Optimal. Leaf size=76 \[ \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{6 x^{3/2}}+\frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{3 \sqrt {x}}-\frac {\cosh ^{-1}\left (\sqrt {x}\right )}{2 x^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6017, 12, 278,
271} \begin {gather*} \frac {\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{6 x^{3/2}}-\frac {\cosh ^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{3 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 271
Rule 278
Rule 6017
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx &=-\frac {\cosh ^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {1}{2} \int \frac {1}{2 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}} \, dx\\ &=-\frac {\cosh ^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {1}{4} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}} \, dx\\ &=\frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{6 x^{3/2}}-\frac {\cosh ^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {1}{6} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}} \, dx\\ &=\frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{6 x^{3/2}}+\frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{3 \sqrt {x}}-\frac {\cosh ^{-1}\left (\sqrt {x}\right )}{2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 49, normalized size = 0.64 \begin {gather*} \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x} (1+2 x)-3 \cosh ^{-1}\left (\sqrt {x}\right )}{6 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 35, normalized size = 0.46
method | result | size |
derivativedivides | \(-\frac {\mathrm {arccosh}\left (\sqrt {x}\right )}{2 x^{2}}+\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (2 x +1\right )}{6 x^{\frac {3}{2}}}\) | \(35\) |
default | \(-\frac {\mathrm {arccosh}\left (\sqrt {x}\right )}{2 x^{2}}+\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (2 x +1\right )}{6 x^{\frac {3}{2}}}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 30, normalized size = 0.39 \begin {gather*} \frac {\sqrt {x - 1}}{3 \, \sqrt {x}} + \frac {\sqrt {x - 1}}{6 \, x^{\frac {3}{2}}} - \frac {\operatorname {arcosh}\left (\sqrt {x}\right )}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 32, normalized size = 0.42 \begin {gather*} \frac {{\left (2 \, x + 1\right )} \sqrt {x - 1} \sqrt {x} - 3 \, \log \left (\sqrt {x - 1} + \sqrt {x}\right )}{6 \, x^{2}} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {acosh}{\left (\sqrt {x} \right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.55, size = 62, normalized size = 0.82 \begin {gather*} -\frac {\log \left (\sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \sqrt {x}\right )}{2 \, x^{2}} + \frac {2 \, {\left (3 \, {\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} + 1\right )}}{3 \, {\left ({\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} + 1\right )}^{3}} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {acosh}\left (\sqrt {x}\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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