Optimal. Leaf size=42 \[ -\frac {1}{6 x^{3/2}}-\frac {1}{2 \sqrt {x}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6038, 53, 65,
212} \begin {gather*} -\frac {1}{6 x^{3/2}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {1}{2 \sqrt {x}}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 212
Rule 6038
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx &=-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {1}{4} \int \frac {1}{(1-x) x^{5/2}} \, dx\\ &=-\frac {1}{6 x^{3/2}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {1}{4} \int \frac {1}{(1-x) x^{3/2}} \, dx\\ &=-\frac {1}{6 x^{3/2}}-\frac {1}{2 \sqrt {x}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {1}{4} \int \frac {1}{(1-x) \sqrt {x}} \, dx\\ &=-\frac {1}{6 x^{3/2}}-\frac {1}{2 \sqrt {x}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{6 x^{3/2}}-\frac {1}{2 \sqrt {x}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 58, normalized size = 1.38 \begin {gather*} -\frac {1}{6 x^{3/2}}-\frac {1}{2 \sqrt {x}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {1}{4} \log \left (1-\sqrt {x}\right )+\frac {1}{4} \log \left (1+\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 37, normalized size = 0.88
method | result | size |
derivativedivides | \(-\frac {\mathrm {arccoth}\left (\sqrt {x}\right )}{2 x^{2}}-\frac {\ln \left (\sqrt {x}-1\right )}{4}+\frac {\ln \left (\sqrt {x}+1\right )}{4}-\frac {1}{6 x^{\frac {3}{2}}}-\frac {1}{2 \sqrt {x}}\) | \(37\) |
default | \(-\frac {\mathrm {arccoth}\left (\sqrt {x}\right )}{2 x^{2}}-\frac {\ln \left (\sqrt {x}-1\right )}{4}+\frac {\ln \left (\sqrt {x}+1\right )}{4}-\frac {1}{6 x^{\frac {3}{2}}}-\frac {1}{2 \sqrt {x}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 36, normalized size = 0.86 \begin {gather*} -\frac {3 \, x + 1}{6 \, x^{\frac {3}{2}}} - \frac {\operatorname {arcoth}\left (\sqrt {x}\right )}{2 \, x^{2}} + \frac {1}{4} \, \log \left (\sqrt {x} + 1\right ) - \frac {1}{4} \, \log \left (\sqrt {x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 38, normalized size = 0.90 \begin {gather*} \frac {3 \, {\left (x^{2} - 1\right )} \log \left (\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right ) - 2 \, {\left (3 \, x + 1\right )} \sqrt {x}}{12 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 160 vs.
\(2 (36) = 72\).
time = 1.28, size = 160, normalized size = 3.81 \begin {gather*} \frac {3 x^{\frac {7}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{6 x^{\frac {7}{2}} - 6 x^{\frac {5}{2}}} - \frac {3 x^{\frac {5}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{6 x^{\frac {7}{2}} - 6 x^{\frac {5}{2}}} - \frac {3 x^{\frac {3}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{6 x^{\frac {7}{2}} - 6 x^{\frac {5}{2}}} + \frac {3 \sqrt {x} \operatorname {acoth}{\left (\sqrt {x} \right )}}{6 x^{\frac {7}{2}} - 6 x^{\frac {5}{2}}} - \frac {3 x^{3}}{6 x^{\frac {7}{2}} - 6 x^{\frac {5}{2}}} + \frac {2 x^{2}}{6 x^{\frac {7}{2}} - 6 x^{\frac {5}{2}}} + \frac {x}{6 x^{\frac {7}{2}} - 6 x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 114 vs.
\(2 (26) = 52\).
time = 0.39, size = 114, normalized size = 2.71 \begin {gather*} \frac {2 \, {\left (\frac {3 \, {\left (\sqrt {x} + 1\right )}^{2}}{{\left (\sqrt {x} - 1\right )}^{2}} + \frac {3 \, {\left (\sqrt {x} + 1\right )}}{\sqrt {x} - 1} + 2\right )}}{3 \, {\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} + 1\right )}^{3}} + \frac {2 \, {\left (\frac {{\left (\sqrt {x} + 1\right )}^{3}}{{\left (\sqrt {x} - 1\right )}^{3}} + \frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )} \log \left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )}{{\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} + 1\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.49, size = 45, normalized size = 1.07 \begin {gather*} \frac {\ln \left (1-\frac {1}{\sqrt {x}}\right )}{4\,x^2}-\frac {\frac {x}{2}+\frac {1}{6}}{x^{3/2}}-\frac {\ln \left (\frac {1}{\sqrt {x}}+1\right )}{4\,x^2}-\frac {\mathrm {atan}\left (\sqrt {x}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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