Optimal. Leaf size=164 \[ \frac{1}{4} x^4 \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{(1-x)^4}{28 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}-\frac{3 (1-x)^3}{20 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}+\frac{(1-x)^2}{4 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}-\frac{1-x}{4 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}} \]
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Rubi [A] time = 0.0304525, antiderivative size = 164, 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 = {6345, 12, 43} \[ \frac{1}{4} x^4 \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{(1-x)^4}{28 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}-\frac{3 (1-x)^3}{20 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}+\frac{(1-x)^2}{4 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}-\frac{1-x}{4 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 6345
Rule 12
Rule 43
Rubi steps
\begin{align*} \int x^3 \text{sech}^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{1}{4} x^4 \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{1-x} \int \frac{x^3}{2 \sqrt{1-x}} \, dx}{4 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1}{4} x^4 \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{1-x} \int \frac{x^3}{\sqrt{1-x}} \, dx}{8 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1}{4} x^4 \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{1-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{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=-\frac{1-x}{4 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}+\frac{(1-x)^2}{4 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}-\frac{3 (1-x)^3}{20 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}+\frac{(1-x)^4}{28 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}+\frac{1}{4} x^4 \text{sech}^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.04074, size = 84, normalized size = 0.51 \[ \frac{1}{4} x^4 \text{sech}^{-1}\left (\sqrt{x}\right )-\frac{1}{140} \sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}} \left (5 x^{7/2}+5 x^3+6 x^{5/2}+6 x^2+8 x^{3/2}+8 x+16 \sqrt{x}+16\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.139, size = 54, normalized size = 0.3 \begin{align*}{\frac{{x}^{4}}{4}{\rm arcsech} \left (\sqrt{x}\right )}-{\frac{5\,{x}^{3}+6\,{x}^{2}+8\,x+16}{140}\sqrt{-{ \left ( -1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}}\sqrt{x}\sqrt{{ \left ( 1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982, size = 78, normalized size = 0.48 \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{arsech}\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 = 1.83839, size = 144, normalized size = 0.88 \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{arsech}\left (\sqrt{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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