Optimal. Leaf size=114 \[ \frac{1}{4} x^4 \text{csch}^{-1}\left (\sqrt{x}\right )-\frac{(-x-1)^{7/2} \sqrt{x}}{28 \sqrt{-x}}-\frac{3 (-x-1)^{5/2} \sqrt{x}}{20 \sqrt{-x}}-\frac{(-x-1)^{3/2} \sqrt{x}}{4 \sqrt{-x}}-\frac{\sqrt{-x-1} \sqrt{x}}{4 \sqrt{-x}} \]
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Rubi [A] time = 0.0323269, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6346, 12, 43} \[ \frac{1}{4} x^4 \text{csch}^{-1}\left (\sqrt{x}\right )-\frac{(-x-1)^{7/2} \sqrt{x}}{28 \sqrt{-x}}-\frac{3 (-x-1)^{5/2} \sqrt{x}}{20 \sqrt{-x}}-\frac{(-x-1)^{3/2} \sqrt{x}}{4 \sqrt{-x}}-\frac{\sqrt{-x-1} \sqrt{x}}{4 \sqrt{-x}} \]
Antiderivative was successfully verified.
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Rule 6346
Rule 12
Rule 43
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 = 0.0327313, size = 47, normalized size = 0.41 \[ \frac{1}{140} \sqrt{\frac{1}{x}+1} \left (5 x^3-6 x^2+8 x-16\right ) \sqrt{x}+\frac{1}{4} x^4 \text{csch}^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.128, size = 43, normalized size = 0.4 \begin{align*}{\frac{{x}^{4}}{4}{\rm arccsch} \left (\sqrt{x}\right )}+{\frac{ \left ( 1+x \right ) \left ( 5\,{x}^{3}-6\,{x}^{2}+8\,x-16 \right ) }{140}{\frac{1}{\sqrt{{\frac{1+x}{x}}}}}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02526, size = 78, normalized size = 0.68 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.58256, size = 142, normalized size = 1.25 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{arcsch}\left (\sqrt{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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