Optimal. Leaf size=172 \[ \frac{5 (1-x)}{48 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{3/2}}+\frac{5 (1-x)}{72 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{5/2}}+\frac{1-x}{18 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{7/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{3 x^3}+\frac{5 \sqrt{1-x} \tanh ^{-1}\left (\sqrt{1-x}\right )}{48 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}} \]
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Rubi [A] time = 0.039765, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6345, 12, 51, 63, 206} \[ \frac{5 (1-x)}{48 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{3/2}}+\frac{5 (1-x)}{72 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{5/2}}+\frac{1-x}{18 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{7/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{3 x^3}+\frac{5 \sqrt{1-x} \tanh ^{-1}\left (\sqrt{1-x}\right )}{48 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 6345
Rule 12
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{x^4} \, dx &=-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{3 x^3}-\frac{\sqrt{1-x} \int \frac{1}{2 \sqrt{1-x} x^4} \, dx}{3 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{3 x^3}-\frac{\sqrt{1-x} \int \frac{1}{\sqrt{1-x} x^4} \, dx}{6 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1-x}{18 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{7/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{3 x^3}-\frac{\left (5 \sqrt{1-x}\right ) \int \frac{1}{\sqrt{1-x} x^3} \, dx}{36 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1-x}{18 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{7/2}}+\frac{5 (1-x)}{72 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{5/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{3 x^3}-\frac{\left (5 \sqrt{1-x}\right ) \int \frac{1}{\sqrt{1-x} x^2} \, dx}{48 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1-x}{18 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{7/2}}+\frac{5 (1-x)}{72 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{5/2}}+\frac{5 (1-x)}{48 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{3/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{3 x^3}-\frac{\left (5 \sqrt{1-x}\right ) \int \frac{1}{\sqrt{1-x} x} \, dx}{96 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1-x}{18 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{7/2}}+\frac{5 (1-x)}{72 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{5/2}}+\frac{5 (1-x)}{48 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{3/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{3 x^3}+\frac{\left (5 \sqrt{1-x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x}\right )}{48 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1-x}{18 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{7/2}}+\frac{5 (1-x)}{72 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{5/2}}+\frac{5 (1-x)}{48 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{3/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{3 x^3}+\frac{5 \sqrt{1-x} \tanh ^{-1}\left (\sqrt{1-x}\right )}{48 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.128421, size = 140, normalized size = 0.81 \[ \frac{\sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}} \left (15 x^{5/2}+15 x^2+10 x^{3/2}+10 x+8 \sqrt{x}+8\right )+15 x^3 \log \left (\sqrt{x} \sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}}+\sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}}+1\right )-\frac{15}{2} x^3 \log (x)-48 \text{sech}^{-1}\left (\sqrt{x}\right )}{144 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.141, size = 91, normalized size = 0.5 \begin{align*} -{\frac{1}{3\,{x}^{3}}{\rm arcsech} \left (\sqrt{x}\right )}+{\frac{1}{144}\sqrt{-{ \left ( -1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}}\sqrt{{ \left ( 1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}} \left ( 15\,{\it Artanh} \left ({\frac{1}{\sqrt{1-x}}} \right ){x}^{3}+15\,\sqrt{1-x}{x}^{2}+10\,\sqrt{1-x}x+8\,\sqrt{1-x} \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{1-x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989615, size = 157, normalized size = 0.91 \begin{align*} -\frac{15 \, x^{\frac{5}{2}}{\left (\frac{1}{x} - 1\right )}^{\frac{5}{2}} - 40 \, x^{\frac{3}{2}}{\left (\frac{1}{x} - 1\right )}^{\frac{3}{2}} + 33 \, \sqrt{x} \sqrt{\frac{1}{x} - 1}}{144 \,{\left (x^{3}{\left (\frac{1}{x} - 1\right )}^{3} - 3 \, x^{2}{\left (\frac{1}{x} - 1\right )}^{2} + 3 \, x{\left (\frac{1}{x} - 1\right )} - 1\right )}} - \frac{\operatorname{arsech}\left (\sqrt{x}\right )}{3 \, x^{3}} + \frac{5}{96} \, \log \left (\sqrt{x} \sqrt{\frac{1}{x} - 1} + 1\right ) - \frac{5}{96} \, \log \left (\sqrt{x} \sqrt{\frac{1}{x} - 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92981, size = 153, normalized size = 0.89 \begin{align*} \frac{{\left (15 \, x^{2} + 10 \, x + 8\right )} \sqrt{x} \sqrt{-\frac{x - 1}{x}} + 3 \,{\left (5 \, x^{3} - 16\right )} \log \left (\frac{x \sqrt{-\frac{x - 1}{x}} + \sqrt{x}}{x}\right )}{144 \, x^{3}} \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 \frac{\operatorname{arsech}\left (\sqrt{x}\right )}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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