Optimal. Leaf size=90 \[ \frac {\sqrt {-x-1}}{8 \sqrt {-x} x^{3/2}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {3 \sqrt {-x-1}}{16 \sqrt {-x} \sqrt {x}}+\frac {3 \sqrt {x} \tan ^{-1}\left (\sqrt {-x-1}\right )}{16 \sqrt {-x}} \]
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Rubi [A] time = 0.03, antiderivative size = 90, 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 = {6346, 12, 51, 63, 204} \[ \frac {\sqrt {-x-1}}{8 \sqrt {-x} x^{3/2}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {3 \sqrt {-x-1}}{16 \sqrt {-x} \sqrt {x}}+\frac {3 \sqrt {x} \tan ^{-1}\left (\sqrt {-x-1}\right )}{16 \sqrt {-x}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 51
Rule 63
Rule 204
Rule 6346
Rubi steps
\begin {align*} \int \frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{x^3} \, dx &=-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {\sqrt {x} \int \frac {1}{2 \sqrt {-1-x} x^3} \, dx}{2 \sqrt {-x}}\\ &=-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {\sqrt {x} \int \frac {1}{\sqrt {-1-x} x^3} \, dx}{4 \sqrt {-x}}\\ &=\frac {\sqrt {-1-x}}{8 \sqrt {-x} x^{3/2}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {\left (3 \sqrt {x}\right ) \int \frac {1}{\sqrt {-1-x} x^2} \, dx}{16 \sqrt {-x}}\\ &=\frac {\sqrt {-1-x}}{8 \sqrt {-x} x^{3/2}}-\frac {3 \sqrt {-1-x}}{16 \sqrt {-x} \sqrt {x}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {\left (3 \sqrt {x}\right ) \int \frac {1}{\sqrt {-1-x} x} \, dx}{32 \sqrt {-x}}\\ &=\frac {\sqrt {-1-x}}{8 \sqrt {-x} x^{3/2}}-\frac {3 \sqrt {-1-x}}{16 \sqrt {-x} \sqrt {x}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {\left (3 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {-1-x}\right )}{16 \sqrt {-x}}\\ &=\frac {\sqrt {-1-x}}{8 \sqrt {-x} x^{3/2}}-\frac {3 \sqrt {-1-x}}{16 \sqrt {-x} \sqrt {x}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {3 \sqrt {x} \tan ^{-1}\left (\sqrt {-1-x}\right )}{16 \sqrt {-x}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 47, normalized size = 0.52 \[ \frac {3 x^2 \sinh ^{-1}\left (\frac {1}{\sqrt {x}}\right )+\sqrt {\frac {1}{x}+1} (2-3 x) \sqrt {x}-8 \text {csch}^{-1}\left (\sqrt {x}\right )}{16 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 53, normalized size = 0.59 \[ -\frac {{\left (3 \, x - 2\right )} \sqrt {x} \sqrt {\frac {x + 1}{x}} - {\left (3 \, x^{2} - 8\right )} \log \left (\frac {x \sqrt {\frac {x + 1}{x}} + \sqrt {x}}{x}\right )}{16 \, x^{2}} \]
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 \[ \int \frac {\operatorname {arcsch}\left (\sqrt {x}\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 57, normalized size = 0.63 \[ -\frac {\mathrm {arccsch}\left (\sqrt {x}\right )}{2 x^{2}}-\frac {\sqrt {1+x}\, \left (-3 \arctanh \left (\frac {1}{\sqrt {1+x}}\right ) x^{2}+3 x \sqrt {1+x}-2 \sqrt {1+x}\right )}{16 \sqrt {\frac {1+x}{x}}\, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 92, normalized size = 1.02 \[ -\frac {3 \, x^{\frac {3}{2}} {\left (\frac {1}{x} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {x} \sqrt {\frac {1}{x} + 1}}{16 \, {\left (x^{2} {\left (\frac {1}{x} + 1\right )}^{2} - 2 \, x {\left (\frac {1}{x} + 1\right )} + 1\right )}} - \frac {\operatorname {arcsch}\left (\sqrt {x}\right )}{2 \, x^{2}} + \frac {3}{32} \, \log \left (\sqrt {x} \sqrt {\frac {1}{x} + 1} + 1\right ) - \frac {3}{32} \, \log \left (\sqrt {x} \sqrt {\frac {1}{x} + 1} - 1\right ) \]
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 \[ \int \frac {\mathrm {asinh}\left (\frac {1}{\sqrt {x}}\right )}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {acsch}{\left (\sqrt {x} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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