Optimal. Leaf size=89 \[ \frac {\sqrt {-1-x} \sqrt {x}}{3 \sqrt {-x}}+\frac {2 (-1-x)^{3/2} \sqrt {x}}{9 \sqrt {-x}}+\frac {(-1-x)^{5/2} \sqrt {x}}{15 \sqrt {-x}}+\frac {1}{3} x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6481, 12, 45}
\begin {gather*} \frac {1}{3} x^3 \text {csch}^{-1}\left (\sqrt {x}\right )+\frac {(-x-1)^{5/2} \sqrt {x}}{15 \sqrt {-x}}+\frac {2 (-x-1)^{3/2} \sqrt {x}}{9 \sqrt {-x}}+\frac {\sqrt {-x-1} \sqrt {x}}{3 \sqrt {-x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 45
Rule 6481
Rubi steps
\begin {align*} \int x^2 \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {1}{3} x^3 \text {csch}^{-1}\left (\sqrt {x}\right )-\frac {\sqrt {x} \int \frac {x^2}{2 \sqrt {-1-x}} \, dx}{3 \sqrt {-x}}\\ &=\frac {1}{3} x^3 \text {csch}^{-1}\left (\sqrt {x}\right )-\frac {\sqrt {x} \int \frac {x^2}{\sqrt {-1-x}} \, dx}{6 \sqrt {-x}}\\ &=\frac {1}{3} x^3 \text {csch}^{-1}\left (\sqrt {x}\right )-\frac {\sqrt {x} \int \left (\frac {1}{\sqrt {-1-x}}+2 \sqrt {-1-x}+(-1-x)^{3/2}\right ) \, dx}{6 \sqrt {-x}}\\ &=\frac {\sqrt {-1-x} \sqrt {x}}{3 \sqrt {-x}}+\frac {2 (-1-x)^{3/2} \sqrt {x}}{9 \sqrt {-x}}+\frac {(-1-x)^{5/2} \sqrt {x}}{15 \sqrt {-x}}+\frac {1}{3} x^3 \text {csch}^{-1}\left (\sqrt {x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 42, normalized size = 0.47 \begin {gather*} \frac {1}{45} \sqrt {1+\frac {1}{x}} \sqrt {x} \left (8-4 x+3 x^2\right )+\frac {1}{3} x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.15, size = 38, normalized size = 0.43
method | result | size |
derivativedivides | \(\frac {x^{3} \mathrm {arccsch}\left (\sqrt {x}\right )}{3}+\frac {\left (1+x \right ) \left (3 x^{2}-4 x +8\right )}{45 \sqrt {\frac {1+x}{x}}\, \sqrt {x}}\) | \(38\) |
default | \(\frac {x^{3} \mathrm {arccsch}\left (\sqrt {x}\right )}{3}+\frac {\left (1+x \right ) \left (3 x^{2}-4 x +8\right )}{45 \sqrt {\frac {1+x}{x}}\, \sqrt {x}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 46, normalized size = 0.52 \begin {gather*} \frac {1}{15} \, x^{\frac {5}{2}} {\left (\frac {1}{x} + 1\right )}^{\frac {5}{2}} + \frac {1}{3} \, x^{3} \operatorname {arcsch}\left (\sqrt {x}\right ) - \frac {2}{9} \, x^{\frac {3}{2}} {\left (\frac {1}{x} + 1\right )}^{\frac {3}{2}} + \frac {1}{3} \, \sqrt {x} \sqrt {\frac {1}{x} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 50, normalized size = 0.56 \begin {gather*} \frac {1}{3} \, x^{3} \log \left (\frac {x \sqrt {\frac {x + 1}{x}} + \sqrt {x}}{x}\right ) + \frac {1}{45} \, {\left (3 \, x^{2} - 4 \, x + 8\right )} \sqrt {x} \sqrt {\frac {x + 1}{x}} \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 x^{2} \operatorname {acsch}{\left (\sqrt {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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,\mathrm {asinh}\left (\frac {1}{\sqrt {x}}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________