Optimal. Leaf size=115 \[ -\frac {5 \sqrt {-x-1}}{72 \sqrt {-x} x^{3/2}}+\frac {\sqrt {-x-1}}{18 \sqrt {-x} x^{5/2}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{3 x^3}+\frac {5 \sqrt {-x-1}}{48 \sqrt {-x} \sqrt {x}}-\frac {5 \sqrt {x} \tan ^{-1}\left (\sqrt {-x-1}\right )}{48 \sqrt {-x}} \]
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Rubi [A] time = 0.03, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ -\frac {5 \sqrt {-x-1}}{72 \sqrt {-x} x^{3/2}}+\frac {\sqrt {-x-1}}{18 \sqrt {-x} x^{5/2}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{3 x^3}+\frac {5 \sqrt {-x-1}}{48 \sqrt {-x} \sqrt {x}}-\frac {5 \sqrt {x} \tan ^{-1}\left (\sqrt {-x-1}\right )}{48 \sqrt {-x}} \]
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^4} \, dx &=-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{3 x^3}+\frac {\sqrt {x} \int \frac {1}{2 \sqrt {-1-x} x^4} \, dx}{3 \sqrt {-x}}\\ &=-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{3 x^3}+\frac {\sqrt {x} \int \frac {1}{\sqrt {-1-x} x^4} \, dx}{6 \sqrt {-x}}\\ &=\frac {\sqrt {-1-x}}{18 \sqrt {-x} x^{5/2}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{3 x^3}-\frac {\left (5 \sqrt {x}\right ) \int \frac {1}{\sqrt {-1-x} x^3} \, dx}{36 \sqrt {-x}}\\ &=\frac {\sqrt {-1-x}}{18 \sqrt {-x} x^{5/2}}-\frac {5 \sqrt {-1-x}}{72 \sqrt {-x} x^{3/2}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{3 x^3}+\frac {\left (5 \sqrt {x}\right ) \int \frac {1}{\sqrt {-1-x} x^2} \, dx}{48 \sqrt {-x}}\\ &=\frac {\sqrt {-1-x}}{18 \sqrt {-x} x^{5/2}}-\frac {5 \sqrt {-1-x}}{72 \sqrt {-x} x^{3/2}}+\frac {5 \sqrt {-1-x}}{48 \sqrt {-x} \sqrt {x}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{3 x^3}-\frac {\left (5 \sqrt {x}\right ) \int \frac {1}{\sqrt {-1-x} x} \, dx}{96 \sqrt {-x}}\\ &=\frac {\sqrt {-1-x}}{18 \sqrt {-x} x^{5/2}}-\frac {5 \sqrt {-1-x}}{72 \sqrt {-x} x^{3/2}}+\frac {5 \sqrt {-1-x}}{48 \sqrt {-x} \sqrt {x}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{3 x^3}+\frac {\left (5 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {-1-x}\right )}{48 \sqrt {-x}}\\ &=\frac {\sqrt {-1-x}}{18 \sqrt {-x} x^{5/2}}-\frac {5 \sqrt {-1-x}}{72 \sqrt {-x} x^{3/2}}+\frac {5 \sqrt {-1-x}}{48 \sqrt {-x} \sqrt {x}}-\frac {\text {csch}^{-1}\left (\sqrt {x}\right )}{3 x^3}-\frac {5 \sqrt {x} \tan ^{-1}\left (\sqrt {-1-x}\right )}{48 \sqrt {-x}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 52, normalized size = 0.45 \[ \frac {-15 x^3 \sinh ^{-1}\left (\frac {1}{\sqrt {x}}\right )+\sqrt {\frac {1}{x}+1} \left (15 x^2-10 x+8\right ) \sqrt {x}-48 \text {csch}^{-1}\left (\sqrt {x}\right )}{144 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 58, normalized size = 0.50 \[ \frac {{\left (15 \, x^{2} - 10 \, x + 8\right )} \sqrt {x} \sqrt {\frac {x + 1}{x}} - 3 \, {\left (5 \, x^{3} + 16\right )} \log \left (\frac {x \sqrt {\frac {x + 1}{x}} + \sqrt {x}}{x}\right )}{144 \, x^{3}} \]
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 \[ \int \frac {\operatorname {arcsch}\left (\sqrt {x}\right )}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 67, normalized size = 0.58 \[ -\frac {\mathrm {arccsch}\left (\sqrt {x}\right )}{3 x^{3}}+\frac {\sqrt {1+x}\, \left (-15 \arctanh \left (\frac {1}{\sqrt {1+x}}\right ) x^{3}+15 x^{2} \sqrt {1+x}-10 x \sqrt {1+x}+8 \sqrt {1+x}\right )}{144 \sqrt {\frac {1+x}{x}}\, x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 116, normalized size = 1.01 \[ \frac {15 \, x^{\frac {5}{2}} {\left (\frac {1}{x} + 1\right )}^{\frac {5}{2}} - 40 \, x^{\frac {3}{2}} {\left (\frac {1}{x} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {x} \sqrt {\frac {1}{x} + 1}}{144 \, {\left (x^{3} {\left (\frac {1}{x} + 1\right )}^{3} - 3 \, x^{2} {\left (\frac {1}{x} + 1\right )}^{2} + 3 \, x {\left (\frac {1}{x} + 1\right )} - 1\right )}} - \frac {\operatorname {arcsch}\left (\sqrt {x}\right )}{3 \, x^{3}} - \frac {5}{96} \, \log \left (\sqrt {x} \sqrt {\frac {1}{x} + 1} + 1\right ) + \frac {5}{96} \, \log \left (\sqrt {x} \sqrt {\frac {1}{x} + 1} - 1\right ) \]
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 \[ \int \frac {\mathrm {asinh}\left (\frac {1}{\sqrt {x}}\right )}{x^4} \,d x \]
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 \[ \int \frac {\operatorname {acsch}{\left (\sqrt {x} \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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