Optimal. Leaf size=21 \[ -\sqrt {-1+x} \sqrt {1+x}+x \cosh ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6462, 5879, 75}
\begin {gather*} x \cosh ^{-1}(x)-\sqrt {x-1} \sqrt {x+1} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 75
Rule 5879
Rule 6462
Rubi steps
\begin {align*} \int \text {sech}^{-1}\left (\frac {1}{x}\right ) \, dx &=\int \cosh ^{-1}(x) \, dx\\ &=x \cosh ^{-1}(x)-\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx\\ &=-\sqrt {-1+x} \sqrt {1+x}+x \cosh ^{-1}(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 25, normalized size = 1.19 \begin {gather*} -\sqrt {\frac {-1+x}{1+x}} (1+x)+x \text {sech}^{-1}\left (\frac {1}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.17, size = 29, normalized size = 1.38
method | result | size |
derivativedivides | \(x \,\mathrm {arcsech}\left (\frac {1}{x}\right )-\sqrt {-\left (-1+\frac {1}{x}\right ) x}\, \sqrt {\left (1+\frac {1}{x}\right ) x}\) | \(29\) |
default | \(x \,\mathrm {arcsech}\left (\frac {1}{x}\right )-\sqrt {-\left (-1+\frac {1}{x}\right ) x}\, \sqrt {\left (1+\frac {1}{x}\right ) x}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 16, normalized size = 0.76 \begin {gather*} x \operatorname {arsech}\left (\frac {1}{x}\right ) - \sqrt {x^{2} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.50, size = 22, normalized size = 1.05 \begin {gather*} x \log \left (x + \sqrt {x^{2} - 1}\right ) - \sqrt {x^{2} - 1} \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 \operatorname {asech}{\left (\frac {1}{x} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.35, size = 17, normalized size = 0.81 \begin {gather*} x\,\mathrm {acosh}\left (x\right )-\sqrt {x-1}\,\sqrt {x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________