Optimal. Leaf size=16 \[ \sqrt{x-1}+x \csc ^{-1}\left (\sqrt{x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0039246, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5269, 12, 32} \[ \sqrt{x-1}+x \csc ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5269
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \csc ^{-1}\left (\sqrt{x}\right ) \, dx &=x \csc ^{-1}\left (\sqrt{x}\right )+\int \frac{1}{2 \sqrt{-1+x}} \, dx\\ &=x \csc ^{-1}\left (\sqrt{x}\right )+\frac{1}{2} \int \frac{1}{\sqrt{-1+x}} \, dx\\ &=\sqrt{-1+x}+x \csc ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0043057, size = 16, normalized size = 1. \[ \sqrt{x-1}+x \csc ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.116, size = 24, normalized size = 1.5 \begin{align*} x{\rm arccsc} \left (\sqrt{x}\right )+{(x-1){\frac{1}{\sqrt{{\frac{x-1}{x}}}}}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01964, size = 27, normalized size = 1.69 \begin{align*} x \operatorname{arccsc}\left (\sqrt{x}\right ) + \sqrt{x} \sqrt{-\frac{1}{x} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 3.08308, size = 45, normalized size = 2.81 \begin{align*} x \operatorname{arccsc}\left (\sqrt{x}\right ) + \sqrt{x - 1} \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 \operatorname{acsc}{\left (\sqrt{x} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1163, size = 20, normalized size = 1.25 \begin{align*} x \arcsin \left (\frac{1}{\sqrt{x}}\right ) - i + \sqrt{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]