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 = 0.0343055, antiderivative size = 115, 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 = {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 6346
Rule 12
Rule 51
Rule 63
Rule 204
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*}
Mathematica [A] time = 0.0446572, size = 52, normalized size = 0.45 \[ \frac{\sqrt{\frac{1}{x}+1} \left (15 x^2-10 x+8\right ) \sqrt{x}-15 x^3 \sinh ^{-1}\left (\frac{1}{\sqrt{x}}\right )-48 \text{csch}^{-1}\left (\sqrt{x}\right )}{144 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.117, size = 67, normalized size = 0.6 \begin{align*} -{\frac{1}{3\,{x}^{3}}{\rm arccsch} \left (\sqrt{x}\right )}-{\frac{1}{144}\sqrt{1+x} \left ( 15\,{\it Artanh} \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 ){\frac{1}{\sqrt{{\frac{1+x}{x}}}}}{x}^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991373, size = 157, normalized size = 1.37 \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{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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.58923, size = 150, normalized size = 1.3 \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{arcsch}\left (\sqrt{x}\right )}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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