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 = 0.0290022, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 5682
Rule 5675
Rule 30
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 = 0.0128693, 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]
________________________________________________________________________________________
Maple [A] time = 0., size = 26, normalized size = 0.8 \begin{align*}{\frac{x{\it Arcsinh} \left ( x \right ) }{2}\sqrt{{x}^{2}+1}}+{\frac{ \left ({\it Arcsinh} \left ( x \right ) \right ) ^{2}}{4}}-{\frac{{x}^{2}}{4}}-{\frac{1}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.65747, size = 38, normalized size = 1.19 \begin{align*} -\frac{1}{4} \, x^{2} + \frac{1}{2} \,{\left (\sqrt{x^{2} + 1} x + \operatorname{arsinh}\left (x\right )\right )} \operatorname{arsinh}\left (x\right ) - \frac{1}{4} \, \operatorname{arsinh}\left (x\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.40515, size = 115, normalized size = 3.59 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x^{2} + 1} \operatorname{asinh}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x^{2} + 1} \operatorname{arsinh}\left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]