Optimal. Leaf size=27 \[ \sinh ^{-1}(x)-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {1+x^2}}\right )}{\sqrt {2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {399, 221, 385,
212} \begin {gather*} \sinh ^{-1}(x)-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {x^2+1}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 221
Rule 385
Rule 399
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^2}}{2+x^2} \, dx &=\int \frac {1}{\sqrt {1+x^2}} \, dx-\int \frac {1}{\sqrt {1+x^2} \left (2+x^2\right )} \, dx\\ &=\sinh ^{-1}(x)-\text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\frac {x}{\sqrt {1+x^2}}\right )\\ &=\sinh ^{-1}(x)-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {1+x^2}}\right )}{\sqrt {2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.09, size = 44, normalized size = 1.63 \begin {gather*} \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {2+x^2-x \sqrt {1+x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 23, normalized size = 0.85
method | result | size |
default | \(\arcsinh \left (x \right )-\frac {\arctanh \left (\frac {x \sqrt {2}}{2 \sqrt {x^{2}+1}}\right ) \sqrt {2}}{2}\) | \(23\) |
trager | \(\ln \left (x +\sqrt {x^{2}+1}\right )-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+4 x \sqrt {x^{2}+1}+2 \RootOf \left (\textit {\_Z}^{2}-2\right )}{x^{2}+2}\right )}{4}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 67 vs.
\(2 (22) = 44\).
time = 0.49, size = 67, normalized size = 2.48 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (\frac {9 \, x^{2} - 2 \, \sqrt {2} {\left (3 \, x^{2} + 2\right )} - 2 \, \sqrt {x^{2} + 1} {\left (3 \, \sqrt {2} x - 4 \, x\right )} + 6}{x^{2} + 2}\right ) - \log \left (-x + \sqrt {x^{2} + 1}\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 {\sqrt {x^{2} + 1}}{x^{2} + 2}\, 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. 64 vs.
\(2 (22) = 44\).
time = 0.82, size = 64, normalized size = 2.37 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (\frac {{\left (x - \sqrt {x^{2} + 1}\right )}^{2} - 2 \, \sqrt {2} + 3}{{\left (x - \sqrt {x^{2} + 1}\right )}^{2} + 2 \, \sqrt {2} + 3}\right ) - \log \left (-x + \sqrt {x^{2} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.17, size = 77, normalized size = 2.85 \begin {gather*} \mathrm {asinh}\left (x\right )+\frac {\sqrt {2}\,\left (\ln \left (x-\sqrt {2}\,1{}\mathrm {i}\right )-\ln \left (1+\sqrt {2}\,x\,1{}\mathrm {i}+\sqrt {x^2+1}\,1{}\mathrm {i}\right )\right )}{4}-\frac {\sqrt {2}\,\left (\ln \left (x+\sqrt {2}\,1{}\mathrm {i}\right )-\ln \left (1-\sqrt {2}\,x\,1{}\mathrm {i}+\sqrt {x^2+1}\,1{}\mathrm {i}\right )\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________