Optimal. Leaf size=61 \[ \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{\sqrt {b}} \]
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Rubi [A] time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2169, 2165} \[ \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 2165
Rule 2169
Rubi steps
\begin {align*} \int \frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x}} \, dx &=\sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}-\frac {1}{2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \int \frac {1}{\sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{\sqrt {b}}+\sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 62, normalized size = 1.02 \[ \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}+\frac {\left (\tanh ^{-1}(\tanh (a+b x))-b x\right ) \log \left (\sqrt {b} \sqrt {\tanh ^{-1}(\tanh (a+b x))}+b \sqrt {x}\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 93, normalized size = 1.52 \[ \left [\frac {a \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, \sqrt {b x + a} b \sqrt {x}}{2 \, b}, -\frac {a \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - \sqrt {b x + a} b \sqrt {x}}{b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 36, normalized size = 0.59 \[ -\frac {a \log \left ({\left | -\sqrt {b} \sqrt {x} + \sqrt {b x + a} \right |}\right )}{\sqrt {b}} + \sqrt {b x + a} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 75, normalized size = 1.23 \[ \sqrt {x}\, \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}+\frac {\ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) a}{\sqrt {b}}+\frac {\ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}}{\sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}}{\sqrt {x}} \,d x \]
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 \[ \int \frac {\sqrt {\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}}{\sqrt {x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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