Optimal. Leaf size=32 \[ -\frac {x^2}{4}+\frac {1}{2} \sqrt {x^2+1} x \sinh ^{-1}(x)+\frac {1}{4} \sinh ^{-1}(x)^2 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5682, 5675, 30} \[ -\frac {x^2}{4}+\frac {1}{2} \sqrt {x^2+1} x \sinh ^{-1}(x)+\frac {1}{4} \sinh ^{-1}(x)^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 5675
Rule 5682
Rubi steps
\begin {align*} \int \sqrt {1+x^2} \sinh ^{-1}(x) \, dx &=\frac {1}{2} x \sqrt {1+x^2} \sinh ^{-1}(x)-\frac {\int x \, dx}{2}+\frac {1}{2} \int \frac {\sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx\\ &=-\frac {x^2}{4}+\frac {1}{2} x \sqrt {1+x^2} \sinh ^{-1}(x)+\frac {1}{4} \sinh ^{-1}(x)^2\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 28, normalized size = 0.88 \[ \frac {1}{4} \left (-x^2+2 \sqrt {x^2+1} x \sinh ^{-1}(x)+\sinh ^{-1}(x)^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.68, size = 40, normalized size = 1.25 \[ \frac {1}{2} \, \sqrt {x^{2} + 1} x \log \left (x + \sqrt {x^{2} + 1}\right ) - \frac {1}{4} \, x^{2} + \frac {1}{4} \, \log \left (x + \sqrt {x^{2} + 1}\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x^{2} + 1} \operatorname {arsinh}\relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 26, normalized size = 0.81 \[ \frac {x \arcsinh \relax (x ) \sqrt {x^{2}+1}}{2}+\frac {\arcsinh \relax (x )^{2}}{4}-\frac {x^{2}}{4}-\frac {1}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.61, size = 28, normalized size = 0.88 \[ -\frac {1}{4} \, x^{2} + \frac {1}{2} \, {\left (\sqrt {x^{2} + 1} x + \operatorname {arsinh}\relax (x)\right )} \operatorname {arsinh}\relax (x) - \frac {1}{4} \, \operatorname {arsinh}\relax (x)^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \mathrm {asinh}\relax (x)\,\sqrt {x^2+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x^{2} + 1} \operatorname {asinh}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________