Integrand size = 10, antiderivative size = 114 \[ \int x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx=-\frac {\sqrt {-1-x} \sqrt {x}}{4 \sqrt {-x}}-\frac {(-1-x)^{3/2} \sqrt {x}}{4 \sqrt {-x}}-\frac {3 (-1-x)^{5/2} \sqrt {x}}{20 \sqrt {-x}}-\frac {(-1-x)^{7/2} \sqrt {x}}{28 \sqrt {-x}}+\frac {1}{4} x^4 \text {csch}^{-1}\left (\sqrt {x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6481, 12, 45} \[ \int x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{4} x^4 \text {csch}^{-1}\left (\sqrt {x}\right )-\frac {(-x-1)^{7/2} \sqrt {x}}{28 \sqrt {-x}}-\frac {3 (-x-1)^{5/2} \sqrt {x}}{20 \sqrt {-x}}-\frac {(-x-1)^{3/2} \sqrt {x}}{4 \sqrt {-x}}-\frac {\sqrt {-x-1} \sqrt {x}}{4 \sqrt {-x}} \]
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Rule 12
Rule 45
Rule 6481
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.41 \[ \int x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{140} \sqrt {1+\frac {1}{x}} \sqrt {x} \left (-16+8 x-6 x^2+5 x^3\right )+\frac {1}{4} x^4 \text {csch}^{-1}\left (\sqrt {x}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.35
method | result | size |
parts | \(\frac {x^{4} \operatorname {arccsch}\left (\sqrt {x}\right )}{4}+\frac {\sqrt {\frac {1+x}{x}}\, \sqrt {x}\, \left (5 x^{3}-6 x^{2}+8 x -16\right )}{140}\) | \(40\) |
derivativedivides | \(\frac {x^{4} \operatorname {arccsch}\left (\sqrt {x}\right )}{4}+\frac {\left (1+x \right ) \left (5 x^{3}-6 x^{2}+8 x -16\right )}{140 \sqrt {\frac {1+x}{x}}\, \sqrt {x}}\) | \(43\) |
default | \(\frac {x^{4} \operatorname {arccsch}\left (\sqrt {x}\right )}{4}+\frac {\left (1+x \right ) \left (5 x^{3}-6 x^{2}+8 x -16\right )}{140 \sqrt {\frac {1+x}{x}}\, \sqrt {x}}\) | \(43\) |
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Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.48 \[ \int x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{4} \, x^{4} \log \left (\frac {x \sqrt {\frac {x + 1}{x}} + \sqrt {x}}{x}\right ) + \frac {1}{140} \, {\left (5 \, x^{3} - 6 \, x^{2} + 8 \, x - 16\right )} \sqrt {x} \sqrt {\frac {x + 1}{x}} \]
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\[ \int x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx=\int x^{3} \operatorname {acsch}{\left (\sqrt {x} \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.51 \[ \int x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{28} \, x^{\frac {7}{2}} {\left (\frac {1}{x} + 1\right )}^{\frac {7}{2}} - \frac {3}{20} \, x^{\frac {5}{2}} {\left (\frac {1}{x} + 1\right )}^{\frac {5}{2}} + \frac {1}{4} \, x^{4} \operatorname {arcsch}\left (\sqrt {x}\right ) + \frac {1}{4} \, x^{\frac {3}{2}} {\left (\frac {1}{x} + 1\right )}^{\frac {3}{2}} - \frac {1}{4} \, \sqrt {x} \sqrt {\frac {1}{x} + 1} \]
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\[ \int x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx=\int { x^{3} \operatorname {arcsch}\left (\sqrt {x}\right ) \,d x } \]
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Timed out. \[ \int x^3 \text {csch}^{-1}\left (\sqrt {x}\right ) \, dx=\int x^3\,\mathrm {asinh}\left (\frac {1}{\sqrt {x}}\right ) \,d x \]
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