Optimal. Leaf size=49 \[ 2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right )-\frac {2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x}} \]
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Rubi [A] time = 0.03, antiderivative size = 49, 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 = {2168, 2165} \[ 2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right )-\frac {2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 2165
Rule 2168
Rubi steps
\begin {align*} \int \frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{x^{3/2}} \, dx &=-\frac {2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x}}+b \int \frac {1}{\sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx\\ &=2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right )-\frac {2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 52, normalized size = 1.06 \[ 2 \sqrt {b} \log \left (\sqrt {b} \sqrt {\tanh ^{-1}(\tanh (a+b x))}+b \sqrt {x}\right )-\frac {2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 89, normalized size = 1.82 \[ \left [\frac {\sqrt {b} x \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, \sqrt {b x + a} \sqrt {x}}{x}, -\frac {2 \, {\left (\sqrt {-b} x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + \sqrt {b x + a} \sqrt {x}\right )}}{x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 57, normalized size = 1.16 \[ -\sqrt {b} \log \left ({\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{2}\right ) + \frac {4 \, a \sqrt {b}}{{\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{2} - a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.26, size = 149, normalized size = 3.04 \[ -\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}}{\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) \sqrt {x}}+\frac {2 b \sqrt {x}\, \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{\arctanh \left (\tanh \left (b x +a \right )\right )-b x}+\frac {2 \sqrt {b}\, \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) a}{\arctanh \left (\tanh \left (b x +a \right )\right )-b x}+\frac {2 \sqrt {b}\, \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{\arctanh \left (\tanh \left (b x +a \right )\right )-b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 40, normalized size = 0.82 \[ 2 \, \sqrt {b} \log \left (\frac {b \sqrt {x}}{\sqrt {a b}} + \sqrt {\frac {b x}{a} + 1}\right ) - \frac {2 \, \sqrt {b x + a}}{\sqrt {x}} \]
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 \[ \int \frac {\sqrt {\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}}{x^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}}{x^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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