Optimal. Leaf size=21 \[ \frac {\sqrt {\frac {-1+x}{x}}}{\sqrt {1+\frac {1}{x}}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6310, 6315, 37}
\begin {gather*} \frac {\sqrt {\frac {x-1}{x}}}{\sqrt {\frac {1}{x}+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 6310
Rule 6315
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(x)}}{(1+x)^2} \, dx &=\int \frac {e^{\coth ^{-1}(x)}}{\left (1+\frac {1}{x}\right )^2 x^2} \, dx\\ &=-\text {Subst}\left (\int \frac {1}{\sqrt {1-x} (1+x)^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\sqrt {\frac {-1+x}{x}}}{\sqrt {1+\frac {1}{x}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 18, normalized size = 0.86 \begin {gather*} \frac {\sqrt {1-\frac {1}{x^2}} x}{1+x} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(36\) vs.
\(2(17)=34\).
time = 0.08, size = 37, normalized size = 1.76
method | result | size |
trager | \(\sqrt {-\frac {1-x}{1+x}}\) | \(15\) |
gosper | \(\frac {-1+x}{\sqrt {\frac {-1+x}{1+x}}\, \left (1+x \right )}\) | \(21\) |
risch | \(\frac {-1+x}{\sqrt {\frac {-1+x}{1+x}}\, \left (1+x \right )}\) | \(21\) |
default | \(\frac {\sqrt {x^{2}-1}\, \left (-1+x \right )}{\left (1+x \right ) \sqrt {\left (1+x \right ) \left (-1+x \right )}\, \sqrt {\frac {-1+x}{1+x}}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 11, normalized size = 0.52 \begin {gather*} \sqrt {\frac {x - 1}{x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 11, normalized size = 0.52 \begin {gather*} \sqrt {\frac {x - 1}{x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 3.46, size = 8, normalized size = 0.38 \begin {gather*} \sqrt {\frac {x - 1}{x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.40, size = 22, normalized size = 1.05 \begin {gather*} \frac {2}{{\left (x - \sqrt {x^{2} - 1} + 1\right )} \mathrm {sgn}\left (x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.18, size = 11, normalized size = 0.52 \begin {gather*} \sqrt {1-\frac {2}{x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________