Optimal. Leaf size=33 \[ \frac {2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2198}
\begin {gather*} \frac {2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2198
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx &=\frac {2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 32, normalized size = 0.97 \begin {gather*} -\frac {2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x} \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 29, normalized size = 0.88
method | result | size |
derivativedivides | \(-\frac {2 \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) \sqrt {x}}\) | \(29\) |
default | \(-\frac {2 \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) \sqrt {x}}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 15, normalized size = 0.45 \begin {gather*} -\frac {2 \, \sqrt {b x + a}}{a \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 15, normalized size = 0.45 \begin {gather*} -\frac {2 \, \sqrt {b x + a}}{a \sqrt {x}} \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 {1}{x^{\frac {3}{2}} \sqrt {\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 30, normalized size = 0.91 \begin {gather*} \frac {4 \, \sqrt {b}}{{\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{2} - a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.62, size = 101, normalized size = 3.06 \begin {gather*} \frac {4\,\sqrt {\frac {\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}-\frac {\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}}}{\sqrt {x}\,\left (\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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