Optimal. Leaf size=35 \[ -\frac {1}{2} \sqrt {x} \sqrt {x+1}+x \sinh ^{-1}\left (\sqrt {x}\right )+\frac {1}{2} \sinh ^{-1}\left (\sqrt {x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5900, 12, 1958, 50, 54, 215} \[ -\frac {1}{2} \sqrt {x} \sqrt {x+1}+x \sinh ^{-1}\left (\sqrt {x}\right )+\frac {1}{2} \sinh ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 50
Rule 54
Rule 215
Rule 1958
Rule 5900
Rubi steps
\begin {align*} \int \sinh ^{-1}\left (\sqrt {x}\right ) \, dx &=x \sinh ^{-1}\left (\sqrt {x}\right )-\int \frac {1}{2} \sqrt {\frac {x}{1+x}} \, dx\\ &=x \sinh ^{-1}\left (\sqrt {x}\right )-\frac {1}{2} \int \sqrt {\frac {x}{1+x}} \, dx\\ &=x \sinh ^{-1}\left (\sqrt {x}\right )-\frac {1}{2} \int \frac {\sqrt {x}}{\sqrt {1+x}} \, dx\\ &=-\frac {1}{2} \sqrt {x} \sqrt {1+x}+x \sinh ^{-1}\left (\sqrt {x}\right )+\frac {1}{4} \int \frac {1}{\sqrt {x} \sqrt {1+x}} \, dx\\ &=-\frac {1}{2} \sqrt {x} \sqrt {1+x}+x \sinh ^{-1}\left (\sqrt {x}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{2} \sqrt {x} \sqrt {1+x}+\frac {1}{2} \sinh ^{-1}\left (\sqrt {x}\right )+x \sinh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 33, normalized size = 0.94 \[ \frac {1}{2} \left ((2 x+1) \sinh ^{-1}\left (\sqrt {x}\right )-\sqrt {\frac {x}{x+1}} (x+1)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.77, size = 28, normalized size = 0.80 \[ \frac {1}{2} \, {\left (2 \, x + 1\right )} \log \left (\sqrt {x + 1} + \sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x + 1} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.52, size = 40, normalized size = 1.14 \[ x \log \left (\sqrt {x + 1} + \sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x^{2} + x} - \frac {1}{4} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 24, normalized size = 0.69 \[ \frac {\arcsinh \left (\sqrt {x}\right )}{2}+x \arcsinh \left (\sqrt {x}\right )-\frac {\sqrt {x}\, \sqrt {1+x}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.85, size = 23, normalized size = 0.66 \[ x \operatorname {arsinh}\left (\sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x + 1} \sqrt {x} + \frac {1}{2} \, \operatorname {arsinh}\left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.92, size = 31, normalized size = 0.89 \[ \mathrm {atanh}\left (\frac {\sqrt {x}}{\sqrt {x+1}-1}\right )+x\,\mathrm {asinh}\left (\sqrt {x}\right )-\frac {\sqrt {x}\,\sqrt {x+1}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.30, size = 29, normalized size = 0.83 \[ - \frac {\sqrt {x} \sqrt {x + 1}}{2} + x \operatorname {asinh}{\left (\sqrt {x} \right )} + \frac {\operatorname {asinh}{\left (\sqrt {x} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________