\).
time = 0.33, size = 29, normalized size = 7.25 \begin {gather*} -\arctan \left (\frac {\sqrt {-x^{2} - 8 \, x - 15} {\left (x + 4\right )}}{x^{2} + 8 \, x + 15}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\left (- x - 3\right ) \left (x + 5\right )}}\, dx \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. 24 vs.
\(2 (4) = 8\).
time = 2.37, size = 24, normalized size = 6.00 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} - 8 \, x - 15} {\left (x + 4\right )} + \frac {1}{2} \, \arcsin \left (x + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.36, size = 4, normalized size = 1.00 \begin {gather*} \mathrm {asin}\left (x+4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________