Optimal. Leaf size=136 \[ \frac {3 (1-x)}{16 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} x^{3/2}}+\frac {1-x}{8 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} x^{5/2}}-\frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {3 \sqrt {1-x} \tanh ^{-1}\left (\sqrt {1-x}\right )}{16 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} \sqrt {x}} \]
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Rubi [A] time = 0.03, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6345, 12, 51, 63, 206} \[ \frac {3 (1-x)}{16 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} x^{3/2}}+\frac {1-x}{8 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} x^{5/2}}-\frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {3 \sqrt {1-x} \tanh ^{-1}\left (\sqrt {1-x}\right )}{16 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} \sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 51
Rule 63
Rule 206
Rule 6345
Rubi steps
\begin {align*} \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x^3} \, dx &=-\frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {\sqrt {1-x} \int \frac {1}{2 \sqrt {1-x} x^3} \, dx}{2 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}\\ &=-\frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {\sqrt {1-x} \int \frac {1}{\sqrt {1-x} x^3} \, dx}{4 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}\\ &=\frac {1-x}{8 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} x^{5/2}}-\frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {\left (3 \sqrt {1-x}\right ) \int \frac {1}{\sqrt {1-x} x^2} \, dx}{16 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}\\ &=\frac {1-x}{8 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} x^{5/2}}+\frac {3 (1-x)}{16 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} x^{3/2}}-\frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {\left (3 \sqrt {1-x}\right ) \int \frac {1}{\sqrt {1-x} x} \, dx}{32 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}\\ &=\frac {1-x}{8 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} x^{5/2}}+\frac {3 (1-x)}{16 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} x^{3/2}}-\frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {\left (3 \sqrt {1-x}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x}\right )}{16 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}\\ &=\frac {1-x}{8 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} x^{5/2}}+\frac {3 (1-x)}{16 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} x^{3/2}}-\frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {3 \sqrt {1-x} \tanh ^{-1}\left (\sqrt {1-x}\right )}{16 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 125, normalized size = 0.92 \[ \frac {1}{16} \left (-\frac {8 \text {sech}^{-1}\left (\sqrt {x}\right )}{x^2}+\frac {\sqrt {\frac {1-\sqrt {x}}{\sqrt {x}+1}} \left (3 x^{3/2}+3 x+2 \sqrt {x}+2\right )}{x^2}+3 \log \left (\sqrt {x} \sqrt {\frac {1-\sqrt {x}}{\sqrt {x}+1}}+\sqrt {\frac {1-\sqrt {x}}{\sqrt {x}+1}}+1\right )-\frac {3 \log (x)}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 54, normalized size = 0.40 \[ \frac {{\left (3 \, x + 2\right )} \sqrt {x} \sqrt {-\frac {x - 1}{x}} + {\left (3 \, x^{2} - 8\right )} \log \left (\frac {x \sqrt {-\frac {x - 1}{x}} + \sqrt {x}}{x}\right )}{16 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsech}\left (\sqrt {x}\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 79, normalized size = 0.58 \[ -\frac {\mathrm {arcsech}\left (\sqrt {x}\right )}{2 x^{2}}+\frac {\sqrt {-\frac {-1+\sqrt {x}}{\sqrt {x}}}\, \sqrt {\frac {1+\sqrt {x}}{\sqrt {x}}}\, \left (3 \arctanh \left (\frac {1}{\sqrt {1-x}}\right ) x^{2}+3 x \sqrt {1-x}+2 \sqrt {1-x}\right )}{16 x^{\frac {3}{2}} \sqrt {1-x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 92, normalized size = 0.68 \[ -\frac {3 \, x^{\frac {3}{2}} {\left (\frac {1}{x} - 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {x} \sqrt {\frac {1}{x} - 1}}{16 \, {\left (x^{2} {\left (\frac {1}{x} - 1\right )}^{2} - 2 \, x {\left (\frac {1}{x} - 1\right )} + 1\right )}} - \frac {\operatorname {arsech}\left (\sqrt {x}\right )}{2 \, x^{2}} + \frac {3}{32} \, \log \left (\sqrt {x} \sqrt {\frac {1}{x} - 1} + 1\right ) - \frac {3}{32} \, \log \left (\sqrt {x} \sqrt {\frac {1}{x} - 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {acosh}\left (\frac {1}{\sqrt {x}}\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asech}{\left (\sqrt {x} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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