Optimal. Leaf size=114 \[ -\frac {\sqrt {-1-x} \sqrt {x}}{4 \sqrt {-x}}-\frac {(-1-x)^{3/2} \sqrt {x}}{4 \sqrt {-x}}-\frac {3 (-1-x)^{5/2} \sqrt {x}}{20 \sqrt {-x}}-\frac {(-1-x)^{7/2} \sqrt {x}}{28 \sqrt {-x}}+\frac {1}{4} x^4 \text {csch}^{-1}\left (\sqrt {x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 114, 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}{4} x^4 \text {csch}^{-1}\left (\sqrt {x}\right )-\frac {(-x-1)^{7/2} \sqrt {x}}{28 \sqrt {-x}}-\frac {3 (-x-1)^{5/2} \sqrt {x}}{20 \sqrt {-x}}-\frac {(-x-1)^{3/2} \sqrt {x}}{4 \sqrt {-x}}-\frac {\sqrt {-x-1} \sqrt {x}}{4 \sqrt {-x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 45
Rule 6481
Rubi steps
\begin {align*} \int x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {1}{4} x^4 \text {csch}^{-1}\left (\sqrt {x}\right )-\frac {\sqrt {x} \int \frac {x^3}{2 \sqrt {-1-x}} \, dx}{4 \sqrt {-x}}\\ &=\frac {1}{4} x^4 \text {csch}^{-1}\left (\sqrt {x}\right )-\frac {\sqrt {x} \int \frac {x^3}{\sqrt {-1-x}} \, dx}{8 \sqrt {-x}}\\ &=\frac {1}{4} x^4 \text {csch}^{-1}\left (\sqrt {x}\right )-\frac {\sqrt {x} \int \left (-\frac {1}{\sqrt {-1-x}}-3 \sqrt {-1-x}-3 (-1-x)^{3/2}-(-1-x)^{5/2}\right ) \, dx}{8 \sqrt {-x}}\\ &=-\frac {\sqrt {-1-x} \sqrt {x}}{4 \sqrt {-x}}-\frac {(-1-x)^{3/2} \sqrt {x}}{4 \sqrt {-x}}-\frac {3 (-1-x)^{5/2} \sqrt {x}}{20 \sqrt {-x}}-\frac {(-1-x)^{7/2} \sqrt {x}}{28 \sqrt {-x}}+\frac {1}{4} x^4 \text {csch}^{-1}\left (\sqrt {x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 47, normalized size = 0.41 \begin {gather*} \frac {1}{140} \sqrt {1+\frac {1}{x}} \sqrt {x} \left (-16+8 x-6 x^2+5 x^3\right )+\frac {1}{4} x^4 \text {csch}^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.15, size = 43, normalized size = 0.38
method | result | size |
derivativedivides | \(\frac {x^{4} \mathrm {arccsch}\left (\sqrt {x}\right )}{4}+\frac {\left (1+x \right ) \left (5 x^{3}-6 x^{2}+8 x -16\right )}{140 \sqrt {\frac {1+x}{x}}\, \sqrt {x}}\) | \(43\) |
default | \(\frac {x^{4} \mathrm {arccsch}\left (\sqrt {x}\right )}{4}+\frac {\left (1+x \right ) \left (5 x^{3}-6 x^{2}+8 x -16\right )}{140 \sqrt {\frac {1+x}{x}}\, \sqrt {x}}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 58, normalized size = 0.51 \begin {gather*} \frac {1}{28} \, x^{\frac {7}{2}} {\left (\frac {1}{x} + 1\right )}^{\frac {7}{2}} - \frac {3}{20} \, x^{\frac {5}{2}} {\left (\frac {1}{x} + 1\right )}^{\frac {5}{2}} + \frac {1}{4} \, x^{4} \operatorname {arcsch}\left (\sqrt {x}\right ) + \frac {1}{4} \, x^{\frac {3}{2}} {\left (\frac {1}{x} + 1\right )}^{\frac {3}{2}} - \frac {1}{4} \, \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 = 55, normalized size = 0.48 \begin {gather*} \frac {1}{4} \, x^{4} \log \left (\frac {x \sqrt {\frac {x + 1}{x}} + \sqrt {x}}{x}\right ) + \frac {1}{140} \, {\left (5 \, x^{3} - 6 \, x^{2} + 8 \, x - 16\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^{3} \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^3\,\mathrm {asinh}\left (\frac {1}{\sqrt {x}}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________